%I A001045 M2482 N0983
%S A001045 0,1,1,3,5,11,21,43,85,171,341,683,1365,2731,5461,10923,21845,43691,
%T A001045 87381,174763,349525,699051,1398101,2796203,5592405,11184811,22369621,
%U A001045 44739243,89478485,178956971,357913941,715827883,1431655765,2863311531
%N A001045 Jacobsthal sequence (or Jacobsthal numbers): a(n) = a(n-1) + 2a(n-2),
with a(0) = 0, a(1) = 1.
%C A001045 Number of ways to tile a 3 X (n-1) rectangle with 1 X 1 and 2 X 2 square
tiles.
%C A001045 Also, number of ways to tile a 2 X (n-1) rectangle with 1 X 2 dominoes
and 2 X 2 squares. - Toby Gottfried (toby(AT)gottfriedville.net),
Nov 02, 2008.
%C A001045 Also a(n) counts each of the following four things: n-ary quasigroups
of order 3 with automorphism group of order 3, n-ary quasigroups
of order 3 with automorphism group of order 6, (n-1)-ary quasigroups
of order 3 with automorphism group of order 2 and (n-2)-ary quasigroups
of order 3. See the McKay-Wanless (2008) paper. - Ian Wanless (ian.wanless(AT)sci.monash.edu.au),
Apr 28 2008
%C A001045 Also the number of ways to tie a necktie using n+2 turns. So three turns
make an "oriental", four make a "four in hand" and for 5 turns there
are 3 methods: "Kelvin", "Nicky" and "Pratt". The formula also arises
from a special random walk on a triangular grid with side conditions
(see Fink and Mao, 1999). - arne.ring(AT)epost.de, Mar 18 2001
%C A001045 Also the number of compositions of n+1 ending with an odd part (a(2)=3
because 3, 21, 111 are the only compositions of 3 ending with an
odd part). Also the number of compositions of n+2 ending with an
even part (a(2)=3 because 4, 22, 112 are the only compositions of
4 ending with an even part). - Emeric Deutsch (deutsch(AT)duke.poly.edu),
May 08 2001
%C A001045 Arises in study of sorting by merge insertions and in analysis of a method
for computing GCDs - see Knuth reference.
%C A001045 Roberto E. Martinez II (remartin(AT)fas.harvard.edu), Jan 07 2002: Number
of perfect matchings of a 2 X n grid upon replacing unit squares
with tetrahedra (C_4 to K_4):
%C A001045 o----o----o----o...
%C A001045 | \/ | \/ | \/ |
%C A001045 | /\ | /\ | /\ |
%C A001045 o----o----o----o...
%C A001045 Also the numerators of the reduced fractions in the alternating sum 1/
2 - 1/4 + 1/8 - 1/16 + 1/32 - 1/64 + ... - Joshua Zucker (joshua.zucker(AT)stanfordalumni.org),
Feb 07 2002
%C A001045 Also, if A(n),B(n),C(n) are the angles of the n-orthic triangle of ABC
then A(1) = Pi - 2A, A(n) = s(n)Pi + (-2)^nA where s(n) = (-1)^(n-1)
* a(n) [1-orthic triangle = the orthic triangle of ABC, n-orthic
triangle = the orthic triangle of the (n-1)-orthic triangle] - Antreas
P. Hatzipolakis (xpolakis(AT)otenet.gr), Jun 05 2002
%C A001045 Also the number of words of length n+1 in the two letters s and t that
reduce to the identity 1 by using the relations sss=1, tt=1 and stst=1.
The generators s and t and the three stated relations generate the
group S3. - John W. Layman (layman(AT)math.vt.edu), Jun 14 2002
%C A001045 Sums of pair of consecutive terms give all powers of 2 in increasing
order. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Aug 15 2002
%C A001045 Excess clockwise moves (over anti-clockwise) needed to move a tower of
size n to the clockwise peg is -(-1)^n(2^n - (-1)^n)/3; a(n)=its
unsigned version. - Wouter Meeussen (wouter.meeussen(AT)pandora.be),
Sep 01 2002
%C A001045 Also the absolute value of the number represented in base -2 by the string
of n 1's, the negabinary repunit. The Mersenne numbers (A000225 and
its subsequences) are the binary repunits. - Rick L. Shepherd(AT)prodigy.net
(rshepherd2(AT)hotmail.com), Sep 16 2002
%C A001045 Note that 3a(n)+(-1)^n=2^n is significant for Pascal' triangle A007318.
It arises from a Jacobsthal decomposition of Pascal's triangle illustrated
by 1+7+21+35+35+21+7+1 = (7+35+1)+(1+35+7)+(21+21) = 43 + 43 + 42
= 3a(7)-1; 1+8+28+56+70+56+29+8+1 = (1+56+28)+(28+56+1)+(8+70+8)
= 85 + 85 + 86 = 3a(8)+1. - Paul Barry (pbarry(AT)wit.ie), Feb 20
2003
%C A001045 Number of positive integers requiring exactly n signed bits in the non-adjacent
form representation.
%C A001045 Counts walks between adjacent vertices of a triangle - Paul Barry (pbarry(AT)wit.ie),
Nov 17 2003
%C A001045 Comment from Slavik Jablan, Dec 26, 2003: Every amphichiral rational
knot written in Conway notation is a palindromic sequence of numbers,
not beginning or ending with 1. For example, for 4 <= n <= 12, the
amphichiral rational knots are: 2 2, 2 1 1 2, 4 4, 3 1 1 3, 2 2 2
2, 4 1 1 4, 3 1 1 1 1 3, 2 3 3 2, 2 1 2 2 1 2, 2 1 1 1 1 1 1 2, 6
6, 5 1 1 5, 4 2 2 4, 3 3 3 3, 2 4 4 2, 3 2 1 1 2 3, 3 1 2 2 1 3,
2 2 2 2 2 2, 2 2 1 1 1 1 2 2, 2 1 2 1 1 2 1 2, 2 1 1 1 1 1 1 1 1
2. The number of amphichiral knots for n=2k (k=1, 2, 3, ...) we obtain
the 0, 1, 1, 3, 5, 11, 21, 43, 85, 171, 341, 683, ...
%C A001045 a(n+2) counts the binary sequences of total length n made up of codewords
from C={0,10,11} - Paul Barry (pbarry(AT)wit.ie), Jan 23 2004
%C A001045 Number of permutations with no fixed points avoiding 231 and 132.
%C A001045 The n-th entry (n>1) of the sequence is equal to the 2,2-entry of the
n-th power of the unnormalized 4 by 4 Haar matrix: [1 1 1 0 / 1 1
-1 0 / 1 1 0 1 / 1 1 0 -1]. - Simone Severini (ss54(AT)york.ac.uk),
Oct 27 2004
%C A001045 a(n) = number of Motzkin (n+1)-sequences whose flatsteps all occur at
level 1 and whose height is <=2. For example, a(4)=5 counts UDUFD,
UFDUD, UFFFD, UFUDD, UUDFD. - David Callan (callan(AT)stat.wisc.edu),
Dec 09 2004
%C A001045 a(n+1) gives row sums of A059260. - Paul Barry (pbarry(AT)wit.ie), Jan
26 2005
%C A001045 If (m + n) is odd, then 3*(a(m) + a(n)) is always of the form a^2 + 2*b^2,
where a and b both equal powers of 2; consequently every factor of
(a(m) + a(n)) is always of the form a^2 + 2*b^2. - Matthew Vandermast
(ghodges14(AT)comcast.net), Jul 12 2003
%C A001045 Number of "0,0" in f_{n+1}, where f_0 = "1" and f_{n+1} = a sequenece
formed by changing all "1"s in f_n to "1,0" and all "0"s in f_n to
"0,1" . - Fung Cheok Yin (cheokyin_restart(AT)yahoo.com.hk), Sep
22 2006
%C A001045 All prime Jacobsthal numbers A049883[n] = {3,5,11,43,683,2731,43691,...}
have prime indices except a(4) = 5. All prime Jacobsthal numbers
with prime indices (all but a(4) = 5) are of the form (2^p + 1)/3
- the Wagstaff primes A000979[n]. Indices of prime Jacobsthal numbers
are listed in A107036[n] = {3,4,5,7,11,13,17,19,23,31,43,61,...}.
For n>1 A107036[n] = A000978[n] Numbers n such that (2^n + 1)/3 is
prime. - Alexander Adamchuk (alex(AT)kolmogorov.com), Oct 03 2006
%C A001045 Correspondence: a(n)=b(n)*2^(n-1), where b(n) is the sequence of the
arithmetic means of previous two terms defined by b(n)=1/2*(b(n-1)+b(n-2))
with initial values b(0)=0, b(1)=1; The g.f. for b(n) is B(x):=x/
(1-(x^1+x^2)/2), so the g.f. A(x) for a(n) suffices A(x)=B(2*x)/2.
Because b(n) converges to the limit lim (1-x)*B(x)=1/3*(b(0)+2*b(1))=2/
3 (for x-->1), it follows that a(n)/2^(n-1) also converges to 2/3
(see also A103770). - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de),
Feb 04 2006
%C A001045 Inverse: floor(log_2(a(n))=n-2 for n>=2. Also: log_2(a(n)+a(n-1))=n-1
for n>=1(see also A130249). Characterization: x is a Jacobsthal number
if and only if there is a power of 4 (=c) such that x is a root of
p(x)=9x(x-c)+(c-1)(2c+1) (see also the indicator sequence A105348).
- Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), May 17 2007
%C A001045 This sequence counts the odd coefficients in the expansion of (1+x+x^2)^(2^n-1),
n>=0. - Tewodros Amdeberhan (tewodros(AT)math.mit.edu), Oct 18 2007,
Jan 08 2008
%C A001045 2^(n+1) = 2*A005578(n) + 2*a(n) + 2*A000975(n-1); e.g. 2^6 = 64 = 2*A005578(5)
+ 2*a(5) + 2*A000975(4) = (2*11 + 2*11 + 2*10). Let A005578(n), a(n),
A000975(n-1) = triangle (a, b, c). Then ((S-c), (S-b), (S-a)) = (A005578(n-1),
a(n-1), A000975(n-2)). Example: (a, b, c) = (11, 11, 10) = (A005578(5),
a(5), A000975(4). Then ((S-c), (S-b), (S-a)) = (6, 5, 5) = (A005578(4),
a(4), A000975(3)). - Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec
24 2007
%C A001045 Sequence is identical to the absolute values of its inverse binomial
transform. A similar result holds for [0,A001045*2^n]. - Paul Curtz
(bpcrtz(AT)free.fr), Jan 17 2008
%C A001045 From a(2) on (i.e., 1,3,5,11,21,...) also: least odd number such that
the subsets of {a(2),...,a(n)} sum to 2^(n-1) different values, cf.
A138000 and A064934. It is interesting to note the pattern of numbers
occuring (or not occuring) as such a sum (A003158). - M. F. Hasler
(www.univ-ag.fr/~mhasler), Apr 09 2008
%C A001045 a(n) = term (5,1) of n-th power of the 5x5 matrix shown in A121231 [From
Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 03 2008]
%C A001045 A147612(a(n)) = 1. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com),
Nov 08 2008]
%C A001045 General form: k=2^n-k. [From Vladimir Orlovsky (4vladimir(AT)gmail.com),
Dec 11 2008]
%C A001045 a(n+1) = Sum(A153778(i): 2^n <= i < 2^(n+1)). [From Reinhard Zumkeller
(reinhard.zumkeller(AT)gmail.com), Jan 01 2009]
%C A001045 Contribution from John Fossaceca (john(AT)fossaceca.net), Jan 31 2009:
(Start)
%C A001045 It appears that a(n) is also the number of integers between 2^n and 2^(n+1)
%C A001045 that are divisible by 3 with no remainder (End)
%C A001045 Number of pairs of consecutive odious (or evil) numbers between 2^(n+1)
and 2^(n+2), inclusive. [From T. D. Noe (noe(AT)sspectra.com), Feb
05 2009]
%C A001045 Equals eigensequence of triangle A156319 [From Gary W. Adamson (qntmpkt(AT)yahoo.com),
Feb 07 2009]
%C A001045 Starting with offset 1 = row sums of triangle A156667. [From Gary W.
Adamson (qntmpkt(AT)yahoo.com), Feb 12 2009]
%C A001045 A three-dimensional interpretation of a(n+1) is that it gives the number
of ways of filling a 2 by 2 by n hole with 1 by 2 by 2 bricks. [From
Martin Griffiths (griffm(AT)essex.ac.uk), Mar 28 2009]
%C A001045 Starting with offset 1 = INVERTi transform of A002605: (1, 2, 6, 16,
44,...). [From Gary W. Adamson (qntmpkt(AT)yahoo.com), May 12 2009]
%C A001045 Convolved with (1, 2, 2, 2,...) = A000225: (1, 3, 7, 15, 31,...). [From
Gary W. Adamson (qntmpkt(AT)yahoo.com), May 23 2009]
%C A001045 The product of a pair of successive terms is always a trianguler number.
- Giuseppe Ottonello, Jun 14 2009
%D A001045 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences,
Academic Press, 1995 (includes this sequence).
%D A001045 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973
(includes this sequence).
%D A001045 Paul Barry, A Catalan Transform and Related Transformations on Integer
Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
%D A001045 Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal
Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.
%D A001045 L. Carlitz, R. Scoville and V. E. Hoggatt, Jr., Representations for a
special sequence, Fib. Quart., 10 (1972), 499-518, 550.
%D A001045 D. E. Daykin, D. J. Kleitman and D. B. West, The number of meets between
two subsets of a lattice, J. Combin. Theory, A 26 (1979), 135-156.
%D A001045 Th. Fink and Y. Mao. The 85 ways to tie a tie, Fourth Estate, London,
1999; Die 85 Methoden eine Krawatte zu binden. Hoffmann und Kampe,
Hamburg, 1999.
%D A001045 International Mathematical Olympiad 2001, Hong Kong Preliminary Selection
Contest Problem #16.
%D A001045 Jablan S. and Sazdanovic R., LinKnot: Knot Theory by Computer, World
Scientific Press, 2007. See p. 80.
%D A001045 D. E. Knuth, Art of Computer Programming, Vol. 3, Sect. 5.3.1, Eq. 13.
%D A001045 T. Koshy, Fibonacci and Lucas numbers with applications, Wiley, 2001,
p. 98.
%D A001045 S. L. Levine, Suppose more rabbits are born, Fib. Quart., 26 (1988),
306-311.
%D A001045 B. D. McKay and I. M. Wanless, A census of small latin hypercubes, SIAM
J. Discrete Math. 22, (2008) 719-736.
%D A001045 G. Myerson and A. J. van der Poorten, Some problems concerning recurrence
sequences, Amer. Math. Monthly, 102 (1995), 698-705.
%D A001045 S. Roman, Introduction to Coding and Information Theory, Springer Verlag,
1996, 41-42
%D A001045 Two-Year College Math. Jnl., 28 (1997), p. 76.
%D A001045 Robert M. Young, "Excursions in Calculus", MAA, 1992, p. 239
%D A001045 G. B. M. Zerr, Problem 64, American Mathematical Monthly, vol. 3, no.
12, 1896 (p. 311).
%H A001045 T. D. Noe, <a href="b001045.txt">Table of n, a(n) for n = 0..500</a>
%H A001045 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to
linear recurrences with constant coefficients</a>
%H A001045 Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Fxtbook</a>
%H A001045 W. Bosma, <a href="http://almira.math.u-bordeaux.fr/jtnb/2001-1/jtnb13-1.html#jourelec">
Signed bits and fast exponentiation</a>
%H A001045 D. D. Frey and J. A. Sellers, Jacobsthal Numbers and Alternating Sign
Matrices, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
J. Integer Seqs., Vol. 3 (2000), #00.2.3</a>
%H A001045 S. Heubach, <a href="http://www.calstatela.edu/faculty/sheubac/CGTC30.PDF">
Tiling an m X n area with squares of size up to k X k (m <=5)</a>
, Congressus Numerantium 140 (1999), pp. 43-64.
%H A001045 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=142">
Encyclopedia of Combinatorial Structures 142</a>
%H A001045 Lee Hae-hwang, <a href="a026644.html">Illustration of initial terms in
terms of rosemary plants</a>
%H A001045 T. Mansour and A. Robertson, <a href="http://arXiv.org/abs/math.CO/0204005">
Refined restricted permutations...</a>.
%H A001045 G. Myerson and A. J. van der Poorten, <a href="http://www-centre.mpce.mq.edu.au/
alfpapers/a106.pdf">Some problems concerning recurrence sequences</
a>, Amer. Math. Monthly, 102 (1995), 698-705.
%H A001045 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">
Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</
a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al,
1992.
%H A001045 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">
1031 Generating Functions and Conjectures</a>, Universit\'{e} du
Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%H A001045 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
JacobsthalNumber.html">Link to a section of The World of Mathematics.</
a>
%H A001045 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
Negabinary.html">Link to a section of The World of Mathematics.</
a>
%H A001045 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
Repunit.html">Link to a section of The World of Mathematics.</a>
%H A001045 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
Rule28.html">Rule 28</a>
%H A001045 Thomas Wieder, <a href="http://www.thomas-wieder.privat.t-online.de/">
HomePage</a>.
%H A001045 <a href="Sindx_Ch.html#Cheby">Index entries for sequences related to
Chebyshev polynomials.</a>
%F A001045 a(n) = 2^(n-1) - a(n-1). a(n) = 2*a(n-1) - (-1)^n = {2^n - (-1)^n}/3.
%F A001045 G.f.: x/(1-x-2*x^2). E.g.f.: (exp(2*x)-exp(-x))/3.
%F A001045 a(2n)=2*a(2n-1)-1 for n>=1, a(2n+1)=2*a(2n)+1 for n>=0. - Lee Hae-hwang
(mathmaniac(AT)empal.com), Oct 11 2002; corrected by Mario Catalani
(mario.catalani(AT)unito.it), Dec 04 2002
%F A001045 Also a(n) is the coefficient of x^(n-1) in the bivariate Fibonacci polynomials
F(n)(x, y)=xF(n-1)(x, y)+yF(n-2)(x, y), with y=2x^2. - Mario Catalani
(mario.catalani(AT)unito.it), Dec 04 2002
%F A001045 a(n)=sum{k=1..n, binomial(n, k)(-1)^(n+k)*3^(k-1) }. - Paul Barry (pbarry(AT)wit.ie),
Apr 02 2003
%F A001045 The ratios a(n)/2^(n-1) converge to 2/3 and every fraction after 1/2
is the arithmetic mean of the two preceding fractions. - Gary W.
Adamson (qntmpkt(AT)yahoo.com), Jul 05 2003
%F A001045 a(n)=U(n-1, i/(2sqrt(2)))(-i*sqrt(2))^(n-1) with i^2=-1 - Paul Barry
(pbarry(AT)wit.ie), Nov 17 2003
%F A001045 a(n+1)=sum(k=0, ceil(n/2), 2^k*binomial(n-k, k)) - Benoit Cloitre (benoit7848c(AT)orange.fr),
Mar 06 2004
%F A001045 a(2n) = A002450(n) = (4^n - 1)/3; a(2n+1) = A007583(n) = (2^(2n+1) +
1)/3. - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Mar 27 2004
%F A001045 a(n) = round(2^n/3) = (2^n + (-1)^(n-1))/3 so lim n->inf 2^n/a(n) = 3
- Gerald McGarvey (Gerald.McGarvey(AT)comcast.net), Jul 21 2004
%F A001045 a(0)=0, a(n)=2a(n-1)-(-1)^n, n>0; a(n)=sum{k=0..n-1, (-1)^k*2^(n-k-1)}=sum{k=0..n-1,
2^k(-1)^(n-k-1)}. - Paul Barry (pbarry(AT)wit.ie), Jul 30 2004
%F A001045 a(n+1)=sum{k=0..n, binomial(k, n-k)2^(n-k)} - Paul Barry (pbarry(AT)wit.ie),
Oct 07 2004
%F A001045 a(n)=sum{k=0..n-1, W(n-k, k)(-1)^(n-k)binomial(2k, k)}, W(n, k) as in
A004070. - Paul Barry (pbarry(AT)wit.ie), Dec 17 2004
%F A001045 a(n)=sum{k=0..n, k*binomial(n-1, (n-k)/2)(1+(-1)^(n+k))floor((2k+1)/3)};
a(n+1)=sum{k=0..n, k*binomial(n-1, (n-k)/2)(1+(-1)^(n+k))(A042965(k)+0^k)};
- Paul Barry (pbarry(AT)wit.ie), Jan 17 2005
%F A001045 a(n+1)=ceiling(2^n/3)+floor(2^n/3)=(ceiling(2^n/3))^2-(floor(2^n/3))^2;
a(n+1)=A005578(n)+A000975(n-1)=A005578(n)^2-A000975(n-1)^2; - Paul
Barry (pbarry(AT)wit.ie), Jan 17 2005
%F A001045 a(n+1)=sum{k=0..n, sum{j=0..n, (-1)^(n-j)*binomial(j, k)}}; - Paul Barry
(pbarry(AT)wit.ie), Jan 26 2005
%F A001045 Let M=[1, 1, 0;1, 0, 1;0, 1, 1], then a(n) = (M^n)[2, 1], also matrix
characteristic polynomial x^3 - 2*x^2 - x + 2 defines the three step
recursion a(0)=0, a(1)=1, a(2)=1, a(n)=2a(n-1)+a(n-2)-2a(n-3) for
n>2 - Lambert Klasen (lambert.klasen(AT)gmx.net), Jan 28 2005
%F A001045 a(n)=ceiling(2^(n+1)/3)-ceiling(2^n/3)=A005578(n+1)-A005578(n); - Paul
Barry (pbarry(AT)wit.ie), Oct 08 2005
%F A001045 a(n)=floor(2^(n+1)/3)-floor(2^n/3)=A000975(n)-A000975(n-1); - Paul Barry
(pbarry(AT)wit.ie), Oct 08 2005
%F A001045 a(n)=Sum{k=0..floor(n, 3), binomial(n, f(n-1)+3k)} a(n)=Sum{k=0..floor(n/
3), binomial(n, f(n-2)+3k)}, where f(n)=(0, 2, 1, 0, 2, 1, ...)=A080424(n).
- Paul Barry (pbarry(AT)wit.ie), Feb 20 2003
%F A001045 a(2n)=Product(d divides n, cyclotomic(d,4))/3. a(2n+1)=Product(d divides
2n+1, cyclotomic(2d,2))/3. - Miklos Kristof (kristmikl(AT)freemail.hu),
Mar 07 2007
%F A001045 Further comments and formulae from Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de),
Apr 23 2007: (Start) The a(n) are closely related to nested square
roots; this is 2*sin(2^(-n)*pi/2*a(n))=sqr(2-sqr(2-sqr(2-sqr(...sqr(2)))...){n-times
the '2', n>=0}.
%F A001045 Also true: 2*cos(2^(-n)*pi*a(n))=sqr(2-sqr(2-sqr(2-sqr(...sqr(2)))...){(n-1)-times
the '2', n>=1} as well as
%F A001045 2*sin(2^(-n)*3/2*pi*a(n))=sqr(2+sqr(2+sqr(2+sqr(...sqr(2)))...){n-times
the '2', n>=0} and
%F A001045 2*cos(2^(-n)*3*pi*a(n))=-sqr(2+sqr(2+sqr(2+sqr(...sqr(2)))...){(n-1)-times
the '2', n>=1}.
%F A001045 a(n)=2^(n+1)/pi*arcsin(b(n+1)/2) where b(n) is defined recursively by
b(0)=2, b(n)=sqr(2-b(n-1)).
%F A001045 There is a similar formula regarding the arccos function, this is a(n)=2^n/
pi*arccos(b(n)/2).
%F A001045 With respect to the sequence c(n) defined recursively by c(0)=-2, c(n)=sqr(2+c(n-1))
the following fomulas hold true: a(n)=2^n/3*(1-(-1)^n*(1-2/pi*arcsin(c(n+1)/
2)); a(n)=2^n/3*(1-(-1)^n*(1-1/pi*arccos(-c(n)/2)). (End)
%F A001045 Sum_{k, 0<=k<=n}A039599(n,k)*a(k)=A049027(n), for n>=1 . - Philippe DELEHAM
(kolotoko(AT)wanadoo.fr), Jun 10 2007
%F A001045 Sum_[k, 0<=k<=n}A039599(n,k)*a(k+1)=A067336(n). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr),
Jun 10 2007
%F A001045 Row sums of triangle A134317. - Gary W. Adamson (qntmpkt(AT)yahoo.com),
Oct 19 2007
%F A001045 Let T = the 3 X 3 matrix [1,1,0; 1,0,1; 0,1,1]. Then T^n * [1,0,0,] =
[A005578(n), a(n), A000975(n-1]. - Gary W. Adamson (qntmpkt(AT)yahoo.com),
Dec 24 2007
%F A001045 a(n)+a(n+5)=11*2^n. - Paul Curtz (bpcrtz(AT)free.fr), Jan 17 2008
%F A001045 a(n)=sum(K(2, k)*a(n - k),k=1..n), where K(n,k) = k if 0 <= k AND k <=
n and K(n,k)=0 else. (When using such a K-coefficient several different
arguments to K or several different definitions of K may lead to
the same integer sequence. For example, the Fibonacci sequence can
be generated in several ways using the K-coefficient.) - Thomas Wieder
(thomas.wieder(AT)t-online.de), Jan 13 2008
%F A001045 a(n)+a(n+2k+1)=a(2k+1)*2^n. - Paul Curtz (bpcrtz(AT)free.fr), Feb 12
2008
%F A001045 a(n) = lower left term in the 2 X 2 matrix [0,2; 1,1]^n - Gary W. Adamson
(qntmpkt(AT)yahoo.com), Mar 02 2008
%F A001045 a(n+1)=Sum_{k, 0<=k<=n} A109466(n,k)*(-2)^(n-k). [From Philippe DELEHAM
(kolotoko(AT)wanadoo.fr), Oct 26 2008]
%F A001045 For n > 0, a(n) = b(n) - b(n-1), where b(n) is defined by the sequence
A000975. [From Kailasam Viswanathan Iyer (kvi(AT)nitt.edu), May 12
2009]
%F A001045 a(n) = sqrt( 8*a(n-1)*a(n-2) + 1 ). E.g. sqrt(3*5*8+1)=11, sqrt(5*11*8+1)=21.
- Giuseppe Ottonello, Jun 14 2009
%e A001045 a(2) = 3 because the tiling of the 3x2 rectangle has either only 1 X
1 tiles, or one 2 X 2 tile in one of two positions (together with
2 1 X 1 tiles)
%p A001045 a:=n->sum(binomial(n-k, k)*2^k, k=0..n): - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com),
Sep 30 2006
%p A001045 A001045:=-1/(z+1)/(2*z-1); [S. Plouffe in his 1992 dissertation.]
%p A001045 a := proc(n::integer) # A001045 Jacobsthal sequence: a(n) = a(n-1) +
2a(n-2), with a(0) = 0, a(1) = 1. local k; option remember; if n
= 0 then 1 else add(K(2,k)*procname(n - k),k=1..n) end if end proc;
K := proc(n::integer, k::integer) local KC; if 0 <= k and k <= n
then KC := k else KC := 0 end if; end proc; - Thomas Wieder (thomas.wieder(AT)t-online.de),
Jan 13 2008
%p A001045 a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=a[n-1]+2*a[n-2]od: seq(a[n],
n=0..33);# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec
15 2008]
%p A001045 with(combstruct): SeqSeqSeqL := [T, {T=Sequence(S, card > 0), S=Sequence(U,
card > 0), U=Sequence(Z, card >1)}, unlabeled]: seq(count(SeqSeqSeqL,
size=j), j=1..34); # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com),
Apr 04 2009]
%t A001045 f[n_] := (2^n - (-1)^n)/3; Table[ f[n], {n, 0, 33}] (from Robert G. Wilson
v (rgwv(at)rgwv.com), Dec 05 2005)
%t A001045 Array[(2^# - (-1)^#)/3 &, 33, 0] - Joseph Biberstine (jrbibers(AT)indiana.edu),
Dec 26 2006
%t A001045 ...and/or...k=0;lst={k};Do[k=2^n-k;AppendTo[lst, k], {n, 0, 5!}];lst
[From Vladimir Orlovsky (4vladimir(AT)gmail.com), Dec 11 2008]
%o A001045 (PARI) a(n)=if(n<0,0,(2^n-(-1)^n)/3)
%o A001045 (PARI) a(n)=if(n==0,0,if(n==1,1,if(n==2,1,2*a(n-1)+a(n-2)-2*a(n-3))))
for(i=0,15,print1(a(i),",")) M=[1,1,0;1,0,1;0,1,1];for(i=0,15,print1((M^i)[2,
1],",")) (Klasen)
%o A001045 sage: from sage.combinat.sloane_functions import recur_gen2 sage: it
= recur_gen2(0,1,1,2) sage: [it.next() for i in range(30)] - Zerinvary
Lajos (zerinvarylajos(AT)yahoo.com), Jun 25 2008
%o A001045 (Other) sage: [lucas_number1(n,1,-2) for n in xrange(0, 34)] # [From
Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 22 2009]
%o A001045 (Other) sage: [abs(gaussian_binomial(n,1,-2)) for n in xrange(0,34)]
# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 28 2009]
%Y A001045 Partial sums of this sequence give A000975, where there are additional
comments from B. E. Williams and Bill Blewett on the tie problem.
Cf. A049883, A026644.
%Y A001045 A002487(A001045(n))=A000045(n).
%Y A001045 Row sums of A059260. Equals A026644(n) + 1 for n > 1.
%Y A001045 a(n)= A073370(n-1, 0), n>=1 (first column of triangle).
%Y A001045 Apart from initial term, equals A026644(n+1) + 1.
%Y A001045 See also A081857.
%Y A001045 Cf. A000978, A000979. Cf. A049883 = primes in this sequence, A107036
= indices of primes, A129738.
%Y A001045 Cf. A019322 A066845, A105348, A130249, A130250, 130253, A134317, A005578,
A002083, A002487, A113405, A138000, A064934, A003158,
%Y A001045 Cf. A147613. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com),
Nov 08 2008]
%Y A001045 A156319 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Feb 07 2009]
%Y A001045 A156667 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Feb 12 2009]
%Y A001045 A002605 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), May 12 2009]
%Y A001045 A000225 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), May 23 2009]
%Y A001045 Sequence in context: A122997 A146042 A152046 this_sequence A154917 A167167
A077925
%Y A001045 Adjacent sequences: A001042 A001043 A001044 this_sequence A001046 A001047
A001048
%K A001045 nonn,nice,easy,core
%O A001045 0,4
%A A001045 N. J. A. Sloane (njas(AT)research.att.com).
%E A001045 More terms from James A. Sellers (sellersj(AT)math.psu.edu), Dec 23 1999.
Zerr reference from Len Smiley (smiley(AT)math.uaa.alaska.edu), May
21 2001.
%E A001045 More terms from Simone Severini (ss54(AT)york.ac.uk), Oct 27 2004
%E A001045 Further terms from Lambert Klasen (lambert.klasen(AT)gmx.net), Jan 28
2005
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