Search: id:A001045 Results 1-1 of 1 results found. %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, Table of n, a(n) for n = 0..500 %H A001045 Index entries for sequences related to linear recurrences with constant coefficients %H A001045 Joerg Arndt, Fxtbook %H A001045 W. Bosma, Signed bits and fast exponentiation %H A001045 D. D. Frey and J. A. Sellers, Jacobsthal Numbers and Alternating Sign Matrices, J. Integer Seqs., Vol. 3 (2000), #00.2.3 %H A001045 S. Heubach, Tiling an m X n area with squares of size up to k X k (m <=5) , Congressus Numerantium 140 (1999), pp. 43-64. %H A001045 INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 142 %H A001045 Lee Hae-hwang, Illustration of initial terms in terms of rosemary plants %H A001045 T. Mansour and A. Robertson, Refined restricted permutations.... %H A001045 G. Myerson and A. J. van der Poorten, Some problems concerning recurrence sequences, Amer. Math. Monthly, 102 (1995), 698-705. %H A001045 S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992. %H A001045 S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992. %H A001045 Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics. %H A001045 Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics. %H A001045 Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics. %H A001045 Eric Weisstein's World of Mathematics, Rule 28 %H A001045 Thomas Wieder, HomePage. %H A001045 Index entries for sequences related to Chebyshev polynomials. %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 Search completed in 0.006 seconds