%I A000129 M1413 N0552
%S A000129 0,1,2,5,12,29,70,169,408,985,2378,5741,13860,33461,80782,195025,
%T A000129 470832,1136689,2744210,6625109,15994428,38613965,93222358,225058681,
%U A000129 543339720,1311738121,3166815962,7645370045,18457556052,44560482149
%N A000129 Pell numbers: a(0) = 0, a(1) = 1; for n > 1, a(n) = 2*a(n-1) + a(n-2).
%C A000129 Sometimes also called lambda numbers.
%C A000129 Also denominators of continued fraction convergents to sqrt(2): 1, 3/
2, 7/5, 17/12, 41/29, 99/70, 239/169, 577/408, 1393/985, 3363/2378,
8119/5741, 19601/13860, 47321/33461, 114243/80782, ... = A001333/
A000129
%C A000129 Number of lattice paths from (0,0) to the line x=n-1 consisting of U=(1,
1), D=(1,-1) and H=(2,0) steps (i.e. left factors of Grand Schroeder
paths); for example, a(3)=5, counting the paths H, UD, UU, DU and
DD. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Oct 27 2002
%C A000129 a(2*n) with b(2*n) := A001333(2*n), n>=1, give all (positive integer)
solutions to Pell equation b^2 - 2*a^2 = +1 (see Emerson reference).
a(2*n+1) with b(2*n+1) := A001333(2*n+1), n>=0, give all (positive
integer) solutions to Pell equation b^2 - 2*a^2 = -1.
%C A000129 Bisection: a(2*n+1)= T(2*n+1,sqrt(2))/sqrt(2)= A001653(n), n>=0 and a(2*n)=
2*S(n-1,6)= 2*A001109(n),n>=0, with T(n,x), resp. S(n,x), Chebyshev's
polynomials of the first,resp. second kind. S(-1,x)=0. See A053120,
resp. A049310. - W. Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de),
Jan 10 2003
%C A000129 Consider the mapping f(a/b) = (a + 2b)/(a + b). Taking a = b = 1 to start
with and carrying out this mapping repeatedly on each new (reduced)
rational number gives the following sequence 1/1, 3/2,7/5,17/12,41/
29,... converging to 2^(1/2). Sequence contains the denominators.
- Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Mar 22 2003
%C A000129 This is also the Horadam sequence (0,1,1,2). a(n) / a(n-1) converges
to 2^1/2 + 1 as n approaches infinity. - Ross La Haye (rlahaye(AT)new.rr.com),
Aug 18 2003
%C A000129 Number of 132-avoiding two-stack sortable permutations.
%C A000129 y satisfying x^2 - 2*y^2=-+1. Corresponding x given by A001333(n). -
Lekraj Beedassy (blekraj(AT)yahoo.com), Jun 24 2004
%C A000129 For n>0, the number of (s(0), s(1), ..., s(n)) such that 0 < s(i) < 4
and |s(i) - s(i-1)| <= 1 for i = 1,2,....,n, s(0) = 2, s(n) = 3.
- Herbert Kociemba (kociemba(AT)t-online.de), Jun 02 2004
%C A000129 Number of (s(0), s(1), ..., s(n)) such that 0 < s(i) < 4 and |s(i) -
s(i-1)| <= 1 for i = 1,2,....,n, s(0) = 1, s(n) = 2. - Herbert Kociemba
(kociemba(AT)t-online.de), Jun 02 2004
%C A000129 Counts walks of length n from a vertex of a triangle to another vertex
to which a loop has been added. - Mario Catalani (mario.catalani(AT)unito.it),
Jul 23 2004
%C A000129 Apart from initial terms, Pisot sequence P(2,5). See A008776 for definition
of Pisot sequences. - David W. Wilson (davidwwilson(AT)comcast.net)
%C A000129 Sums of antidiagonals of A038207 [Pascal's triangle squared] - Ross La
Haye (rlahaye(AT)new.rr.com), Oct 28 2004
%C A000129 The Pell primality test is "If N is an odd prime, then P(N)-kronecker(2,
N) is divisible by N". "Most" composite numbers fail this test, so
it makes a useful pseudoprimality test. The odd composite numbers
which are Pell pseudoprimes (i.e. that pass the above test) are in
A099011. - Jack Brennen (jb(AT)brennen.net), Nov 13, 2004
%C A000129 a(n) = sum of n-th row of triangle in A008288 = A094706(n)+A000079(n).
- Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Dec 03 2004
%C A000129 Pell trapezoids (cf. A084158); for n>0, A001109(n)= (a(n-1)+a(n+1))*a(n)/
2; e.g. 1189=(12+70)*29/2 - Charlie Marion (charliemath(AT)optonline.net),
Apr 1 2006
%C A000129 (0!a(1),1!a(2),2!a(3),3!a(4),...) and (1,-2,-2,0,0,0,...) form a reciprocal
pair under the list partition transform and associated operations
described in A133314. - Tom Copeland (tcjpn(AT)msn.com), Oct 29 2007
%C A000129 Let C = (sqrt(2)+1) = 2.414213562..., then for n>1, C^n = a(n)*(1/C)
+ a(n+1). Example: C^3 = 14.0710678... = 5*(.414213562...) + 12.
Let X = the 2 X 2 matrix [0, 1; 1, 2]; then X^n * [1, 0] = [a(n-1),
a(n); a(n), a(n+1)]. a(n) = numerator of n-th convergent to (Sqrt(2)-1)
= .41421356... = [2, 2, 2,...], the convergents being [1/2, 2/5,
5/12,...]. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 21 2007
%C A000129 A = sqrt(2) = 2/2 + 2/5 + 2/(5*29) + 2/(29*169) + 2/(169*985) + ...;
B = ((5/2) - sqrt(2)) = 2/2 + 2/(2*12) + 2/(12*70) + 2/(70*408) +
2/(408*2378) + ...; A+B = 5/2. C = 1/2 = 2/(1*5) + 2/(2*12) + 2/(5*29)
+ 2/(12*70) + 2/(29*169) + ... - Gary W. Adamson (qntmpkt(AT)yahoo.com),
Mar 16 2008
%C A000129 Prime Pell numbers with an odd index gives the RMS value (A141812) of
prime RMS numbers (A140480). [From Ctibor O. Zizka (ctibor.zizka(AT)seznam.cz),
Aug 13 2008]
%C A000129 Comment from Clark Kimberling (ck6(AT)evansville.edu), Aug 27 2008 (Start):
Related convergents (numerator/denominator):
%C A000129 lower principal convergents: A002315/A001653
%C A000129 upper principal convergents: A001541/A001542
%C A000129 intermediate convergents: A052542/A001333
%C A000129 lower intermediate convergents: A005319/A001541
%C A000129 upper intermediate convergents: A075870/A002315
%C A000129 principal and intermediate convergents: A143607/A002965
%C A000129 lower principal and intermediate convergents: A143608/A079496
%C A000129 upper principal and intermediate convergents: A143609/A084068 (End)
%C A000129 Equals row sums of triangle A143808 starting with offset 1. [From Gary
W. Adamson (qntmpkt(AT)yahoo.com), Sep 01 2008]
%C A000129 Recurrence a(n)=2a(n-1)+a(n-2) holds for b(n)= 1, A000129. [From Paul
Curtz (bpcrtz(AT)free.fr), Oct 23 2008]
%C A000129 Binomial transform of the sequence:= 0,1,0,2,0,4,0,8,0,16,..., powers
of 2 alternating with zeros. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr),
Oct 28 2008]
%C A000129 a(n) is also the sum of the nth row of the triangle formed by starting
with the top two rows of Pascal's triangle and then each next row
has a 1 at both ends and the interior values are the sum of the three
numbers in the triangle above that position. [From Patrick Costello
(pat.costello(AT)eku.edu), Dec 07 2008]
%C A000129 Starting with offset 1 = eigensequence of triangle A135387 (an infinite
lower triangular matrix with (2,2,2,...) in the main diagonal and
(1,1,1,...) in the subdiagonal. [From Gary W. Adamson (qntmpkt(AT)yahoo.com),
Dec 29 2008]
%C A000129 Starting with offset 1 = row sums of triangle A153345 [From Gary W. Adamson
(qntmpkt(AT)yahoo.com), Dec 24 2008]
%C A000129 Contribution from Charlie Marion (charliemath(AT)optonline.net), Jan
07 2009: (Start)
%C A000129 In general, denominators, a(k,n) and numerators, b(k,n), of continued
%C A000129 fraction convergents to sqrt((k+1)/k) may be found as follows:
%C A000129 let a(k,0) = 1, a(k,1) = 2k; for n>0, a(k,2n) = 2*a(k,2n-1)+a(k,2n-2)
%C A000129 and a(k,2n+1)=(2k)*a(k,2n)+a(k,2n-1);
%C A000129 let b(k,0) = 1, b(k,1) = 2k+1; for n>0, b(k,2n) = 2*b(k,2n-1)+b(k,2n-2)
%C A000129 and b(k,2n+1)=(2k)*b(k,2n)+b(k,2n-1).
%C A000129 For example, the convergents to sqrt(2/1) start 1/1, 3/2, 7/5,
%C A000129 17/12, 41/29.
%C A000129 In general, if a(k,n) and b(k,n) are the denominators and numerators,
%C A000129 respectively, of continued fraction convergents to sqrt((k+1)/k)
%C A000129 as defined above, then
%C A000129 k*a(k,2n)^2-a(k,2n-1)*a(k,2n+1)=k=k*a(k,2n-2)*a(k,2n)-a(k,2n-1)^2 and
%C A000129 b(k,2n-1)*b(k,2n+1)-k*b(k,2n)^2=k+1=b(k,2n-1)^2-k*b(k,2n-2)*b(k,2n);
%C A000129 for example, if k=1 and n=3, then a(1,n)=a(n+1) and
%C A000129 1*a(1,6)^2-a(1,5)*a(1,7)=1*169^2-70*408=1;
%C A000129 1*a(1,4)*a(1,6)-a(1,5)^2=1*29*169-70^2=1;
%C A000129 b(1,5)*b(1,7)-1*b(1,6)^2=99*577-1*239^2=2;
%C A000129 b(1,5)^2-1*b(1,4)*b(1,6)=99^2-1*41*239=2.
%C A000129 Cf. A001333, A142238-A142239, A153313-153318.
%C A000129 [From Charlie Marion (charliemath(AT)optonline.net), Jan 07 2009]
%C A000129 (End)
%C A000129 Starting with offset 1 = row sums of triangle A155002, equivalent to
the statement that the Fibonacci series convolved with the Pell series
prefaced with a "1": (1, 1, 2, 5, 12, 29,...) = (1, 2, 5, 12, 29,
...). [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Jan 18 2009]
%C A000129 It appears that P(p) == 8^((p-1/2)) mod p, p = prime; analogous to [Schroeder,
p.90]: Fp == 5^((p-1)/2)) mod p. Example: Given P(11) = 5741, ==
8^5 mod 11. Given P(17) = 11336689, == 8^8 mod 17 since 17 divides
(8^8 - P(l7)). [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Feb
21 2009]
%C A000129 Equals eigensequence of triangle A154325 [From Gary W. Adamson (qntmpkt(AT)yahoo.com),
Feb 12 2009]
%C A000129 Another combinatorial interpretation of a(n+1) arises from a simple tiling
scenario. Namely, a(n+1) gives the number of ways of tiling a 1 by
n rectangle with indistinguishable 1 by 2 rectangles and 1 by 1 squares
that come in two varieties, A and B say. For example, with C representing
the 1 by 2 rectangle, we obtain a(4)=12 from AAA, AAB, ABA, BAA,
ABB, BAB, BBA, BBB, AC, BC, CA and CB. [From Martin Griffiths (griffm(AT)essex.ac.uk),
Apr 25 2009]
%C A000129 a(n+1)=2*a(n)+ a(n-1) a(1=1,a(2)=2 was used by Theon from Smyrna. [From
Sture Sjoestedt (sture.sjostedt(AT)spray.se), May 29 2009]
%C A000129 The nth Pell number counts the perfect matchings of the edge-labeled
graph C_2 x P_(n-1), or equivalently, the number of domino tilings
of a 2 x (n-1) cylindrical grid. [From Sarah-Marie Belcastro (smbelcas(AT)toroidalsnark.net),
Jul 04 2009]
%C A000129 Number of units of a(n) belongs to a periodic sequence: 0, 1, 2, 5, 2,
9, 0, 9, 8, 5, 8, 1. [From Mohamed Bouhamida (bhmd95(AT)yahoo.fr),
Sep 04 2009]
%D A000129 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences,
Academic Press, 1995 (includes this sequence).
%D A000129 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973
(includes this sequence).
%D A000129 P. Bachmann, Niedere Zahlentheorie (1902, 1910), reprinted Chelsea, NY,
1968, vol. 2, p. 76.
%D A000129 Paul Barry, A Catalan Transform and Related Transformations on Integer
Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
%D A000129 Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal
Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.
%D A000129 A. H. Beiler, Recreations in the Theory of Numbers. New York: Dover,
pp. 122-125, 1964.
%D A000129 S.-M. Belcastro, Tilings of 2 x n Grids on Surfaces, preprint. [From
Sarah-Marie Belcastro (smbelcas(AT)toroidalsnark.net), Jul 04 2009]
%D A000129 John Derbyshire, Prime Obsession, Joseph Henry Press, 2004, see p. 16.
%D A000129 E. Deutsch, A formula for the Pell numbers, Problem 10663, Amer. Math.
Monthly 107 (No. 4, 2000), solutions pp. 370-371.
%D A000129 E. I. Emerson, Recurrent sequences in the equation DQ^2=R^2+N, Fib. Quart.,
7 (1969), 231-242, Ex.1, p. 237-8.
%D A000129 S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.1.
%D A000129 A. F. Horadam, Special Properties of the Sequence W(n){a, b; p, q}, Fibonacci
Quarterly, Vol. 5, No. 5, 1967, pp. 424-434.
%D A000129 A. F. Horadam, Pell identities, Fib. Quart., 9 (1971), 245-252, 263.
%D A000129 Problem B-82, Fib. Quart., 4 (1966), 374-375.
%D A000129 P. Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY,
2nd ed., 1989, p. 43.
%D A000129 J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 224.
%D A000129 Ayoub B. Ayoub, "Fibonacci-like sequences and Pell equations", The College
Mathematics Journal, Vol. 38 (2007), pp. 49-53.
%D A000129 Clark Kimberling, "Best lower and upper approximates to irrational numbers,
" Elemente der Mathematik, 52 (1997) 122-126.
%D A000129 Serge Lang, Introduction to Diophantine Approximations, Addison-Wesley,
New York, 1966.
%D A000129 Hao Pan, Arithmetic properties of q-Fibonacci numbers and q-Pell numbers,
Discr. Math., 306 (2006), 2118-2127. [From N. J. A. Sloane (njas(AT)research.att.com),
Jan 29 2009]
%D A000129 Manfred R. Schroeder, "Number Theory in Science and Communication", 5-th
ed., Springer-Verlag, 2009, p. 90. [From Gary W. Adamson (qntmpkt(AT)yahoo.com),
Feb 21 2009]
%D A000129 Mark A. Shattuck, Tiling proofs of some formulas for the Pell numbers
of odd index, Integers, 9 (2009), 53-64.
%D A000129 Extending Theons Ladder to Any Square Root Problem 3858 in Elementa nr4
1996 [From Sture Sjoestedt (sture.sjostedt(AT)spray.se), May 29 2009]
%H A000129 N. J. A. Sloane, <a href="b000129.txt">Table of n, a(n) for n = 0..500</
a>
%H A000129 Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Fxtbook</a>
%H A000129 Tewodros Amdeberhan, <a href="http://www.math.temple.edu/~tewodros/solutions/
solu.html">Solution to problem #10663 (AMM)</a>
%H A000129 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
Sequences realized by oligomorphic permutation groups</a>, J. Integ.
Seqs. Vol. 3 (2000), #00.1.5.
%H A000129 E. S. Egge and T. Mansour, <a href="http://arXiv.org/abs/math.CO/0205206">
132-avoiding two-stack sortable permutations...</a>.
%H A000129 Nick Hobson, <a href="a000129.py.txt">Python program for this sequence</
a>
%H A000129 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=135">
Encyclopedia of Combinatorial Structures 135</a>
%H A000129 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/
RecursiveSequences.html">Recursive Sequences</a>
%H A000129 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 A000129 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 A000129 James A. Sellers, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
Domino Tilings and Products of Fibonacci and Pell Numbers</a>, Journal
of Integer Sequences, Vol. 5 (2002), Article 02.1.2
%H A000129 R. A. Sulanke, <a href="http://math.boisestate.edu/~sulanke/recentpapindex.html">
Moments, Narayana numbers and the cut and paste for lattice paths</
a>
%H A000129 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
PellNumber.html">Link to a section of The World of Mathematics.</
a>
%H A000129 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
PellPolynomial.html">Link to a section of The World of Mathematics.</
a>
%H A000129 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
SquareRoot.html">Link to a section of The World of Mathematics.</
a>
%H A000129 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
PythagorassConstant.html">Pythagoras's Constant</a>
%H A000129 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
SquareTriangularNumber.html">Square Triangular Number</a>
%H A000129 <a href="Sindx_Cor.html#core">Index entries for "core" sequences</a>
%H A000129 <a href="Sindx_Ch.html#Cheby">Index entries for sequences related to
Chebyshev polynomials.</a>
%H A000129 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to
linear recurrences with constant coefficients</a>
%F A000129 G.f.: x/(1-2*x-x^2).
%F A000129 a(n) = 2*a(n-1)+a(n-2), a(0)=0, a(1)=1.
%F A000129 a(n)=( (1+sqrt(2))^n -(1-sqrt(2))^n )/(2*sqrt(2))
%F A000129 a(n) = integer nearest a(n-1)/(sqrt(2) - 1), where a(0) = 1 - from Clark
Kimberling (ck6(AT)evansville.edu)
%F A000129 a(n)= Sum_{i, j, k >= 0: i+j+2k=n} (i+j+k)!/(i!*j!*k!).
%F A000129 a(n)^2 + a(n+1)^2 = a(2n+1) (1999 Putnam examination).
%F A000129 a(2n) = 2*a(n)*A001333(n). - John McNamara, Oct 30, 2002
%F A000129 a(n) = ((-i)^(n-1))*S(n-1, 2*i), with S(n, x) := U(n, x/2) Chebyshev's
polynomials of the second kind. See A049310. S(-1, x)=0, S(-2, x)=
-1.
%F A000129 Binomial transform of expansion of sinh(sqrt(2)x)/sqrt(2). E.g.f.: exp(x)sinh(sqrt(2)x)/
sqrt(2). - Paul Barry (pbarry(AT)wit.ie), May 09 2003
%F A000129 a(n)=sum{k=0, ..floor(n/2), C(n, 2k+1)2^k}. - Paul Barry (pbarry(AT)wit.ie),
May 13 2003
%F A000129 a(n-2) + a(n) = (1 + sqrt2)^(n-1) + (1 - sqrt2)^(n-1) = A002203(n-1).
[A002203(n)]^2 - 8[a(n)]^2 = 4(-1)^n - Gary W. Adamson (qntmpkt(AT)yahoo.com),
Jun 15 2003
%F A000129 G.f. : x(1+x)/(1-x-3x^2-x^3); a(n)=a(n-1)+3a(n-2)+a(n-2); - Mario Catalani
(mario.catalani(AT)unito.it), Jul 23 2004
%F A000129 a(n+1)=Sum(C(n-k, k)2^(n-2k), k=0, .., Floor[n/2]). - Mario Catalani
(mario.catalani(AT)unito.it), Jul 23 2004
%F A000129 Apart from initial terms, inverse binomial transform of A052955. - Paul
Barry, May 23 2004
%F A000129 a(n)^2+a(n+2k+1)^2=A001653(k)*A001653(n+k);e.g., 5^2+70^2=5*985 - Charlie
Marion (charliemath(AT)optonline.net) Aug 03 2005
%F A000129 a(n+1)=sum{k=0..n, binomial((n+k)/2, (n-k)/2)(1+(-1)^(n-k))2^k/2}; -
Paul Barry (pbarry(AT)wit.ie), Aug 28 2005
%F A000129 a(n) = a(n - 1) + A001333(n - 1) = A001333(n) - a(n - 1) = A001109(n)/
A001333(n) = sqrt(A001110(n)/A001333(n)^2) = ceiling(sqrt(A001108(n)/
2)) - Henry Bottomley (se16(AT)btinternet.com), Apr 18 2000
%F A000129 a(n)=F(n, 2), the n-th Fibonacci polynomial evaluated at x=2. - T. D.
Noe (noe(AT)sspectra.com), Jan 19 2006
%F A000129 Define c(2n) = -A001108(n), c(2n+1) = -A001108(n+1) and d(2n) = d(2n+1)
= A001652(n), then ((-1)^n)*(c(n) + d(n)) = a(n). - Proof given by
Max Alekseyev (maxale(AT)gmail.com) - Creighton Dement (creighton.k.dement(AT)uni-oldenburg.de),
Jul 21 2005
%F A000129 a(r+s) = a(r)*a(s+1) + a(r-1)*a(s). - Lekraj Beedassy (blekraj(AT)yahoo.com),
Sep 03 2006
%F A000129 a(n)=(b(n+1)+b(n-1))/n where {b(n)} is the sequence A006645 - Sergio
Falcon (sfalcon(AT)dma.ulpgc.es), Nov 22 2006
%F A000129 Comments from Miklos Kristof (kristmikl(AT)freemail.hu), Mar 19 2007:
(Start)
%F A000129 Let F(n)=a(n)=Pell numbers, L(n)=A002203=companion Pell numbers (A002203):
%F A000129 For a>=b and odd b F(a+b)+F(a-b)=L(a)*F(b).
%F A000129 For a>=b and even b F(a+b)+F(a-b)=F(a)*L(b).
%F A000129 For a>=b and odd b F(a+b)-F(a-b)=F(a)*L(b).
%F A000129 For a>=b and even b F(a+b)-F(a-b)=L(a)*F(b).
%F A000129 F(n+m)+(-1)^m*F(n-m)=F(n)*L(m)
%F A000129 F(n+m)-(-1)^m*F(n-m)=L(n)*F(m)
%F A000129 F(n+m+k)+(-1)^k*F(n+m-k)+(-1)^m*(F(n-m+k)+(-1)^k*F(n-m-k))=F(n)*L(m)*L(k)
%F A000129 F(n+m+k)-(-1)^k*F(n+m-k)+(-1)^m*(F(n-m+k)-(-1)^k*F(n-m-k))=L(n)*L(m)*F(k)
%F A000129 F(n+m+k)+(-1)^k*F(n+m-k)-(-1)^m*(F(n-m+k)+(-1)^k*F(n-m-k))=L(n)*F(m)*L(k)
%F A000129 F(n+m+k)-(-1)^k*F(n+m-k)-(-1)^m*(F(n-m+k)-(-1)^k*F(n-m-k))=8*F(n)*F(m)*F(k)
(End)
%F A000129 a(n+1)*a(n)=2*sum{k=0..n, a(k)^2} (a similar relation holds for A001333)
- Creighton Dement (creighton.k.dement(AT)uni-oldenburg.de), Aug
28 2007
%F A000129 a(n+1) = sum(k=0,...,n) binomial(n+1,2k+1) * 2^k = sum(k=0,...,n) A034867(n,
k) * 2^k = (1/n!)sum(k=0,...,n) A131980(n,k) * 2^k . - Tom Copeland
(tcjpn(AT)msn.com), Nov 30 2007
%F A000129 Equals row sums of unsigned triangle A133156. - Gary W. Adamson (qntmpkt(AT)yahoo.com),
Apr 21 2008
%F A000129 a(n) (n>=3) is the determinant of the (n-1) by (n-1) tridiagonal matrix
with diagonal entries 2, superdiagonal entries 1 and subdiagonal
entries -1. [From Emeric Deutsch (deutsch(AT)duke.poly.edu), Aug
29 2008]
%F A000129 a(n)=5*a(n-2)+2*a(n-3), a(n)=6*a(n-2)-a(n-4). - Mohamed Bouhamida (bhmd95(AT)yahoo.fr),
Sep 04 2008
%F A000129 Comments from Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Jan
02 2009 (Start): fract((1+sqrt(2))^n)) = (1/2)*(1+(-1)^n)-(-1)^n*(1+sqrt(2))^(-n)
= (1/2)*(1+(-1)^n)-(1-sqrt(2))^n.
%F A000129 See A001622 for a general formula concerning the fractional parts of
powers of numbers x>1, which suffice x-x^(-1)=floor(x).
%F A000129 a(n) = nint((1+sqrt(2))^n) for n>0. (End)
%F A000129 a(n)=((4+sqrt18)*(1+sqrt2)^n)+(4-sqrt18)*(1-sqrt2)^n)/4 offset 0. [From
Al Hakanson (hawkuu(AT)gmail.com), Aug 08 2009]
%p A000129 A000129 := proc(n) option remember; if n <=1 then n; else 2*A000129(n-1)+A000129(n-2);
fi; end;
%p A000129 with(numtheory):pel := cfrac (sin(Pi/4),100): seq(nthnumer(pel,i), i=0..29
); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Feb 07 2007
%p A000129 A000129:=-1/(-1+2*z+z**2); [S. Plouffe in his 1992 dissertation.]
%p A000129 (Maple) a := n -> (Matrix([[2,1],[1,0]])^n)[1,2]; seq (a(n), n=0..29);
[From Alois P. Heinz (heinz(AT)hs-heilbronn.de), Aug 01 2008]
%t A000129 CoefficientList[Series[x/(1 - 2*x - x^2), {x, 0, 60}], x] - Stefan Steinerberger
(stefan.steinerberger(AT)gmail.com), Apr 08 2006
%t A000129 Expand[Table[((1 + Sqrt[2])^n - (1 - Sqrt[2])^n)/(2Sqrt[2]), {n, 0, 30}]]
- Artur Jasinski (grafix(AT)csl.pl), Dec 10 2006
%t A000129 a=1;b=0;c=0;lst={b};Do[c=a+b+c;AppendTo[lst,c];a=b;b=c,{n,5!}];lst [From
Vladimir Orlovsky (4vladimir(AT)gmail.com), Mar 23 2009]
%o A000129 (PARI) a(n)=if(n<0,0,contfracpnqn(vector(n,i,1+(i>1)))[2,1])
%o A000129 (Other) sage: [lucas_number1(n,2,-1) for n in xrange(0, 30)] # [From
Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 22 2009]
%o A000129 (PARI) { default(realprecision, 2000); for (n=0, 4000, a=contfracpnqn(vector(n,
i, 1+(i>1)))[2, 1]; if (a > 10^(10^3 - 6), break); write("b000129.txt",
n, " ", a); ); } [From Harry J. Smith (hjsmithh(AT)sbcglobal.net),
Jun 12 2009]
%Y A000129 Partial sums of A001333, also A000129(n)+A000129(n+1) = A001333(n+1).
%Y A000129 a(n) = A054456(n-1, 0), n>=1 (first column of triangle).
%Y A000129 Cf. A002203, A096669, A096670, A097075, A097076, A051927, A005409.
%Y A000129 A077985 is a signed version.
%Y A000129 INVERT transform of Fibonacci numbers (A000045).
%Y A000129 Cf. A038207.
%Y A000129 The following sequences (and others) belong to the same family: A001333,
A000129, A026150, A002605, A046717, A015518, A084057, A063727, A002533,
A002532, A083098, A083099, A083100, A015519.
%Y A000129 Cf. A034867, A131980.
%Y A000129 Cf. A133156.
%Y A000129 A143808 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 01 2008]
%Y A000129 Cf. A135387, A153346. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec
29 2008]
%Y A000129 Cf. A001622, A006497, A014176, A098316.
%Y A000129 A155002 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Jan 18 2009]
%Y A000129 A154325 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Feb 12 2009]
%Y A000129 Sequence in context: A067687 A130009 A048624 this_sequence A141682 A077985
A054198
%Y A000129 Adjacent sequences: A000126 A000127 A000128 this_sequence A000130 A000131
A000132
%K A000129 nonn,easy,core,cofr,nice,frac
%O A000129 0,3
%A A000129 N. J. A. Sloane (njas(AT)research.att.com).
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