A072256 a(n) = 10*a(n-1) - a(n-2) for n > 1, a(0) = a(1) = 1.

1, 1, 9, 89, 881, 8721, 86329, 854569, 8459361, 83739041, 828931049, 8205571449, 81226783441, 804062262961, 7959395846169, 78789896198729, 779939566141121, 7720605765212481, 76426118085983689, 756540575094624409, 7488979632860260401, 74133255753507979601, 733843577902219535609

Offset: 0

Author

Lekraj Beedassy, Jul 08 2002

Keywords

• nonn
• easy

Comments

Any k in the sequence is followed by 5*k + 2*sqrt(2*(3*k^2 - 1)).

Gives solutions for x in 3*x^2 - 2*y^2 = 1. Corresponding y is given by A054320(n-1). [corrected by Jon E. Schoenfield, Jun 08 2018]

Number of 01-avoiding words of length n on alphabet {0,1,2,3,4,5,6,7,8,9} which do not end in 0. - Tanya Khovanova, Jan 10 2007

For n >= 2, a(n) equals the permanent of the (2n-2) X (2n-2) tridiagonal matrix with sqrt(8)'s along the main diagonal, and 1's along the superdiagonal and the subdiagonal. - John M. Campbell, Jul 08 2011

Except for the first term, positive values of x (or y) satisfying x^2 - 10xy + y^2 + 8 = 0. - Colin Barker, Feb 09 2014

The aerated sequence [b(n)]n>=1 = [1, 0, 9, 0, 89, 0, 881, 0, ...] is a fourth-order linear divisibility sequence; that is, if n | m then b(n) | b(m). It is the case P1 = 0, P2 = -12, Q = 1 of the 3-parameter family of divisibility sequences found by Williams and Guy. - Peter Bala, May 12 2025

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Christian Aebi and Grant Cairns, Equable Parallelograms on the Eisenstein Lattice, arXiv:2401.08827 [math.CO], 2024. See p. 14.
• Jean-Paul Allouche, Jeffrey Shallit, and Manon Stipulanti, Combinatorics on words and generating Dirichlet series of automatic sequences, arXiv:2401.13524 [math.CO], 2024.
• S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (p. 283).
• Bruno Deschamps, Sur les bonnes valeurs initiales de la suite de Lucas-Lehmer, Journal of Number Theory, Volume 130, Issue 12, December 2010, Pages 2658-2670.
• Editors, L'Intermédiaire des Mathématiciens, Query 4500: The equation x(x+1)/2 = y*(y+1)/3, L'Intermédiaire des Mathématiciens, 22 (1915), 255-260 (I).
• Editors, L'Intermédiaire des Mathématiciens, Query 4500: The equation x(x+1)/2 = y*(y+1)/3, L'Intermédiaire des Mathématiciens, 22 (1915), 255-260 (II).
• Editors, L'Intermédiaire des Mathématiciens, Query 4500: The equation x(x+1)/2 = y*(y+1)/3, L'Intermédiaire des Mathématiciens, 22 (1915), 255-260 (III).
• Editors, L'Intermédiaire des Mathématiciens, Query 4500: The equation x(x+1)/2 = y*(y+1)/3, L'Intermédiaire des Mathématiciens, 22 (1915), 255-260 (IV).
• Alex Fink, Richard K. Guy, and Mark Krusemeyer, Partitions with parts occurring at most thrice, Contributions to Discrete Mathematics, Vol 3, No 2 (2008), pp. 76-114. See Section 13.
• Tanya Khovanova, Recursive Sequences
• J.-C. Novelli and J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv preprint arXiv:1403.5962 [math.CO], 2014.
• H. C. Williams and R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory 7 (5) (2011) 1255-1277.
• H. C. Williams and R. K. Guy, Some Monoapparitic Fourth Order Linear Divisibility Sequences, Integers, Volume 12A (2012) The John Selfridge Memorial Volume.
• Index entries for sequences related to Chebyshev polynomials
• Index entries for linear recurrences with constant coefficients, signature (10,-1).

Crossrefs

Cf. A031138, A046172, A054320, A108299.

Row 10 of array A094954.

First differences of A004189.

Essentially the same as A138288.

Programs

GAP

a:=[1,1];; for n in [3..20] do a[n]:=10*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Jan 14 2020

Magma

[n le 2 select 1 else 10*Self(n-1)-Self(n-2): n in [1..25]]; // Vincenzo Librandi, Feb 10 2014

Maple

seq( simplify(ChebyshevU(n,5) -9*ChebyshevU(n-1,5)), n=0..20); # G. C. Greubel, Jan 14 2020

Mathematica

a[n_]:= a[n]= 10a[n-1] -a[n-2]; a[0]=a[1]=1; Table[ a[n], {n, 0, 20}]
CoefficientList[Series[(1-9x)/(1-10x+x^2), {x, 0, 30}], x] (* Vincenzo Librandi, Feb 10 2014 *)
Table[ChebyshevU[n, 5] -9*ChebyshevU[n-1, 5], {n,0,20}] (* G. C. Greubel, Jan 14 2020 *)
LinearRecurrence[{10,-1},{1,1},20] (* Harvey P. Dale, Jun 17 2022 *)

PARI

a(n)=([0,1; -1,10]^n*[1;1])[1,1] \\ Charles R Greathouse IV, May 10 2016

PARI

vector(21, n, polchebyshev(n-1,2,5) -9*polchebyshev(n-2,2,5) ) \\ G. C. Greubel, Jan 14 2020

Formula

a(n) = (3-sqrt(6))/6 * (5+2*sqrt(6))^n + (3+sqrt(6))/6 * (5-2*sqrt(6))^n.

a(n) = (2*A031138(n) + 1)/3 = sqrt((2*A054320(n-1)^2 + 1)/3), n >= 1.

a(n) = U(n-1, 5)-U(n-2, 5) = T(2*n-1, sqrt(3))/sqrt(3) with Chebyshev's U- and T- polynomials and U(-1, x) := 0, U(-2, x) := -1, T(-1, x) := x.

G.f.: (1-9*x)/(1-10*x+x^2).

6*a(n)^2 - 2 is a square. Limit_{n->oo} a(n)/a(n-1) = 5 + 2*sqrt(6). - Gregory V. Richardson, Oct 10 2002

Let q(n, x) = Sum_{i=0..n} x^(n-i)*binomial(2*n-i, i); then q(n, 8) = a(n+1). - Benoit Cloitre, Nov 10 2002

a(n)*a(n+3) = 80 + a(n+1)*a(n+2). - Ralf Stephan, May 29 2004

a(n) = L(n-1,10), where L is defined as in A108299; see also A054320 for L(n,-10). - Reinhard Zumkeller, Jun 01 2005

a(n) = A138288(n-1) for n > 0. - Reinhard Zumkeller, Mar 12 2008

a(n) = sqrt(A046172(n)). - Paul Weisenhorn, May 15 2009

a(n) = ceiling(((3-sqrt(6))*(5+2*sqrt(6))^n)/6). - Paul Weisenhorn, May 23 2020

E.g.f.: exp(5*x)*(3*cosh(2*sqrt(6)*x) - sqrt(6)*sinh(2*sqrt(6)*x))/3. - Stefano Spezia, Oct 25 2023

From Peter Bala, May 08 2025: (Start)

a(n) = (-1)^n * Dir(n-1, -5), where Dir(n, x) denotes the n-th row polynomial of A244419.

For arbitrary x, a(n+x)^2 - 10*a(n+x)*a(n+x+1) + a(n+x+1)^2 = -8 with a(n) := (3-sqrt(6))/6 * (5+2*sqrt(6))^n + (3+sqrt(6))/6 * (5-2*sqrt(6))^n as given above (the particular case x = 0 is noted in the Comments section).

a(n+1/2) = 1/sqrt(3) * A001079(n).

a(n+3/4) + a(n+1/4) = sqrt(2/3) * sqrt(1 + sqrt(3)) * A001079(n).

a(n+3/4) - a(n+1/4) = 4 * sqrt(sqrt(3) - 1) * A004189(n).

a(n) divides a(3*n-1); a(n) divides a(5*n-2); in general, for k >= 0, a(n) divides a((2*k+1)*n - k).

Sum_{n >= 2} 1/(a(n) - 1/a(n)) = 1/8 (telescoping series: for n >= 2, 1/(a(n) - 1/a(n)) = 1/A291181(n-2) - 1/A291181(n-1).)

Product_{n >= 2} ((a(n) + 1)/(a(n) - 1))^(-1)^n = sqrt(3/2) (telescoping product: Product_{n = 2..k} (((a(n) + 1)/(a(n) - 1))^(-1)^n)^2 = 3/2 * (1 - (-1)^(k+1)/(3*A098308(k))).) (End)

Extensions

Edited by Robert G. Wilson v, Jul 17 2002