A015442 a(n) = a(n-1) + 7*a(n-2), a(0)=0, a(1)=1.

0, 1, 1, 8, 15, 71, 176, 673, 1905, 6616, 19951, 66263, 205920, 669761, 2111201, 6799528, 21577935, 69174631, 220220176, 704442593, 2245983825, 7177081976, 22898968751, 73138542583, 233431323840, 745401121921, 2379420388801

Offset

0,4

Comments

One obtains A015523 through a binomial transform, and A197189 by shifting one place left (starting 1,1,8 with offset 0) followed by a binomial transform. - R. J. Mathar, Oct 11 2011

The compositions of n in which each positive integer is colored by one of p different colors are called p-colored compositions of n. For n>=2, 8*a(n-1) equals the number of 8-colored compositions of n, with all parts >=2, such that no adjacent parts have the same color. - Milan Janjic, Nov 26 2011

a(n+1) is the number of compositions (ordered partitions) of n into parts 1 and 2, where there are 7 sorts of part 2. - Joerg Arndt, Jan 16 2024

Pisano period lengths: 1, 3, 8, 6, 4, 24, 1, 6, 24, 12, 60, 24, 12, 3, 8, 6, 288, 24, 120, 12, ... - R. J. Mathar, Aug 10 2012

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Joerg Arndt, Matters Computational (The Fxtbook), section 14.8 "Strings with no two consecutive nonzero digits", pp.317-318

Milan Janjic, On Linear Recurrence Equations Arising from Compositions of Positive Integers, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.7.

Index entries for linear recurrences with constant coefficients, signature (1,7).

Formula

O.g.f.: x/(1-x-7x^2). - R. J. Mathar, May 06 2008

a(n) = ( ((1+sqrt(29))/2)^(n+1) - ((1-sqrt(29))/2)^(n+1) )/sqrt(29).

a(n) = 8*a(n-2) + 7*a(n-3) with characteristic polynomial x^3 - 8*x - 7. - Roger L. Bagula, May 30 2007

a(n+1) = Sum_{k=0..n} A109466(n,k)*(-7)^(n-k). - Philippe Deléham, Oct 26 2008

a(n) = (Sum_{1<=k<=n, k odd} C(n,k)*29^((k-1)/2))/2^(n-1). - Vladimir Shevelev, Feb 05 2014

a(n) = sqrt(-7)^(n-1)*S(n-1, 1/sqrt(-7)), with the Chebyshev polynomial S(n, x), and S(-1, x) = 1 (see A049310). - Wolfdieter Lang, Nov 26 2023

Mathematica

LinearRecurrence[{1, 7}, {0, 1}, 30] (* Vincenzo Librandi, Oct 17 2012 *)
nxt[{a_, b_}]:={b, 7a+b}; NestList[nxt, {0, 1}, 30][[All, 1]] (* Harvey P. Dale, Feb 25 2022 *)

Prog

(Sage) [lucas_number1(n, 1, -7) for n in range(0, 27)] # Zerinvary Lajos, Apr 22 2009
(Magma) I:=[0, 1]; [n le 2 select I[n] else Self(n-1) + 7*Self(n-2): n in [1..40]]; // Vincenzo Librandi, Oct 17 2012
(PARI) concat(0, Vec(1/(1-x-7*x^2)+O(x^99))) \\ Charles R Greathouse IV, Mar 12 2014

Crossrefs

Cf. A015440, A015441, A049310.

Sequence in context: A048732 A185038 A193490 * A253211 A275246 A177199

Adjacent sequences: A015439 A015440 A015441 * A015443 A015444 A015445

Keyword

nonn,easy,changed

Author

Olivier Gérard

Status

approved