A015453 Generalized Fibonacci numbers.

1, 1, 8, 57, 407, 2906, 20749, 148149, 1057792, 7552693, 53926643, 385039194, 2749201001, 19629446201, 140155324408, 1000716717057, 7145172343807, 51016923123706, 364263634209749, 2600862362591949, 18570300172353392

Offset

0,3

Comments

Row m=7 of A135597.

For n >= 1, row sums of triangle for numbers 7^k*C(m,k) with duplicated diagonals. - Vladimir Shevelev, Apr 13 2012

For n >= 1, a(n) equals the numbers of words of length n-1 on alphabet {0,1,2,3,5,6,7} containing no subwords ii, (i=0,1,...,6). - Milan Janjic, Jan 31 2015

Links

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

Taras Goy and Mark Shattuck, Determinants of Toeplitz-Hessenberg Matrices with Generalized Leonardo Number Entries, Ann. Math. Silesianae (2023). See p. 18.

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

Tanya Khovanova, Recursive Sequences

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

Formula

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

a(n) = Sum_{k=0..n} 6^k*A055830(n,k). - Philippe Deléham, Oct 18 2006

G.f.: (1-6*x)/(1-7*x-x^2). - Philippe Deléham, Nov 20 2008

For n >= 2, a(n) = F_(n)(7) + F_(n+1)(7), where F_(n)(x) is Fibonacci polynomial (cf. A049310): F_(n)(x) = Sum_{i=0..floor((n-1)/2)} C(n-i-1,i)*x^(n-2*i-1). - Vladimir Shevelev, Apr 13 2012

a(n) = A054413(n) - 6*A054413(n-1). - R. J. Mathar, Jul 06 2012

a(n) = Sum_{k=0..n} A046854(n-1,k)*7^k. - R. J. Mathar, Feb 14 2024

Mathematica

LinearRecurrence[{7, 1}, {1, 1}, 30] (* Vincenzo Librandi, Nov 08 2012 *)
CoefficientList[Series[(1-6*x)/(1-7*x-x^2), {x, 0, 50}], x] (* G. C. Greubel, Dec 19 2017 *)

Prog

(Magma) [n le 2 select 1 else 7*Self(n-1) + Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 08 2012
(PARI) my(x='x+O('x^30)); Vec((1-6*x)/(1-7*x-x^2)) \\ G. C. Greubel, Dec 19 2017
(Sage) [lucas_number1(n+1, 7, -1) - 6*lucas_number1(n, 7, -1) for n in (0..30)] # G. C. Greubel, Dec 24 2021

Crossrefs

Cf. A054413, A055830.

Row m=7 of A135597.

Sequence in context: A097114 A022038 A277671 * A181246 A281355 A281912

Adjacent sequences: A015450 A015451 A015452 * A015454 A015455 A015456

Keyword

nonn,easy

Author

Olivier Gérard

Status

approved