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

Author

Olivier Gérard

Keywords

• nonn
• easy

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)

Crossrefs

Cf. A054413, A055830.

Row m=7 of A135597.

Programs

Magma

[n le 2 select 1 else 7*Self(n-1) + Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 08 2012

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 *)

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

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