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A015453 Generalized Fibonacci numbers. 5
1, 1, 8, 57, 407, 2906, 20749, 148149, 1057792, 7552693, 53926643, 385039194, 2749201001, 19629446201, 140155324408, 1000716717057, 7145172343807, 51016923123706, 364263634209749, 2600862362591949, 18570300172353392 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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

M. 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<=k<=n}6^k*A055830(n,k) . - Philippe Deléham, Oct 18 2006

a(n) = (5/106)*sqrt(53)*[7/2-(1/2)*sqrt(53)]^n+(1/2)*[7/2+(1/2)*sqrt(53)]^n+(1/2)*[7/2-(1/2) *sqrt(53)]^n-(5/106)*[7/2+(1/2)*sqrt(53)]^n*sqrt(53), with n>=0. - Paolo P. Lava, Jun 25 2008

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

MATHEMATICA

LinearRecurrence[{7, 1}, {1, 1}, 30] (* Vincenzo Librandi, Nov 08 2012 *)

PROG

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

CROSSREFS

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

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Last modified June 27 19:44 EDT 2017. Contains 288790 sequences.