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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, 2014; http://matinf.pmfbl.org/wp-content/uploads/2015/01/za-arhiv-18.-1.pdf

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

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 May 23 23:53 EDT 2017. Contains 286937 sequences.