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A015451 a(n) = 6*a(n-1) + a(n-2) for n > 1, with a(0) = a(1) = 1. 12
1, 1, 7, 43, 265, 1633, 10063, 62011, 382129, 2354785, 14510839, 89419819, 551029753, 3395598337, 20924619775, 128943316987, 794584521697, 4896450447169, 30173287204711, 185936173675435, 1145790329257321 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

a(n) = term (1,1) in the 2x2 matrix [1,2; 3,5]^n. - Gary W. Adamson, May 30 2008

a(n)/a(n-1) tends to sqrt(10) + 3 = 6.16227766... - Gary W. Adamson, May 30 2008

For n >= 1, row sums of triangle for numbers 6^k*C(m,k) with duplicated diagonals. - Vladimir Shevelev, Apr 13 2012

Z[sqrt(10)] is not a unique factorization domain, since, for example, 6 = 2 * 3 = (-1)(2 - sqrt(10))(2 + sqrt(10)) = (4 - sqrt(10))(4 + sqrt(10)). However, the latter two factorizations are not distinct, because 3 + sqrt(10) is a unit in Z[sqrt(10)], and (2 - sqrt(10))(-3 - sqrt(10)) = 4 + sqrt(10). In fact, (2 - sqrt(10))(-3 - sqrt(10))^n gives an algebraic integer b + a(n) * sqrt(10) which, when multiplied by its associate (and by -1 when n is even) is equal to 6. - Alonso del Arte, Mar 15 2014

For n>=1, a(n) equals the numbers of words of length n-1 on alphabet {0,1,2,3,5,6} containing no subwords 00, 11, 22 ,33, 44, 55. - 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 (6,1).

FORMULA

a(n) = Sum_{k, 0 <= k <= n}5^k*A055830(n, k). - Philippe Deléham, Oct 18 2006

a(n) = (1/10)*[3 - sqrt(10)]^n*sqrt(10) - (1/10)*[3 + sqrt(10)]^n*sqrt(10) + (1/2)*[3 + sqrt(10)]^n + (1/2) *[3 - sqrt(10)]^n, with n. >= 0 - Paolo P. Lava, Jun 25 2008

G.f.: (1-5*x)/(1-6*x-x^2). - Philippe Deléham, Nov 20 2008

For n >= 2, a(n) = F_n(6) + F_(n+1)(6), 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

MAPLE

a[0]:=1: a[1]:=1: for n from 2 to 26 do a[n]:=6*a[n-1]+a[n-2] od: seq(a[n], n=0..20); # Zerinvary Lajos, Jul 26 2006

MATHEMATICA

LinearRecurrence[{6, 1}, {1, 1}, 30] (* Vincenzo Librandi, Nov 08 2012 *)

PROG

(MAGMA) [n le 2 select 1 else 6*Self(n-1) + Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 08 2012

CROSSREFS

Sequence in context: A003464 A022036 A277670 * A194779 A126502 A277188

Adjacent sequences:  A015448 A015449 A015450 * A015452 A015453 A015454

KEYWORD

nonn,easy

AUTHOR

Olivier Gérard

STATUS

approved

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Last modified March 25 15:14 EDT 2017. Contains 284082 sequences.