A015445 Generalized Fibonacci numbers: a(n) = a(n-1) + 9*a(n-2).

1, 1, 10, 19, 109, 280, 1261, 3781, 15130, 49159, 185329, 627760, 2295721, 7945561, 28607050, 100117099, 357580549, 1258634440, 4476859381, 15804569341, 56096303770, 198337427839, 703204161769, 2488241012320, 8817078468241, 31211247579121, 110564953793290

Offset: 0

Author

Olivier Gérard

Keywords

• nonn
• easy

Comments

The compositions of n in which each natural number is colored by one of p different colors are called p-colored compositions of n. For n >= 2, 10*a(n-2) equals the number of 10-colored compositions of n with all parts >= 2, such that no adjacent parts have the same color. - Milan Janjic, Nov 26 2011

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• J. Borowska, L. Lacinska, Recurrence form of determinant of a heptadiagonal symmetric Toeplitz matrix, J. Appl. Math. Comp. Mech. 13 (2014) 19-16, remark 2 for permanent of tridiagonal Toeplitz matrices a=1, b=3.
• Index entries for linear recurrences with constant coefficients, signature (1,9).

Crossrefs

Cf. A015443, A015442, A026595, A128099.

Programs

Magma

[ n eq 1 select 1 else n eq 2 select 1 else Self(n-1)+9*Self(n-2): n in [1..30] ]; // Vincenzo Librandi, Aug 23 2011

Maple

m:=25; S:=series(1/(1-x-9*x^2), x, m+1): seq(coeff(S, x, j), j=0..m); # G. C. Greubel, Feb 18 2020

Mathematica

CoefficientList[Series[1/(1-x-9*x^2), {x,0,25}], x] (* or *) LinearRecurrence[{1,9}, {1,1}, 25] (* G. C. Greubel, Apr 30 2017 *)

PARI

a(n)=([0,1; 9,1]^n*[1;1])[1,1] \\ Charles R Greathouse IV, Oct 03 2016

Sage

[lucas_number1(n,1,-9) for n in range(1, 25)] # Zerinvary Lajos, Apr 22 2009

Formula

a(n) = (((1+sqrt(37))/2)^(n+1) - ((1-sqrt(37))/2)^(n+1))/sqrt(37).

a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k)*9^k. - Paul Barry, Jul 20 2004

a(n) = Sum_{k=0..n} binomial((n+k)/2, (n-k)/2)*(1+(-1)^(n-k))*3^(n-k)/2. - Paul Barry, Aug 28 2005

a(n) = Sum_{k=0..n} A109466(n,k)*(-9)^(n-k). - Philippe Deléham, Oct 26 2008

a(n) = (-703*(1/2-sqrt(37)/2)^n + 199*sqrt(37)*(1/2-sqrt(37)/2)^n-333*(1/2+sqrt(37)/2)^n + 171*sqrt(37)*(1/2+sqrt(37)/2)^n)/(74*(5*sqrt(37)-14)). - Alexander R. Povolotsky, Oct 13 2010

a(n) = Sum_{k=1..n+1, k odd} C(n+1,k)*37^((k-1)/2)/2^n. - Vladimir Shevelev, Feb 05 2014

G.f.: 1/(1-x-9*x^2). - Philippe Deléham, Feb 19 2020

a(n) = J(n, 9/2), where J(n,x) are the Jacobsthal polynomials. - G. C. Greubel, Feb 18 2020

E.g.f.: exp(x/2)*(sqrt(37)*cosh(sqrt(37)*x/2) + sinh(sqrt(37)*x/2))/sqrt(37). - Stefano Spezia, Feb 19 2020

Extensions

Edited by N. J. A. Sloane, Oct 11 2010