A172172 Sums of NW-SE diagonals of triangle A172171.

0, 1, 10, 20, 39, 68, 116, 193, 318, 520, 847, 1376, 2232, 3617, 5858, 9484, 15351, 24844, 40204, 65057, 105270, 170336, 275615, 445960, 721584, 1167553, 1889146, 3056708, 4945863, 8002580, 12948452, 20951041, 33899502, 54850552, 88750063, 143600624, 232350696

Offset

0,3

Comments

This is the sequence A(0,1;1,1;9) of the family of sequences [a,b:c,d:k] considered by G. Detlefs, and treated as A(a,b;c,d;k) in the W. Lang link given below. - Wolfdieter Lang, Oct 18 2010

Links

Colin Barker, Table of n, a(n) for n = 0..1000

Wolfdieter Lang, Notes on certain inhomogeneous three term recurrences.

Index entries for linear recurrences with constant coefficients, signature (2,0,-1).

Formula

a(n) = a(n-1) + a(n-2) + 9 with a(0)=0 and a(1)=1.

From Wolfdieter Lang, Oct 18 2010: (Start)

O.g.f.: x*(1+8*x)/((1-x)*(1-x-x^2)).

a(n) = 2*a(n-1) - a(n-3), a(0)=0, a(1)=1, a(2)=10 (Observation by G. Detlefs).

(End)

a(n+1) - a(n) = A022099(n). - R. J. Mathar, Apr 22 2013

a(n) = -9 + ( (11 + 9*sqrt(5))*(1 + sqrt(5))^n - (11 - 9*sqrt(5))*(1 - sqrt(5))^n )/(2^(n+1)*sqrt(5)). - Colin Barker, Jul 13 2017

a(n) = Fibonacci(n+3) + 7*Fibonacci(n+1) - 9. - G. C. Greubel, Apr 25 2022

Mathematica

CoefficientList[Series[x*(1+8*x)/((1-x)*(1-x-x^2)), {x, 0, 50}], x] (* G. C. Greubel, Jul 13 2017 *)

Prog

(PARI) concat(0, Vec(x*(1+8*x)/((1-x)*(1-x-x^2)) + O(x^50))) \\ Colin Barker, Jul 13 2017
(Magma) [Lucas(n+2) +6*Fibonacci(n+1) -9: n in [0..50]]; // G. C. Greubel, Apr 25 2022
(SageMath) [fibonacci(n+3) +7*fibonacci(n+1) -9 for n in (0..50)] # G. C. Greubel, Apr 25 2022

Crossrefs

Cf. A000032, A000045, A172171.

Sequence in context: A188334 A266087 A047881 * A275245 A020953 A033846

Adjacent sequences: A172169 A172170 A172171 * A172173 A172174 A172175

Keyword

nonn,easy

Author

Mark Dols, Jan 28 2010

Extensions

Wrong offset 1 changed into 0 Wolfdieter Lang, Oct 18 2010

Status

approved