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

Author

Mark Dols, Jan 28 2010

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A000032, A000045, A172171.

Programs

Magma

[Lucas(n+2) +6*Fibonacci(n+1) -9: n in [0..50]]; // 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 *)

PARI

concat(0, Vec(x*(1+8*x)/((1-x)*(1-x-x^2)) + O(x^50))) \\ Colin Barker, Jul 13 2017

SageMath

[fibonacci(n+3) +7*fibonacci(n+1) -9 for n in (0..50)] # G. C. Greubel, Apr 25 2022

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

Extensions

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