A172172 - OEIS

%I #23 Apr 25 2022 08:04:22

%S 0,1,10,20,39,68,116,193,318,520,847,1376,2232,3617,5858,9484,15351,

%T 24844,40204,65057,105270,170336,275615,445960,721584,1167553,1889146,

%U 3056708,4945863,8002580,12948452,20951041,33899502,54850552,88750063,143600624,232350696

%N Sums of NW-SE diagonals of triangle A172171.

%C 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

%H Colin Barker, <a href="/A172172/b172172.txt">Table of n, a(n) for n = 0..1000</a>

%H Wolfdieter Lang, <a href="/A172172/a172172.pdf">Notes on certain inhomogeneous three term recurrences.</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-1).

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

%F From _Wolfdieter Lang_, Oct 18 2010: (Start)

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

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

%F (End)

%F a(n+1) - a(n) = A022099(n). - _R. J. Mathar_, Apr 22 2013

%F 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

%F a(n) = Fibonacci(n+3) + 7*Fibonacci(n+1) - 9. - _G. C. Greubel_, Apr 25 2022

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

%o (PARI) concat(0, Vec(x*(1+8*x)/((1-x)*(1-x-x^2)) + O(x^50))) \\ _Colin Barker_, Jul 13 2017

%o (Magma) [Lucas(n+2) +6*Fibonacci(n+1) -9: n in [0..50]]; // _G. C. Greubel_, Apr 25 2022

%o (SageMath) [fibonacci(n+3) +7*fibonacci(n+1) -9 for n in (0..50)] # _G. C. Greubel_, Apr 25 2022

%Y Cf. A000032, A000045, A172171.

%K nonn,easy

%O 0,3

%A _Mark Dols_, Jan 28 2010

%E Wrong offset 1 changed into 0 _Wolfdieter Lang_, Oct 18 2010