This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A172172 #23 Apr 25 2022 08:04:22
%S A172172 0,1,10,20,39,68,116,193,318,520,847,1376,2232,3617,5858,9484,15351,
%T A172172 24844,40204,65057,105270,170336,275615,445960,721584,1167553,1889146,
%U A172172 3056708,4945863,8002580,12948452,20951041,33899502,54850552,88750063,143600624,232350696
%N A172172 Sums of NW-SE diagonals of triangle A172171.
%C A172172 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 A172172 Colin Barker, <a href="/A172172/b172172.txt">Table of n, a(n) for n = 0..1000</a>
%H A172172 Wolfdieter Lang, <a href="/A172172/a172172.pdf">Notes on certain inhomogeneous three term recurrences.</a>
%H A172172 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-1).
%F A172172 a(n) = a(n-1) + a(n-2) + 9 with a(0)=0 and a(1)=1.
%F A172172 From _Wolfdieter Lang_, Oct 18 2010: (Start)
%F A172172 O.g.f.: x*(1+8*x)/((1-x)*(1-x-x^2)).
%F A172172 a(n) = 2*a(n-1) - a(n-3), a(0)=0, a(1)=1, a(2)=10 (Observation by G. Detlefs).
%F A172172 (End)
%F A172172 a(n+1) - a(n) = A022099(n). - _R. J. Mathar_, Apr 22 2013
%F A172172 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 A172172 a(n) = Fibonacci(n+3) + 7*Fibonacci(n+1) - 9. - _G. C. Greubel_, Apr 25 2022
%t A172172 CoefficientList[Series[x*(1+8*x)/((1-x)*(1-x-x^2)), {x,0,50}], x] (* _G. C. Greubel_, Jul 13 2017 *)
%o A172172 (PARI) concat(0, Vec(x*(1+8*x)/((1-x)*(1-x-x^2)) + O(x^50))) \\ _Colin Barker_, Jul 13 2017
%o A172172 (Magma) [Lucas(n+2) +6*Fibonacci(n+1) -9: n in [0..50]]; // _G. C. Greubel_, Apr 25 2022
%o A172172 (SageMath) [fibonacci(n+3) +7*fibonacci(n+1) -9 for n in (0..50)] # _G. C. Greubel_, Apr 25 2022
%Y A172172 Cf. A000032, A000045, A172171.
%K A172172 nonn,easy
%O A172172 0,3
%A A172172 _Mark Dols_, Jan 28 2010
%E A172172 Wrong offset 1 changed into 0 _Wolfdieter Lang_, Oct 18 2010