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 A106514 #13 Sep 08 2022 08:45:18
%S A106514 1,3,9,25,67,175,449,1137,2851,7095,17553,43225,106051,259423,633089,
%T A106514 1541985,3749827,9107175,22095249,53559817,129739171,314086735,
%U A106514 760009793,1838300625,4444999651,10745077143,25968708369,62749602745
%N A106514 Expansion of (1-x)/((1-2*x)*(1-2*x-x^2)).
%C A106514 Convolution of A000079 and A001333.
%C A106514 Row sums of A106513.
%C A106514 Equals eigensequence of a triangle with the Pell series as the left border and the rest 1's. - _Gary W. Adamson_, Jul 24 2010
%H A106514 G. C. Greubel, <a href="/A106514/b106514.txt">Table of n, a(n) for n = 0..1000</a>
%H A106514 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (4,-3,-2).
%F A106514 a(n) = Sum_{k=0..n} 2^(n-k-1)*((1-sqrt(2))^k + (1+sqrt(2))^k).
%F A106514 a(n) = Sum_{k=0..n} Sum_{j=0..floor((n+1)/2)} binomial(n+1, 2*j+k+1)*2^j.
%F A106514 a(n) = Pell(n+3) - Pell(n+2) - 2^(n+1) = A001333(n+2) - 2^(n+1). - _Ralf Stephan_, Jun 02 2007
%F A106514 From _G. C. Greubel_, Aug 05 2021: (Start)
%F A106514 a(n) = A001333(n+2) - 2^(n+1) = (A002203(n+2) - 2^(n+2))/2.
%F A106514 Sum_{j=0..n} a(j) = A094706(n+1). (End)
%t A106514 Table[(LucasL[n+2, 2] - 2^(n+2))/2, {n, 0, 40}] (* _G. C. Greubel_, Aug 05 2021 *)
%o A106514 (Magma) I:=[1,3,9]; [n le 3 select I[n] else 4*Self(n-1) -3*Self(n-2) -2*Self(n-3): n in [1..41]]; // _G. C. Greubel_, Aug 05 2021
%o A106514 (Sage) [(lucas_number2(n+2, 2, -1) - 2^(n+2))/2 for n in (0..40)] # _G. C. Greubel_, Aug 05 2021
%Y A106514 Cf. A000079, A001333, A002203, A094706, A106513.
%K A106514 easy,nonn
%O A106514 0,2
%A A106514 _Paul Barry_, May 05 2005