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 A114220 #31 Jan 21 2025 09:18:16
%S A114220 1,0,1,1,2,3,4,6,7,10,11,15,16,21,22,28,29,36,37,45,46,55,56,66,67,78,
%T A114220 79,91,92,105,106,120,121,136,137,153,154,171,172,190,191,210,211,231,
%U A114220 232,253,254,276,277,300,301,325,326,351,352,378,379,406,407,435,436
%N A114220 a(n) = Sum_{k=0..floor(n/2)} (k - (k-1)*0^(n-2*k)).
%C A114220 Diagonal sums of A114219.
%C A114220 It appears that the sequence terms from a(4) = 2 onwards are the exponents in the expansion of Sum_{n >= 0} q^(3*n-3) * Product_{k >= n} 1 - q^(2*k) = 1 - q^2 + q^3 - q^4 + q^6 - q^7 + q^10 - q^11 + - .... - _Peter Bala_, Jan 17 2025
%H A114220 G. C. Greubel, <a href="/A114220/b114220.txt">Table of n, a(n) for n = 0..1000</a>
%H A114220 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1)
%F A114220 G.f.: (1-x-x^2+2x^3)/((1-x)*(1-x^2)^2).
%F A114220 a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5).
%F A114220 a(n) = (2*n^2-2*n+7 + (9-2*n)*(-1)^n)/16.
%F A114220 a(n) = A055802(n+1), n > 1. - _R. J. Mathar_, Aug 11 2008
%F A114220 E.g.f.: (1/16)*((9 + 2*x)*exp(-x) + (7 + 2*x^2)*exp(x)). - _G. C. Greubel_, Oct 21 2024
%t A114220 CoefficientList[Series[(1-x-x^2+2x^3)/((1-x)(1-x^2)^2), {x,0,80}],x] (* _Harvey P. Dale_, Mar 24 2011 *)
%o A114220 (Magma) [(2*n^2-2*n+7 + (9-2*n)*(-1)^n)/16: n in [0..80]]; // _G. C. Greubel_, Oct 21 2024
%o A114220 (SageMath)
%o A114220 def A114220(n): return (2*n^2-2*n+7 + (9-2*n)*(-1)^n)//16
%o A114220 [A114220(n) for n in range(81)] # _G. C. Greubel_, Oct 21 2024
%Y A114220 Cf. A055802, A114219.
%Y A114220 Column k=2 of A309049 (for n>0).
%K A114220 easy,nonn
%O A114220 0,5
%A A114220 _Paul Barry_, Nov 18 2005