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 A178947 #21 Feb 09 2025 09:44:29
%S A178947 1,4,17,38,81,138,229,340,497,680,921,1194,1537,1918,2381,2888,3489,
%T A178947 4140,4897,5710,6641,7634,8757,9948,11281,12688,14249,15890,17697,
%U A178947 19590,21661,23824,26177,28628,31281,34038,37009,40090,43397,46820,50481,54264,58297
%N A178947 Expansion of x*(1+2*x+8*x^2+4*x^3+3*x^4) / ( (1+x)^2*(x-1)^4 ).
%C A178947 Let S(x) be the generating function of A016777; then the generating function of this sequence is x/2 * (S(x)^2 + S(x^2)): the sequence is obtained by adding half of the convolution square, A100175, and the aerated A016777.
%H A178947 Colin Barker, <a href="/A178947/b178947.txt">Table of n, a(n) for n = 1..1000</a>
%H A178947 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,-4,1,2,-1).
%F A178947 a(2n) = A100175(2n)/2.
%F A178947 a(2n+1) = (A100175(2n+1)+A016777(n))/2.
%F A178947 From _Colin Barker_, Aug 02 2016: (Start)
%F A178947 a(n) = (-1+(-1)^n+(7-3*(-1)^n)*n-6*n^2+6*n^3)/8.
%F A178947 a(n) = (3*n^3-3*n^2+2*n)/4 for n even.
%F A178947 a(n) = (3*n^3-3*n^2+5*n-1)/4 for n odd.
%F A178947 a(n) = 2*a(n-1)+a(n-2)-4*a(n-3)+a(n-4)+2*a(n-5)-a(n-6) for n>6.
%F A178947 (End)
%e A178947 (1/2) * ((1, 8, 30, 76, 155, 276,...) + (1, 0, 4, 0, 7, 0, 10,...)) = (1, 4, 17, 38, 81, 138, 229,...).
%t A178947 LinearRecurrence[{2,1,-4,1,2,-1},{1,4,17,38,81,138},50] (* _Harvey P. Dale_, Jun 12 2018 *)
%Y A178947 Cf. A100175, A016777.
%K A178947 nonn,easy
%O A178947 1,2
%A A178947 _Gary W. Adamson_, Dec 30 2010