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 A153977 #39 Aug 28 2025 22:02:29
%S A153977 2,9,27,65,135,252,434,702,1080,1595,2277,3159,4277,5670,7380,9452,
%T A153977 11934,14877,18335,22365,27027,32384,38502,45450,53300,62127,72009,
%U A153977 83027,95265,108810,123752,140184,158202,177905,199395,222777,248159,275652,305370,337430
%N A153977 One-fourth of partial sums of A153976.
%H A153977 G. C. Greubel, <a href="/A153977/b153977.txt">Table of n, a(n) for n = 1..1000</a>
%H A153977 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F A153977 a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n>5, a(1)=2, a(2)=9, a(3)=27, a(4)=65, a(5)=135. - _Harvey P. Dale_, Aug 02 2011
%F A153977 a(n) = (A000217(n-1)^2 + A000217(n+1)^2 - 1)/4. - _Richard R. Forberg_, Dec 25 2013
%F A153977 Recurrence: (n-1)*(n^2 - n + 6)*a(n) = (n+1)*(n^2 + n + 6)*a(n-1). - _Vaclav Kotesovec_, Dec 26 2013
%F A153977 a(n) = A000217(A000217(n)) + A000217(n). - _Bruno Berselli_, May 28 2015
%F A153977 a(n) = (A000217(n)^2 + 3*A000217(n))/2 where A000217(n) is the n-th triangular number. - _Frederic Isenmann_, Feb 04 2017
%F A153977 Sum_{n>=1} 1/a(n) = 14/9 - 4*tanh(sqrt(23)*Pi/2)*Pi/(3*sqrt(23)). - _Amiram Eldar_, Aug 23 2022
%F A153977 From _Elmo R. Oliveira_, Aug 28 2025: (Start)
%F A153977 G.f.: x*(2 - x + 2*x^2)/(1-x)^5.
%F A153977 E.g.f.: x*(2 + x)^2*(4 + x)*exp(x)/8. (End)
%p A153977 A153977:=n->(1/4)*sum(i^3 + (i+2)^3, i=0..n): seq(A153977(n), n=0..50); # _Wesley Ivan Hurt_, Feb 04 2017
%t A153977 a[n_]:=n^3;lst={};s=0;Do[s+=(a[n]+a[n+2]);AppendTo[lst,s/4],{n,0,6!}];lst
%t A153977 Accumulate[Array[#^3+(#+2)^3&,40,0]]/4 (* or *) LinearRecurrence[ {5,-10,10,-5,1},{2,9,27,65,135},40] (* _Harvey P. Dale_, Aug 02 2011 *)
%o A153977 (PARI) a(n)=(n^4 + 2*n^3 + 7*n^2 + 6*n)/8 \\ _Charles R Greathouse IV_, Feb 06 2017
%Y A153977 Cf. A000217, A000537, A000578, A003215, A005898, A006007, A027602, A153976.
%K A153977 nonn,easy,changed
%O A153977 1,1
%A A153977 _Vladimir Joseph Stephan Orlovsky_, Jan 03 2009