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 A061317 #46 Jul 04 2025 05:04:20
%S A061317 0,3,18,57,132,255,438,693,1032,1467,2010,2673,3468,4407,5502,6765,
%T A061317 8208,9843,11682,13737,16020,18543,21318,24357,27672,31275,35178,
%U A061317 39393,43932,48807,54030,59613,65568,71907,78642,85785,93348,101343,109782
%N A061317 Split positive integers into extending even groups and sum: 1+2, 3+4+5+6, 7+8+9+10+11+12, 13+14+15+16+17+18+19+20, ...
%C A061317 5*a(n+1) is the sum of the products of the 10 distinct combinations of three consecutive numbers starting with n (using 1,2,3 the 10 combinations are 111 112 113 122 123 133 222 223 233 333; 1*1*1 + 1*1*2 + 1*1*3 + 1*2*2 + 1*2*3 + 1*3*3 + 2*2*2 + 2*2*3 + 2*3*3 + 3*3*3 = 90 = 5*a(2)). - _J. M. Bergot_, Mar 28 2014 [expanded by _Jon E. Schoenfield_, Feb 22 2015]
%H A061317 Harry J. Smith, <a href="/A061317/b061317.txt">Table of n, a(n) for n = 0..1000</a>
%H A061317 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A061317 a(n) = 2*n^3 + n.
%F A061317 a(n) = A000217(A002378(n)) - A000217(A002378(n-1)).
%F A061317 a(n) = 3 * A005900(n).
%F A061317 a(n) = A001477(n) * A058331(n).
%F A061317 a(n) = A000578(n) + A034262(n).
%F A061317 G.f.: 3*x*(1+x)^2/(x-1)^4.
%F A061317 a(n) = A110450(n) - A110450(n-1). - _J.S. Seneschal_, Jul 01 2025
%e A061317 1+2 = 3; 3+4+5+6 = 18; 7+8+9+10+11+12 = 57; 13+14+15+16+17+18+19+20 = 132.
%p A061317 A061317:=n->2*n^3+n; seq(A061317(n), n=0..100); # _Wesley Ivan Hurt_, Mar 20 2014
%t A061317 Table[2n^3+n,{n,0,5!}] (* _Vladimir Joseph Stephan Orlovsky_, Dec 04 2010 *)
%t A061317 LinearRecurrence[{4,-6,4,-1},{0,3,18,57},40] (* _Harvey P. Dale_, Aug 23 2015 *)
%t A061317 With[{nn=40},Total/@TakeList[Range[nn+nn^2],2Range[0,nn]]] (* Requires Mathematica version 11 or later *) (* _Harvey P. Dale_, Mar 10 2018 *)
%o A061317 (PARI) a(n) = { 2*n^3 + n } \\ _Harry J. Smith_, Jul 21 2009
%Y A061317 Cf. A005898, A006003.
%Y A061317 Cf. A000217, A002378, A110450.
%K A061317 nonn,easy
%O A061317 0,2
%A A061317 _Henry Bottomley_, Feb 13 2002