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 A139757 #27 Feb 16 2025 08:33:08
%S A139757 1,18,75,196,405,726,1183,1800,2601,3610,4851,6348,8125,10206,12615,
%T A139757 15376,18513,22050,26011,30420,35301,40678,46575,53016,60025,67626,
%U A139757 75843,84700,94221,104430,115351,127008,139425,152626,166635,181476
%N A139757 a(n) = (n+1)*(2n+1)^2.
%C A139757 Also the detour index of the (n+1)-antiprism graph and (n+1)-cocktail party graphs for n>=2. - _Eric W. Weisstein_, Jul 15 2011 and Dec 20 2017
%H A139757 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/AntiprismGraph.html">Antiprism Graph</a>
%H A139757 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CocktailPartyGraph.html">Cocktail Party Graph</a>
%H A139757 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DetourIndex.html">Detour Index</a>
%H A139757 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A139757 a(n) = (2n+1) * A000217(2n+1).
%F A139757 a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4); G.f.: (1+14*x+9*x^2)/(x-1)^4. - _R. J. Mathar_, Sep 19 2010
%F A139757 a(n) = Sum_{i=1..2n-1} (n^2 + n*i - i). - _Wesley Ivan Hurt_, Sep 29 2014
%F A139757 From _Amiram Eldar_, Jun 28 2020: (Start)
%F A139757 Sum_{n>=0} 1/a(n) = Pi^2/4 - log(4).
%F A139757 Sum_{n>=0} (-1)^n/a(n) = 2*G + log(2) - Pi/2, where G is the Catalan constant (A006752). (End)
%p A139757 A139757:=n->(n+1)*(2*n+1)^2: seq(A139757(n), n=0..30); # _Wesley Ivan Hurt_, Sep 29 2014
%t A139757 Table[(n + 1) (2 n + 1)^2, {n, 0, 30}] (* _Wesley Ivan Hurt_, Sep 29 2014 *)
%t A139757 LinearRecurrence[{4, -6, 4, -1}, {18, 75, 196, 405}, {0, 20}] (* _Eric W. Weisstein_, Dec 20 2017 *)
%t A139757 CoefficientList[Series[(1 + 14 x + 9 x^2)/(-1 + x)^4, {x, 0, 20}], x] (* _Eric W. Weisstein_, Dec 20 2017 *)
%o A139757 (Magma) [(n+1)*(2*n+1)^2 : n in [0..30]]; // _Wesley Ivan Hurt_, Sep 29 2014
%o A139757 (PARI) a(n) = (n+1)*(2*n+1)^2; \\ _Altug Alkan_, Dec 20 2017
%Y A139757 Cf. A000217, A006254.
%K A139757 easy,nonn
%O A139757 0,2
%A A139757 _Odimar Fabeny_, May 19 2008
%E A139757 Missing a(0) inserted by _R. J. Mathar_, Sep 19 2010