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A139757 a(n) = (n+1)*(2n+1)^2. 4

%I #26 Sep 08 2022 08:45:33

%S 1,18,75,196,405,726,1183,1800,2601,3610,4851,6348,8125,10206,12615,

%T 15376,18513,22050,26011,30420,35301,40678,46575,53016,60025,67626,

%U 75843,84700,94221,104430,115351,127008,139425,152626,166635,181476

%N a(n) = (n+1)*(2n+1)^2.

%C 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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/AntiprismGraph.html">Antiprism Graph</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CocktailPartyGraph.html">Cocktail Party Graph</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DetourIndex.html">Detour Index</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).

%F a(n) = (2n+1) * A000217(2n+1).

%F 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 a(n) = Sum_{i=1..2n-1} (n^2 + n*i - i). - _Wesley Ivan Hurt_, Sep 29 2014

%F From _Amiram Eldar_, Jun 28 2020: (Start)

%F Sum_{n>=0} 1/a(n) = Pi^2/4 - log(4).

%F Sum_{n>=0} (-1)^n/a(n) = 2*G + log(2) - Pi/2, where G is the Catalan constant (A006752). (End)

%p A139757:=n->(n+1)*(2*n+1)^2: seq(A139757(n), n=0..30); # _Wesley Ivan Hurt_, Sep 29 2014

%t Table[(n + 1) (2 n + 1)^2, {n, 0, 30}] (* _Wesley Ivan Hurt_, Sep 29 2014 *)

%t LinearRecurrence[{4, -6, 4, -1}, {18, 75, 196, 405}, {0, 20}] (* _Eric W. Weisstein_, Dec 20 2017 *)

%t CoefficientList[Series[(1 + 14 x + 9 x^2)/(-1 + x)^4, {x, 0, 20}], x] (* _Eric W. Weisstein_, Dec 20 2017 *)

%o (Magma) [(n+1)*(2*n+1)^2 : n in [0..30]]; // _Wesley Ivan Hurt_, Sep 29 2014

%o (PARI) a(n) = (n+1)*(2*n+1)^2; \\ _Altug Alkan_, Dec 20 2017

%Y Cf. A000217, A006254.

%K easy,nonn

%O 0,2

%A _Odimar Fabeny_, May 19 2008

%E Missing a(0) inserted by _R. J. Mathar_, Sep 19 2010

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Last modified March 28 12:26 EDT 2024. Contains 371254 sequences. (Running on oeis4.)