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%I #47 Sep 08 2022 08:44:51
%S 0,4,32,108,256,500,864,1372,2048,2916,4000,5324,6912,8788,10976,
%T 13500,16384,19652,23328,27436,32000,37044,42592,48668,55296,62500,
%U 70304,78732,87808,97556,108000,119164,131072,143748,157216,171500,186624,202612,219488
%N a(n) = 4*n^3.
%C 2*a(n) = (2*n)^3 is a perfect cube.
%C Number of edges of the product of two complete bipartite graphs, each of order 2n, K_n,n x K_n,n - _Roberto E. Martinez II_, Jan 07 2002
%C This sequence is related to A049451 by a(n) = n*A049451(n) + sum( A049451(i), i=0..n-1 ) for n>0. - _Bruno Berselli_, Dec 19 2013
%C For n>=3, also the detour index of the n-gear graph. - _Eric W. Weisstein_, Dec 20 2017
%H Vincenzo Librandi, <a href="/A033430/b033430.txt">Table of n, a(n) for n = 0..750</a>
%H F. Ellermann, <a href="/A001792/a001792.txt">Illustration of binomial transforms</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DetourIndex.html">Detour Index</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GearGraph.html">Gear Graph</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F G.f. 4*x*(1+4*x+x^2)/ (x-1)^4. - _R. J. Mathar_, Feb 01 2011
%F From _Ilya Gutkovskiy_, May 25 2016: (Start)
%F E.g.f.: 4*x*(1 + 3*x + x^2)*exp(x).
%F Sum_{n>=1} 1/a(n) = zeta(3)/4. (End)
%F Product_{n>=1} a(n)/A280089(n) = Pi. - _Daniel Suteu_, Dec 26 2016
%F From _Bruce J. Nicholson_, Dec 07 2019: (Start)
%F a(n) = 24*A000292(n-1) + 4*n.
%F a(n) = 2*A007588(n) + 2*n. (End)
%t 4 Range[0, 40]^3 (* _Harvey P. Dale_, Sep 07 2016 *)
%t LinearRecurrence[{4, -6, 4, -1}, {0, 4, 32, 108}, 40] (* _Harvey P. Dale_, Sep 07 2016 *)
%t Table[4 n^3, {n, 0, 20}] (* _Eric W. Weisstein_, Dec 20 2017 *)
%t CoefficientList[Series[(4 x (1 + 4 x + x^2))/(-1 + x)^4, {x, 0, 20}], x] (* _Eric W. Weisstein_, Dec 20 2017 *)
%o (Magma) [4*n^3: n in [0..40]]; // _Vincenzo Librandi_, Apr 27 2011
%o (PARI) a(n)=4*n^3 \\ _Charles R Greathouse IV_, Oct 07 2015
%Y Cf. A000578, A049451.
%Y Cf. A000292, A007588.
%K nonn,easy
%O 0,2
%A _Jeff Burch_
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