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%I #43 Feb 18 2022 05:16:39
%S 0,7,26,57,100,155,222,301,392,495,610,737,876,1027,1190,1365,1552,
%T 1751,1962,2185,2420,2667,2926,3197,3480,3775,4082,4401,4732,5075,
%U 5430,5797,6176,6567,6970,7385,7812,8251,8702,9165,9640,10127,10626
%N Second pentagonal numbers with even index: a(n) = n*(6*n+1).
%C Number of edges in the join of the complete tripartite graph of order 3n and the cycle graph of order n, K_n,n,n * C_n - _Roberto E. Martinez II_, Jan 07 2002
%C Sequence found by reading the line (one of the diagonal axes) from 0, in the direction 0, 7, ..., in the square spiral whose vertices are the generalized pentagonal numbers A001318. - _Omar E. Pol_, Sep 08 2011
%C First bisection of A036498. - _Bruno Berselli_, Nov 25 2012
%H G. C. Greubel, <a href="/A049453/b049453.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F G.f.: x*(7+5*x)/(1-x)^3.
%F a(n) = 12*n + a(n-1) - 5 with n>0, a(0)=0. - _Vincenzo Librandi_, Aug 06 2010
%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - _G. C. Greubel_, Jun 07 2017
%F From _Amiram Eldar_, Feb 18 2022: (Start)
%F Sum_{n>=1} 1/a(n) = 6 - sqrt(3)*Pi/2 - 2*log(2) - 3*log(3)/2.
%F Sum_{n>=1} (-1)^(n+1)/a(n) = Pi + log(2) + sqrt(3)*log(2 + sqrt(3)) - 6. (End)
%p seq(binomial(6*n+1,2)/3, n=0..42); # _Zerinvary Lajos_, Jan 21 2007
%t s=0;lst={s};Do[s+=n++ +7;AppendTo[lst, s], {n, 0, 7!, 12}];lst (* _Vladimir Joseph Stephan Orlovsky_, Nov 16 2008 *)
%t Table[n*(6*n + 1), {n,0,50}] (* _G. C. Greubel_, Jun 07 2017 *)
%o (PARI) x='x+O('x^50); concat([0], Vec(x*(7+5*x)/(1-x)^3)) \\ _G. C. Greubel_, Jun 07 2017
%Y Cf. A001318, A005449, A033568, A033570, A036498, A049452, A185019, A194454.
%Y Cf. A254963 (comment).
%K nonn,easy
%O 0,2
%A Joe Keane (jgk(AT)jgk.org)
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