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%I #73 Feb 13 2024 16:55:21
%S 0,5,13,24,38,55,75,98,124,153,185,220,258,299,343,390,440,493,549,
%T 608,670,735,803,874,948,1025,1105,1188,1274,1363,1455,1550,1648,1749,
%U 1853,1960,2070,2183,2299,2418,2540,2665,2793,2924
%N a(n) = n*(3*n + 7)/2.
%C This sequence is mentioned in the Guo-Niu Han's paper, chapter 6: Dictionary of the standard puzzle sequences, p. 19 (see link). - _Omar E. Pol_, Oct 28 2011
%C Number of cards needed to build an n-tier house of cards with a flat, one-card-wide roof. - _Tyler Busby_, Dec 28 2022
%H G. C. Greubel, <a href="/A140090/b140090.txt">Table of n, a(n) for n = 0..5000</a>
%H Guo-Niu Han, <a href="/A196265/a196265.pdf">Enumeration of Standard Puzzles</a>, 2011. [Cached copy]
%H Guo-Niu Han, <a href="https://arxiv.org/abs/2006.14070">Enumeration of Standard Puzzles</a>, arXiv:2006.14070 [math.CO], 2020.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F G.f.: x*(5 - 2*x)/(1 - x)^3. - _Bruno Berselli_, Feb 11 2011
%F a(n) = (3*n^2 + 7*n)/2.
%F a(n) = a(n-1) + 3*n + 2 (with a(0)=0). - _Vincenzo Librandi_, Nov 24 2010
%F E.g.f.: (1/2)*(3*x^2 + 10*x)*exp(x). - _G. C. Greubel_, Jul 17 2017
%F From _Amiram Eldar_, Feb 22 2022: (Start)
%F Sum_{n>=1} 1/a(n) = 117/98 - Pi/(7*sqrt(3)) - 3*log(3)/7.
%F Sum_{n>=1} (-1)^(n+1)/a(n) = 2*Pi/(7*sqrt(3)) + 4*log(2)/7 - 75/98. (End)
%t LinearRecurrence[{3,-3,1},{0,5,13},50] (* _Harvey P. Dale_, Jan 17 2022 *)
%o (PARI) a(n)=n*(3*n+7)/2 \\ _Charles R Greathouse IV_, Sep 24 2015
%Y The generalized pentagonal numbers b*n+3*n*(n-1)/2, for b = 1 through 12, form sequences A000326, A005449, A045943, A115067, A140090, A140091, A059845, A140672, A140673, A140674, A140675, A151542.
%Y Cf. numbers of the form n*(d*n + 10 - d)/2: A008587, A056000, A028347, A014106, A028895, A045944, A186029, A007742, A022267, A033429, A022268, A049452, A186030, A135703, A152734, A139273.
%K easy,nonn
%O 0,2
%A _Omar E. Pol_, May 22 2008
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