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 A140090 #83 Jun 20 2025 20:23:56
%S A140090 0,5,13,24,38,55,75,98,124,153,185,220,258,299,343,390,440,493,549,
%T A140090 608,670,735,803,874,948,1025,1105,1188,1274,1363,1455,1550,1648,1749,
%U A140090 1853,1960,2070,2183,2299,2418,2540,2665,2793,2924
%N A140090 a(n) = n*(3*n + 7)/2.
%C A140090 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 A140090 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 A140090 G. C. Greubel, <a href="/A140090/b140090.txt">Table of n, a(n) for n = 0..5000</a>
%H A140090 Guo-Niu Han, <a href="/A196265/a196265.pdf">Enumeration of Standard Puzzles</a>, 2011. [Cached copy]
%H A140090 Guo-Niu Han, <a href="https://arxiv.org/abs/2006.14070">Enumeration of Standard Puzzles</a>, arXiv:2006.14070 [math.CO], 2020.
%H A140090 Sela Fried, <a href="https://arxiv.org/abs/2406.18923">Counting r X s rectangles in nondecreasing and Smirnov words</a>, arXiv:2406.18923 [math.CO], 2024. See p. 5.
%H A140090 Julie Zhang, Noah A. Rosenberg, and Julia A. Palacios, <a href="https://arxiv.org/abs/2506.10856">The space of multifurcating ranked tree shapes: enumeration, lattice structure, and Markov chains</a>, arXiv:2506.10856 [math.PR], 2025. See p. 33.
%H A140090 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A140090 G.f.: x*(5 - 2*x)/(1 - x)^3. - _Bruno Berselli_, Feb 11 2011
%F A140090 a(n) = (3*n^2 + 7*n)/2.
%F A140090 a(n) = a(n-1) + 3*n + 2 (with a(0)=0). - _Vincenzo Librandi_, Nov 24 2010
%F A140090 E.g.f.: (1/2)*(3*x^2 + 10*x)*exp(x). - _G. C. Greubel_, Jul 17 2017
%F A140090 From _Amiram Eldar_, Feb 22 2022: (Start)
%F A140090 Sum_{n>=1} 1/a(n) = 117/98 - Pi/(7*sqrt(3)) - 3*log(3)/7.
%F A140090 Sum_{n>=1} (-1)^(n+1)/a(n) = 2*Pi/(7*sqrt(3)) + 4*log(2)/7 - 75/98. (End)
%t A140090 LinearRecurrence[{3,-3,1},{0,5,13},50] (* _Harvey P. Dale_, Jan 17 2022 *)
%o A140090 (PARI) a(n)=n*(3*n+7)/2 \\ _Charles R Greathouse IV_, Sep 24 2015
%Y A140090 The generalized pentagonal numbers b*n+3*n*(n-1)/2, for b = 1 through 12, form sequences A000326, A005449, A045943, A115067, this sequence, A140091, A059845, A140672, A140673, A140674, A140675, A151542.
%Y A140090 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 A140090 easy,nonn
%O A140090 0,2
%A A140090 _Omar E. Pol_, May 22 2008