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%I #36 Jul 06 2024 10:30:47
%S 0,8,19,33,50,70,93,119,148,180,215,253,294,338,385,435,488,544,603,
%T 665,730,798,869,943,1020,1100,1183,1269,1358,1450,1545,1643,1744,
%U 1848,1955,2065,2178,2294,2413,2535,2660,2788,2919,3053
%N a(n) = n*(3*n + 13)/2.
%H G. C. Greubel, <a href="/A140672/b140672.txt">Table of n, a(n) for n = 0..5000</a>
%H 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 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3, -3, 1).
%F a(n) = (3*n^2 + 13*n)/2.
%F a(n) = 3*n + a(n-1) + 5 for n>0, a(0)=0. - _Vincenzo Librandi_, Aug 03 2010
%F a(0)=0, a(1)=8, a(2)=19; for n>2, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - _Harvey P. Dale_, Dec 16 2011
%F G.f.: x*(8 - 5*x)/(1 - x)^3. - _Arkadiusz Wesolowski_, Dec 24 2011
%F E.g.f.: (1/2)*(3*x^2 +16*x)*exp(x). - _G. C. Greubel_, Jul 17 2017
%t Table[n (3 n + 13)/2, {n, 0, 50}] (* or *) LinearRecurrence[{3, -3, 1}, {0, 8, 19}, 50] (* _Harvey P. Dale_, Dec 16 2011 *)
%o (PARI) a(n)=n*(3*n+13)/2 \\ _Charles R Greathouse IV_, Sep 24 2015
%o (Magma) [(3*n^2 + 13*n)/2 : n in [0..80]]; // _Wesley Ivan Hurt_, Dec 27 2023
%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.
%K easy,nonn
%O 0,2
%A _Omar E. Pol_, May 22 2008