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a(n) = n*(3*n + 19)/2.
15

%I #33 Jul 06 2024 10:30:56

%S 0,11,25,42,62,85,111,140,172,207,245,286,330,377,427,480,536,595,657,

%T 722,790,861,935,1012,1092,1175,1261,1350,1442,1537,1635,1736,1840,

%U 1947,2057,2170,2286,2405,2527,2652,2780,2911,3045,3182

%N a(n) = n*(3*n + 19)/2.

%H G. C. Greubel, <a href="/A140675/b140675.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 + 19*n)/2.

%F a(n) = 3*n + a(n-1) + 8 for n>0, a(0)=0. - _Vincenzo Librandi_, Aug 03 2010

%F G.f.: x*(11 - 8*x)/(1 - x)^3. - _Arkadiusz Wesolowski_, Dec 24 2011

%F E.g.f.: (1/2)*(3*x^2 + 22*x)*exp(x). - _G. C. Greubel_, Jul 17 2017

%t Table[(n(3n+19))/2,{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{0,11,25},50] (* _Harvey P. Dale_, Apr 26 2018 *)

%o (PARI) a(n)=n*(3*n+19)/2 \\ _Charles R Greathouse IV_, Jun 17 2017

%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 nonn,easy

%O 0,2

%A _Omar E. Pol_, May 22 2008