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a(n) = n*(3*n + 17)/2.
13

%I #33 Jul 06 2024 10:30:53

%S 0,10,23,39,58,80,105,133,164,198,235,275,318,364,413,465,520,578,639,

%T 703,770,840,913,989,1068,1150,1235,1323,1414,1508,1605,1705,1808,

%U 1914,2023,2135,2250,2368,2489,2613,2740,2870,3003,3139

%N a(n) = n*(3*n + 17)/2.

%H G. C. Greubel, <a href="/A140674/b140674.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 + 17*n)/2.

%F a(n) = 7*n + 3*A000217(n). - _Reinhard Zumkeller_, May 28 2008

%F a(n) = 3*n + a(n-1) + 7 (with a(0)=0). - _Vincenzo Librandi_, Aug 03 2010

%F G.f.: x*(10 - 7*x)/(1 - x)^3. - _Arkadiusz Wesolowski_, Dec 24 2011

%F E.g.f.: (1/2)*(3*x^2 + 20*x)*exp(x). - _G. C. Greubel_, Jul 17 2017

%t Table[(3*n^2 + 17*n)/2, {n, 0, 50}] (* _G. C. Greubel_, Jul 17 2017 *)

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

%o (Magma) [n*(3*n + 17)/2: n in [0..70]]; // _Wesley Ivan Hurt_, Apr 21 2021

%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