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A268351 a(n) = 3*n*(9*n - 1)/2.

%I A268351 #41 Feb 16 2025 08:33:30
%S A268351 0,12,51,117,210,330,477,651,852,1080,1335,1617,1926,2262,2625,3015,
%T A268351 3432,3876,4347,4845,5370,5922,6501,7107,7740,8400,9087,9801,10542,
%U A268351 11310,12105,12927,13776,14652,15555,16485,17442,18426,19437,20475,21540,22632,23751,24897,26070,27270
%N A268351 a(n) = 3*n*(9*n - 1)/2.
%C A268351 First trisection of pentagonal numbers (A000326).
%C A268351 More generally, the ordinary generating function for the first trisection of k-gonal numbers is 3*x*(k - 1 + (2*k - 5)*x)/(1 - x)^3.
%H A268351 G. C. Greubel, <a href="/A268351/b268351.txt">Table of n, a(n) for n = 0..1000</a>
%H A268351 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PentagonalNumber.html">Pentagonal Number</a>.
%H A268351 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A268351 G.f.: 3*x*(4 + 5*x)/(1 - x)^3.
%F A268351 a(n) = binomial(9*n,2)/3.
%F A268351 a(n) = A000326(3*n) = 3*A022266(n).
%F A268351 a(n) = A211538(6*n+2).
%F A268351 a(n) = A001318(6*n-1), with A001318(-1)=0.
%F A268351 a(n) = A188623(9*n-2), with A188623(-2)=0.
%F A268351 Sum_{n>=1} 1/a(n) = 0.132848490245209886617568... = (-Pi*cot(Pi/9) + 5*log(3) + 4*cos(Pi/9)*log(cos(Pi/18)) - 4*cos(2*Pi/9)*log(sin(Pi/9)) - 4*log(sin(2*Pi/9))*sin(Pi/18))/3. [Corrected by _Vaclav Kotesovec_, Feb 25 2016]
%F A268351 From _Elmo R. Oliveira_, Dec 27 2024: (Start)
%F A268351 E.g.f.: 3*exp(x)*x*(8 + 9*x)/2.
%F A268351 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2.
%F A268351 a(n) = A022284(n) - n. (End)
%t A268351 Table[3 n (9 n - 1)/2, {n, 0, 45}]
%t A268351 Table[Binomial[9 n, 2]/3, {n, 0, 45}]
%t A268351 LinearRecurrence[{3, -3, 1}, {0, 12, 51}, 45]
%o A268351 (Magma) [3*n*(9*n-1)/2: n in [0..50]]; // _Vincenzo Librandi_, Feb 04 2016
%o A268351 (PARI) a(n)=3*n*(9*n-1)/2 \\ _Charles R Greathouse IV_, Jul 26 2016
%Y A268351 Cf. A000326, A001318, A016766, A022266, A022284, A081266, A188623, A211538.
%K A268351 nonn,easy
%O A268351 0,2
%A A268351 _Ilya Gutkovskiy_, Feb 02 2016
%E A268351 Edited by _Bruno Berselli_, Feb 03 2016