A268351 - OEIS

%I #40 Dec 28 2024 10:18:50

%S 0,12,51,117,210,330,477,651,852,1080,1335,1617,1926,2262,2625,3015,

%T 3432,3876,4347,4845,5370,5922,6501,7107,7740,8400,9087,9801,10542,

%U 11310,12105,12927,13776,14652,15555,16485,17442,18426,19437,20475,21540,22632,23751,24897,26070,27270

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

%C First trisection of pentagonal numbers (A000326).

%C 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 G. C. Greubel, <a href="/A268351/b268351.txt">Table of n, a(n) for n = 0..1000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PentagonalNumber.html">Pentagonal Number</a>.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

%F G.f.: 3*x*(4 + 5*x)/(1 - x)^3.

%F a(n) = binomial(9*n,2)/3.

%F a(n) = A000326(3*n) = 3*A022266(n).

%F a(n) = A211538(6*n+2).

%F a(n) = A001318(6*n-1), with A001318(-1)=0.

%F a(n) = A188623(9*n-2), with A188623(-2)=0.

%F 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 From _Elmo R. Oliveira_, Dec 27 2024: (Start)

%F E.g.f.: 3*exp(x)*x*(8 + 9*x)/2.

%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2.

%F a(n) = A022284(n) - n. (End)

%t Table[3 n (9 n - 1)/2, {n, 0, 45}]

%t Table[Binomial[9 n, 2]/3, {n, 0, 45}]

%t LinearRecurrence[{3, -3, 1}, {0, 12, 51}, 45]

%o (Magma) [3*n*(9*n-1)/2: n in [0..50]]; // _Vincenzo Librandi_, Feb 04 2016

%o (PARI) a(n)=3*n*(9*n-1)/2 \\ _Charles R Greathouse IV_, Jul 26 2016

%Y Cf. A000326, A001318, A016766, A022266, A022284, A081266, A188623, A211538.

%K nonn,easy

%O 0,2

%A _Ilya Gutkovskiy_, Feb 02 2016

%E Edited by _Bruno Berselli_, Feb 03 2016