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A267322 Expansion of (1 + x + x^2 + x^4 + 2*x^5)/(1 - x^3)^3.

%I A267322 #33 Feb 16 2025 08:33:29
%S A267322 1,1,1,3,4,5,6,9,12,10,16,22,15,25,35,21,36,51,28,49,70,36,64,92,45,
%T A267322 81,117,55,100,145,66,121,176,78,144,210,91,169,247,105,196,287,120,
%U A267322 225,330,136,256,376,153,289,425,171,324,477,190,361,532,210,400,590,231,441,651
%N A267322 Expansion of (1 + x + x^2 + x^4 + 2*x^5)/(1 - x^3)^3.
%C A267322 Triangular numbers alternating with squares and pentagonal numbers.
%H A267322 Ilya Gutkovskiy, <a href="/A267322/a267322.pdf">Extended illustration of initial terms</a>
%H A267322 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TriangularNumber.html">Triangular Number</a>
%H A267322 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SquareNumber.html">Square Number</a>
%H A267322 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PentagonalNumber.html">Pentagonal Number</a>
%H A267322 <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,3,0,0,-3,0,0,1)
%F A267322 G.f.: (1 + x + x^2 + x^4 + 2*x^5)/(1 - x^3)^3.
%F A267322 a(n) = 3*a(n-3) - 3*a(n-6) + a(n-9).
%F A267322 a(3k) = A000217(k+1), a(3k+1) = A000290(k+1), a(3k+2) = A000326(k+1).
%F A267322 Sum_{n>=0} 1/a(n) = 2 - Pi/sqrt(3) + Pi^2/6 + 3*log(3) = 5.1269715686...
%F A267322 a(n) = (floor(n/3) + 1)*((n+1)*floor(n/3) - 3*floor(n/3)^2 + 2)/2. - _Bruno Berselli_, Apr 08 2016
%e A267322 Illustration of initial terms:
%e A267322 ==========================================================
%e A267322 n:    0   1   2     3     4     5       6       7       8
%e A267322 ----------------------------------------------------------
%e A267322                                                         o
%e A267322                                                       o o
%e A267322                                 o       o   o o o   o o o
%e A267322                     o   o o   o o     o o   o o o   o o o
%e A267322       o   o   o   o o   o o   o o   o o o   o o o   o o o
%e A267322 ==========================================================
%e A267322       1   1   1     3     4     5       6       9      12
%e A267322 ----------------------------------------------------------
%t A267322 LinearRecurrence[{0, 0, 3, 0, 0, -3, 0, 0, 1}, {1, 1, 1, 3, 4, 5, 6, 9, 12}, 70]
%t A267322 Table[(Floor[n/3] + 1) ((n + 1) Floor[n/3] - 3 Floor[n/3]^2 + 2)/2, {n, 0, 70}] (* _Bruno Berselli_, Apr 08 2016 *)
%t A267322 CoefficientList[Series[(1+x+x^2+x^4+2x^5)/(1-x^3)^3,{x,0,70}],x] (* _Harvey P. Dale_, Dec 31 2023 *)
%o A267322 (PARI) x='x+O('x^99); Vec((1+x+x^2+x^4+2*x^5)/(1-x^3)^3) \\ _Altug Alkan_, Apr 07 2016
%Y A267322 Cf. A000217, A000290, A000326, A123596, A124093, A271391.
%K A267322 nonn,easy
%O A267322 0,4
%A A267322 _Ilya Gutkovskiy_, Apr 07 2016