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

%I

%S 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 81,117,55,100,145,66,121,176,78,144,210,91,169,247,105,196,287,120,

%U 225,330,136,256,376,153,289,425,171,324,477,190,361,532,210,400,590,231,441,651

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

%C Triangular numbers alternating with squares and pentagonal numbers.

%H Ilya Gutkovskiy, <a href="/A267322/a267322.pdf">Extended illustration of initial terms</a>

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

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SquareNumber.html">Square Number</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_09">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,3,0,0,-3,0,0,1)

%F G.f.: (1 + x + x^2 + x^4 + 2*x^5)/(1 - x^3)^3.

%F a(n) = 3*a(n-3) - 3*a(n-6) + a(n-9).

%F a(3k) = A000217(k+1), a(3k+1) = A000290(k+1), a(3k+2) = A000326(k+1).

%F Sum_{n>=0} 1/a(n) = 2 - Pi/sqrt(3) + Pi^2/6 + 3*log(3) = 5.1269715686...

%F a(n) = (floor(n/3) + 1)*((n+1)*floor(n/3) - 3*floor(n/3)^2 + 2)/2. - _Bruno Berselli_, Apr 08 2016

%e Illustration of initial terms:

%e ==========================================================

%e n: 0 1 2 3 4 5 6 7 8

%e ----------------------------------------------------------

%e o

%e o o

%e o o o o o o o o

%e o o o o o o o o o o o o o

%e o o o o o o o o o o o o o o o o o o

%e ==========================================================

%e 1 1 1 3 4 5 6 9 12

%e ----------------------------------------------------------

%t LinearRecurrence[{0, 0, 3, 0, 0, -3, 0, 0, 1}, {1, 1, 1, 3, 4, 5, 6, 9, 12}, 70]

%t 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 *)

%o (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 Cf. A000217, A000290, A000326, A123596, A124093, A271391.

%K nonn,easy

%O 0,4

%A _Ilya Gutkovskiy_, Apr 07 2016

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Last modified January 24 10:24 EST 2020. Contains 331193 sequences. (Running on oeis4.)