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A267322
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Expansion of (1 + x + x^2 + x^4 + 2*x^5)/(1 - x^3)^3.
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1
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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, 81, 117, 55, 100, 145, 66, 121, 176, 78, 144, 210, 91, 169, 247, 105, 196, 287, 120, 225, 330, 136, 256, 376, 153, 289, 425, 171, 324, 477, 190, 361, 532, 210, 400, 590, 231, 441, 651
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OFFSET
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0,4
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COMMENTS
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Triangular numbers alternating with squares and pentagonal numbers.
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LINKS
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FORMULA
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G.f.: (1 + x + x^2 + x^4 + 2*x^5)/(1 - x^3)^3.
a(n) = 3*a(n-3) - 3*a(n-6) + a(n-9).
Sum_{n>=0} 1/a(n) = 2 - Pi/sqrt(3) + Pi^2/6 + 3*log(3) = 5.1269715686...
a(n) = (floor(n/3) + 1)*((n+1)*floor(n/3) - 3*floor(n/3)^2 + 2)/2. - Bruno Berselli, Apr 08 2016
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EXAMPLE
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Illustration of initial terms:
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n: 0 1 2 3 4 5 6 7 8
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o
o o
o o o o o o o o
o o o o o o o o o o o o o
o o o o o o o o o o o o o o o o o o
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1 1 1 3 4 5 6 9 12
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MATHEMATICA
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LinearRecurrence[{0, 0, 3, 0, 0, -3, 0, 0, 1}, {1, 1, 1, 3, 4, 5, 6, 9, 12}, 70]
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 *)
CoefficientList[Series[(1+x+x^2+x^4+2x^5)/(1-x^3)^3, {x, 0, 70}], x] (* Harvey P. Dale, Dec 31 2023 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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