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A269457
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a(n) = 5*(n + 1)*(n + 4)/2.
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1
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10, 25, 45, 70, 100, 135, 175, 220, 270, 325, 385, 450, 520, 595, 675, 760, 850, 945, 1045, 1150, 1260, 1375, 1495, 1620, 1750, 1885, 2025, 2170, 2320, 2475, 2635, 2800, 2970, 3145, 3325, 3510, 3700, 3895, 4095, 4300, 4510, 4725, 4945, 5170, 5400, 5635
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OFFSET
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0,1
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COMMENTS
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More generally, the ordinary generating function for the sequences of the form k*(n + 1)*(n - 1 + k)/2 is (k*(k - 1)/2 + (k*(3 - k)/2)*x)/(1 - x)^3(see links section).
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LINKS
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FORMULA
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G.f.: 5*(2 - x)/(1 -x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
a(n) = Sum_{k=0..n} 5*(k + 2) = Sum_{k=0..n} A008587(k + 2).
Sum_{n>=0} 1/a(n) = 11/45 = 0.24444444444... = A040002.
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EXAMPLE
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a(0) = 0 + 1 + 2 + 3 + 4 = 10;
a(1) = 0 + 1 + 2 + 3 + 4 + 1 + 2 + 3 + 4 + 5 = 25;
a(2) = 0 + 1 + 2 + 3 + 4 + 1 + 2 + 3 + 4 + 5 + 2 + 3 + 4 + 5 + 6 = 45, etc.
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MATHEMATICA
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Table[5 (n + 1) ((n + 4)/2), {n, 0, 45}]
Table[Sum[5 (k + 2), {k, 0, n}], {n, 0, 45}]
LinearRecurrence[{3, -3, 1}, {10, 25, 45}, 46]
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PROG
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(PARI) Vec(5*(2-x)/(1-x)^3 + O(x^100)) \\ Altug Alkan, Mar 04 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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