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A267522
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a(n) = 4*(n + 1)*(n + 2)*(4*n + 3)/3.
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0
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8, 56, 176, 400, 760, 1288, 2016, 2976, 4200, 5720, 7568, 9776, 12376, 15400, 18880, 22848, 27336, 32376, 38000, 44240, 51128, 58696, 66976, 76000, 85800, 96408, 107856, 120176, 133400, 147560, 162688, 178816, 195976, 214200, 233520, 253968, 275576, 298376, 322400
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OFFSET
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0,1
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COMMENTS
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LINKS
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FORMULA
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G.f.: 8*(1 + 3*x)/(1 - x)^4.
E.g.f.: (4/3)*exp(x)*(6 + 36*x + 27*x^2 + 4*x^3).
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
Sum_{n>=0} 1/a(n) = -3*(2*Pi - 12*log(2) + 1)/20 = 0.15518712893...
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EXAMPLE
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a(0) = (0 + 2)*(1 + 3) = 8;
a(1) = (0 + 2)*(1 + 3) + (2 + 4)*(3 + 5) = 56;
a(2) = (0 + 2)*(1 + 3) + (2 + 4)*(3 + 5) + (4 + 6)*(5 + 7) = 176;
a(3) = (0 + 2)*(1 + 3) + (2 + 4)*(3 + 5) + (4 + 6)*(5 + 7) + (6 + 8)*(7 + 9) = 400, etc
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MATHEMATICA
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Table[(4 (n + 1)) (n + 2) ((4 n + 3)/3), {n, 0, 38}]
LinearRecurrence[{4, -6, 4, -1}, {8, 56, 176, 400}, 39]
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PROG
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(PARI) a(n) = 4*(n + 1)*(n + 2)*(4*n + 3)/3; \\ Michel Marcus, Apr 10 2016
(PARI) x='x+O('x^99); Vec(8*(1+3*x)/(1-x)^4) \\ Altug Alkan, Apr 10 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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