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A258582
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a(n) = n*(2*n + 1)*(4*n + 1)/3.
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3
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0, 5, 30, 91, 204, 385, 650, 1015, 1496, 2109, 2870, 3795, 4900, 6201, 7714, 9455, 11440, 13685, 16206, 19019, 22140, 25585, 29370, 33511, 38024, 42925, 48230, 53955, 60116, 66729, 73810, 81375, 89440, 98021, 107134, 116795, 127020, 137825, 149226, 161239, 173880
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OFFSET
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0,2
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COMMENTS
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First bisection of the square pyramidal numbers (A000330).
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LINKS
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FORMULA
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G.f.: x*(5 + 10*x + x^2)/(1 - x)^4.
Sum_{n>0} 1/a(n) = 3*(6 - Pi - 4*log(2)) = 0.25745587...
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>3. - Vincenzo Librandi, Nov 18 2015
a(n) = Sum_{k=1..2*n} k^2 (see the first comment). E.g.f. exp(x)*(5*x+ 20*x^2/2+16*x^3/3!). - Wolfdieter Lang, Mar 13 2017
Sum_{n>=1} (-1)^(n+1)/a(n) = 3*log(2) + 6*sqrt(2)*log(1+sqrt(2)) + 3*(sqrt(2)-1/2)*Pi - 18. - Amiram Eldar, Sep 17 2022
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MAPLE
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MATHEMATICA
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Table[(1/3) n (2 n + 1) (4 n + 1), {n, 0, 45}]
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PROG
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(PARI) vector(100, n, n--; n*(2*n+1)*(4*n+1)/3) \\ Altug Alkan, Nov 06 2015
(PARI) concat(0, Vec((5*x + 10*x^2 + x^3)/(1 - x)^4 + O(x^50))) \\ Altug Alkan, Nov 06 2015
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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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