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A015222
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Even square pyramidal numbers.
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2
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14, 30, 140, 204, 506, 650, 1240, 1496, 2470, 2870, 4324, 4900, 6930, 7714, 10416, 11440, 14910, 16206, 20540, 22140, 27434, 29370, 35720, 38024, 45526, 48230, 56980, 60116, 70210, 73810, 85344, 89440, 102510, 107134, 121836, 127020, 143450, 149226, 167480, 173880
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OFFSET
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1,1
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COMMENTS
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Square pyramidal numbers k*(k + 1)*(2*k + 1)/6 are even if and only when k is congruent to 0 or 3 mod 4. - Artur Jasinski, Oct 22 2008
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LINKS
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FORMULA
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(2*k + 1)/(k + 2)*binomial(k + 2, 5) if k congruent to 0 or 3 mod 4, and
k*(k + 1)*(2*k + 1)/6 if k congruent to 0 or 3 mod 4. (End)
G.f.: 2*x*(7+x*(8+x*(34+x*(8+7*x)))) / ((-1+x)^4*(1+x)^3). - Harvey P. Dale, May 05 2011 [adapted to the offset by Bruno Berselli, May 16 2011]
a(n) = (3 + 4*n - (-1)^n)*(2 + 4*n - (-1)^n)*(1 + 4*n - (-1)^n)/24.
a(n) = 3*a(n-2) - 3*a(n-4) + a(n-6) + 128. (End)
6*a(n) = (1+2*n)*(8*n^2+8*n-6*(-1)^n*n+3-3*(-1)^n). - R. J. Mathar, Oct 17 2012
Sum_{n>=1} 1/a(n) = 18 + 6*sqrt(2)*log(2-sqrt(2)) - 3*(sqrt(2)+5)*log(2).
Sum_{n>=1} (-1)^(n+1)/a(n) = 3*Pi*(sqrt(2)+1/2) - 18. (End)
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MATHEMATICA
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Select[ Table[ n(n+1)(2n+1)/6, {n, 100} ], EvenQ ]
Select[Rest[CoefficientList[Series[(x(x+1))/(x-1)^4, {x, 0, 80}], x]], EvenQ] (* Harvey P. Dale, May 05 2011 *)
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CROSSREFS
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KEYWORD
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nonn,easy,changed
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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