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A281381
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a(n) = n*(n + 1)*(4*n + 5)/2.
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2
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0, 9, 39, 102, 210, 375, 609, 924, 1332, 1845, 2475, 3234, 4134, 5187, 6405, 7800, 9384, 11169, 13167, 15390, 17850, 20559, 23529, 26772, 30300, 34125, 38259, 42714, 47502, 52635, 58125, 63984, 70224, 76857, 83895, 91350, 99234, 107559, 116337, 125580, 135300, 145509, 156219, 167442, 179190, 191475
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OFFSET
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0,2
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COMMENTS
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Shares its digital root, zero together with period 9: repeat [3, 3, 3, 6, 6, 6, 9, 9, 9] with A027480.
Final digits cycle a length period 20: repeat [0, 9, 9, 2, 0, 5, 9, 4, 2, 5, 5, 4, 4, 7, 5, 0, 4, 9, 7, 0].
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LINKS
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FORMULA
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a(n) = 2*n^3 + 9*n^2/2 + 5*n/2.
G.f.: 3*x*(3 + x) / (1 - x)^4.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>3. (End)
E.g.f.: exp(x)*x*(18 + 21*x + 4*x^2)/2.
Sum_{n>0} 1/a(n) = 2*(20*log(8) + 10*Pi - 71)/25 = 0.1603805895595720759728288896228498341201... . (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = 4*sqrt(2)*Pi/5 + 4*(3+sqrt(2))*log(2)/5 - 8*sqrt(2)*log(2-sqrt(2))/5 - 178/25. - Amiram Eldar, Sep 22 2022
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MATHEMATICA
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Table[n (n + 1) (4 n + 5)/2, {n, 0, 45}] (* or *)
CoefficientList[Series[3 x (3 + x)/(1 - x)^4, {x, 0, 45}], x] (* Michael De Vlieger, Jan 21 2017 *)
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PROG
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(PARI) concat(0, Vec(3*x*(3 + x) / (1 - x)^4 + O(x^50))) \\ Colin Barker, Jan 21 2017
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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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