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A102094
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a(n) = (2*n-1)*(2*n+1)^2.
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1
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9, 75, 245, 567, 1089, 1859, 2925, 4335, 6137, 8379, 11109, 14375, 18225, 22707, 27869, 33759, 40425, 47915, 56277, 65559, 75809, 87075, 99405, 112847, 127449, 143259, 160325, 178695, 198417, 219539, 242109, 266175, 291785, 318987, 347829, 378359, 410625
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OFFSET
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1,1
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COMMENTS
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Numbers which are both the sum of 2n+1 consecutive odd integers and, after skipping one odd integer, the sum of the 2n-1 immediately higher consecutive odd integers. See A082108(n-1) for the smallest of the 2n+1 odd integers, and A054569(n+1) for the skipped number. Odd integer counterpart to A059270. - Charlie Marion, Apr 30 2020
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REFERENCES
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G. Boros and V. Moll, Irresistible Integrals: Symbolics, Analysis and Experiments in the Evaluation of Integrals, Cambridge University Press, Cambridge, 2004, p. 123.
J. Ewell, An Eulerian Method for Representing Pi^2 by Series, The Rocky Mountain Journal of Mathematics 1992 v.22, pp. 165-168.
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LINKS
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FORMULA
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Sum_{n>=1} 1/a(n) = (12 - Pi^2)/16.
Sum_{n>=1} n/a(n) = (Pi^2 - 4)/32. - Sign flipped by Bernard Schott, May 06 2020
a(1)=9, a(2)=75, a(3)=245, a(4)=567, a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
G.f.: (9 + 39*x - x^2 + x^3)/(1-x)^4. (End)
E.g.f.: 1 + (-1 + 10*x + 28*x^2 + 8*x^3)*exp(x). - G. C. Greubel, Oct 27 2019
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MAPLE
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MATHEMATICA
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Table[(2n-1)(2n+1)^2, {n, 40}] (* or *) LinearRecurrence[{4, -6, 4, -1}, {9, 75, 245, 567}, 40] (* Harvey P. Dale, Jul 24 2012 *)
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PROG
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(PARI) vector(40, n, (2*n-1)*(2*n+1)^2) \\ G. C. Greubel, Oct 27 2019
(Magma) [(2*n-1)*(2*n+1)^2: n in [1..40]]; // G. C. Greubel, Oct 27 2019
(Sage) [(2*n-1)*(2*n+1)^2 for n in (1..40)] # G. C. Greubel, Oct 27 2019
(GAP) List([1..40], n-> (2*n-1)*(2*n+1)^2); # G. C. Greubel, Oct 27 2019
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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