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A014641
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Odd octagonal numbers: (2n+1)*(6n+1).
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14
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1, 21, 65, 133, 225, 341, 481, 645, 833, 1045, 1281, 1541, 1825, 2133, 2465, 2821, 3201, 3605, 4033, 4485, 4961, 5461, 5985, 6533, 7105, 7701, 8321, 8965, 9633, 10325, 11041, 11781, 12545, 13333, 14145, 14981, 15841, 16725, 17633, 18565, 19521, 20501, 21505
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OFFSET
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0,2
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COMMENTS
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Sequence found by reading the line from 1, in the direction 1, 21, ..., in the square spiral whose vertices are the generalized octagonal numbers A001082. - Omar E. Pol, Jul 18 2012
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LINKS
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FORMULA
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G.f.: (1 + 18*x + 5*x^2)/(1 - 3*x + 3*x^2 - x^3). - Colin Barker, Jan 06 2012
This is the polynomial Qbar(2,n) in Brent. See A160485 for the triangle of coefficients (with signs) of the Qbar polynomials.
a(n) = (1/4^n) * Sum_{k = 0..n} (2*k + 1)^4*binomial(2*n + 1, n - k).
a(n-1) = (2/4^n) * binomial(2*n,n) * ( 1 + 3^4*(n - 1)/(n + 1) + 5^4*(n - 1)*(n - 2)/((n + 1)*(n + 2)) + 7^4*(n - 1)*(n - 2)*(n - 3)/((n + 1)*(n + 2)*(n + 3)) + ... ). (End)
Sum_{n>=0} 1/a(n) = (sqrt(3)*Pi + 3*log(3))/8.
Sum_{n>=0} (-1)^n/a(n) = Pi/8 + sqrt(3)*log(2+sqrt(3))/4. (End)
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MAPLE
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MATHEMATICA
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PROG
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(GAP) List([0..50], n->(2*n+1)*(6*n+1)); # Muniru A Asiru, Feb 05 2019
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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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EXTENSIONS
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STATUS
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approved
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