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A254626
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Indices of triangular numbers (A000217) that are also centered pentagonal numbers (A005891).
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3
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1, 3, 23, 61, 421, 1103, 7563, 19801, 135721, 355323, 2435423, 6376021, 43701901, 114413063, 784198803, 2053059121, 14071876561, 36840651123, 252509579303, 661078661101, 4531100550901, 11862575248703, 81307300336923, 212865275815561, 1459000305513721
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OFFSET
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1,2
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COMMENTS
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Also positive integers x in the solutions to x^2 - 5*y^2 + x + 5*y - 2 = 0, the corresponding values of y being A254627.
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LINKS
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FORMULA
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a(n) = a(n-1) + 18*a(n-2) - 18*a(n-3) - a(n-4) + a(n-5).
G.f.: -x*(x+1)^2*(x^2+1) / ((x-1)*(x^2-4*x-1)*(x^2+4*x-1)).
a(n) = (-2 + (2-r)^n - (-2-r)^n*(-2+r) + 2*(-2+r)^n + r*(-2+r)^n + (2+r)^n)/4 where r = sqrt(5). - Colin Barker, Nov 25 2016
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EXAMPLE
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3 is in the sequence because the 3rd triangular number is 6, which is also the 2nd centered pentagonal number.
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PROG
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(PARI) Vec(-x*(x+1)^2*(x^2+1)/((x-1)*(x^2-4*x-1)*(x^2+4*x-1)) + O(x^100))
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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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