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A308217
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a(n) is the multiplicative inverse of A001844(n) modulo A001844(n+1); where A001844 is the sequence of centered square numbers.
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4
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1, 8, 2, 23, 3, 46, 4, 77, 5, 116, 6, 163, 7, 218, 8, 281, 9, 352, 10, 431, 11, 518, 12, 613, 13, 716, 14, 827, 15, 946, 16, 1073, 17, 1208, 18, 1351, 19, 1502, 20, 1661, 21, 1828, 22, 2003, 23, 2186, 24, 2377, 25, 2576, 26, 2783, 27, 2998, 28, 3221, 29, 3452
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OFFSET
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0,2
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COMMENTS
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The sequence explores the relationship between the terms of A001844, the sums of consecutive squares. The sequence is an interleaving of A033951 (a number spiral arm) and the natural numbers. The gap between the lower values of A308215 and the upper values of A308217 increase by 3n; each successive gap increasing by 6.
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LINKS
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FORMULA
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a(n) satisfies a(n)*(2*n*(n-1)+1) == 1 (mod 2*n*(n+1)+1).
G.f.: (1 + 8*x - x^2 - x^3 + x^5) / ((1 - x)^3*(1 + x)^3).
a(n) = 3*a(n-2) - 3*a(n-4) + a(n-6) for n>5.
a(n) = (9 - 5*(-1)^n + (8-6*(-1)^n)*n - 2*(-1+(-1)^n)*n^2) / 4.
(End)
a(n) = 1 + n/2 if n is even, since 0 < 1+n/2 < A001844(n+1) and (1+n/2)*A001844(n)-1 = (n/2)*A001844(n+1).
a(n) = n^2 + 7/2*(n+1) if n is odd, since 0 < n^2+7/2*(n+1) < A001844(n+1) and (n^2+7/2*(n+1))*A001844(n)-1 = (n^2+3*k/2+1/2)*A001844(n+1).
Colin Barker's conjectures easily follow. (End)
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MAPLE
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PROG
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(PARI) f(n) = 2*n*(n+1)+1; \\ A001844
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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