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A243580
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Integers of the form 8k + 7 that can be written as a sum of four distinct squares of the form m, m + 1, m + 3, m + 5, where m == 2 (mod 4).
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6
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87, 287, 615, 1071, 1655, 2367, 3207, 4175, 5271, 6495, 7847, 9327, 10935, 12671, 14535, 16527, 18647, 20895, 23271, 25775, 28407, 31167, 34055, 37071, 40215, 43487, 46887, 50415, 54071, 57855, 61767, 65807, 69975, 74271, 78695, 83247, 87927, 92735, 97671, 102735, 107927, 113247, 118695, 124271, 129975, 135807, 141767, 147855
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OFFSET
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1,1
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COMMENTS
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If n is of the form 8k + 7 and n = a^2 + b^2 + c^2 + d^2 where [a, b, c, d] has gap pattern 122, then [a, b, c, d] = [2, 3, 5, 7] + [4*i, 4*i, 4*i, 4*i], i >= 0.
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LINKS
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Walter Kehowski, Table of n, a(n) for n = 1..20737
J. Owen Sizemore, Lagrange's Four Square Theorem (web.archive)
R. C. Vaughan, LAGRANGE'S FOUR SQUARE THEOREM
Eric Weisstein's World of Mathematics, Lagrange's Four-Square Theorem
Wikipedia, Lagrange's four-square theorem
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
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FORMULA
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a(n) = 64*n^2 + 8*n + 15.
From Colin Barker, Sep 13 2015: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>3.
G.f.: -x*(15*x^2+26*x+87) / (x-1)^3.
(End)
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EXAMPLE
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a(5) = 64*n^2 + 8*5 + 15 = 1655 and m = 4*5 - 2 = 18 so 1655 = 18^2 + 19^2 + 21^2 + 23^2.
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MAPLE
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A243580 := proc(n::posint) return 64*n^2+8*n+15 end;
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MATHEMATICA
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Table[64n^2 + 8n + 15, {n, 50}] (* Alonso del Arte, Jun 08 2014 *)
LinearRecurrence[{3, -3, 1}, {87, 287, 615}, 50] (* Harvey P. Dale, Mar 27 2019 *)
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PROG
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(PARI) Vec(-x*(15*x^2+26*x+87)/(x-1)^3 + O(x^100)) \\ Colin Barker, Sep 13 2015
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CROSSREFS
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Cf. A008586, A016813, A016825, A004767, A243577, A243578, A243579, A243580, A243581, A243582
Sequence in context: A008879 A186063 A186055 * A219723 A033631 A183724
Adjacent sequences: A243577 A243578 A243579 * A243581 A243582 A243583
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KEYWORD
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nonn,easy
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AUTHOR
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Walter Kehowski, Jun 08 2014
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STATUS
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approved
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