A243580 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).

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

Offset

1,1

Comments

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.

Links

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).

Formula

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)

Example

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.

Maple

A243580 := proc(n::posint) return 64*n^2+8*n+15 end;

Mathematica

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 *)

Prog

(PARI) Vec(-x*(15*x^2+26*x+87)/(x-1)^3 + O(x^100)) \\ Colin Barker, Sep 13 2015

Crossrefs

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

Keyword

nonn,easy

Author

Walter Kehowski, Jun 08 2014

Status

approved