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

Author

Walter Kehowski, Jun 08 2014

Keywords

• nonn
• easy

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.

Examples

			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.
		

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).

Crossrefs

Cf. A008586, A016813, A016825, A004767, A243577, A243578, A243579, A243580, A243581, A243582

Programs

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 *)

PARI

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

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)