A243579 Integers of the form 8k+7 that can be written as a sum of four distinct squares of the form m, m+2, m+4, m+5, where m == 1 (mod 4).

71, 255, 567, 1007, 1575, 2271, 3095, 4047, 5127, 6335, 7671, 9135, 10727, 12447, 14295, 16271, 18375, 20607, 22967, 25455, 28071, 30815, 33687, 36687, 39815, 43071, 46455, 49967, 53607, 57375, 61271, 65295, 69447, 73727, 78135, 82671, 87335, 92127, 97047, 102095, 107271, 112575, 118007, 123567, 129255, 135071, 141015, 147087

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 with gap pattern 221, then [a,b,c,d] = [1,3,5,6]+[4*i,4*i,4*i,4*i] for i>=0.

Examples

			a(5) = 64*5^2-8*5+15 = 1575 and m = 4*5-3 = 17 so 1575 = 17^2+19^2+21^2+22^2.
		

Links

• Walter Kehowski, Table of n, a(n) for n = 1..20737
• J. Owen Sizemore, Lagrange's Four Square Theorem
• 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

Magma

[64*n^2-8*n+15 : n in [1..50]]; // Wesley Ivan Hurt, Nov 28 2021

Maple

A243579 := proc(n::posint) return 64*n^2-8*n+15 end;

PARI

Vec(-x*(15*x^2+42*x+71)/(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+42*x+71) / (1-x)^3. (End)