A243581 Integers of the form 8k + 7 that can be written as a sum of four distinct squares of the form m, m + 2, m + 3, m + 4, where m == 2 (mod 4).

119, 351, 711, 1199, 1815, 2559, 3431, 4431, 5559, 6815, 8199, 9711, 11351, 13119, 15015, 17039, 19191, 21471, 23879, 26415, 29079, 31871, 34791, 37839, 41015, 44319, 47751, 51311, 54999, 58815, 62759, 66831, 71031, 75359, 79815

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] = [3, 5, 6, 7] + [4*i, 4*i, 4*i, 4*i], i >= 0.

Examples

			a(5) = 64*5^2 + 40*5 + 15 = 1815 and 4*5 - 1 = 19 so 1815 = 19^2 + 21^2 + 22^2 + 23^2.
		

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 + 40*n + 15 : n in [1..50] ]; // Wesley Ivan Hurt, Jun 11 2014

Maple

A243581 := proc(n::posint) return 64*n^2+40*n+15 end;

Mathematica

Table[64n^2 + 40n + 15, {n, 50}] (* Alonso del Arte, Jun 08 2014 *)
LinearRecurrence[{3,-3,1},{119,351,711},50] (* Harvey P. Dale, Jul 23 2014 *)

PARI

Vec(-x*(15*x^2-6*x+119)/(x-1)^3 + O(x^100)) \\ Colin Barker, Jun 09 2014

Formula

a(n) = 64*n^2 + 40*n + 15.

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Colin Barker, Jun 09 2014

G.f.: -x*(15*x^2-6*x+119) / (x-1)^3. - Colin Barker, Jun 09 2014