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

1,1

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] = [3, 5, 6, 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

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 + 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

Example

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

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 *)

Prog

(PARI) Vec(-x*(15*x^2-6*x+119)/(x-1)^3 + O(x^100)) \\ Colin Barker, Jun 09 2014
(Magma) [ 64*n^2 + 40*n + 15 : n in [1..50] ]; // Wesley Ivan Hurt, Jun 11 2014

Crossrefs

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

Sequence in context: A207059 A257603 A063348 * A103852 A157040 A256907

Adjacent sequences: A243578 A243579 A243580 * A243582 A243583 A243584

Keyword

nonn,easy

Author

Walter Kehowski, Jun 08 2014

Status

approved