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

1,1

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.

Links

Walter Kehowski, Table of n, a(n) for n = 1..20737

J. Owen Sizemore, Lagrange's Four Square Theorem. [Wayback Machine link]

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

From Elmo R. Oliveira, Jun 09 2026: (Start)

E.g.f.: exp(x)*(64*x^2 + 56*x + 15) - 15.

a(n) = A243577(4*n-2). (End)

Example

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.

Maple

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

Mathematica

A243579[n_] := 8*n*(8*n - 1) + 15; Array[A243579, 50] (* Paolo Xausa, May 19 2026 *)

Prog

(PARI) Vec(-x*(15*x^2+42*x+71)/(x-1)^3 + O(x^100)) \\ Colin Barker, Sep 13 2015
(Magma) [64*n^2-8*n+15 : n in [1..50]]; // Wesley Ivan Hurt, Nov 28 2021

Crossrefs

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

Sequence in context: A325079 A142325 A232475 * A142013 A033240 A298103

Adjacent sequences: A243576 A243577 A243578 * A243580 A243581 A243582

Keyword

nonn,easy,changed

Author

Walter Kehowski, Jun 08 2014

Status

approved