A243578 Integers n of the form 8k+7 that are sum of distinct squares of the form m, m+1, m+2, m+4, where m == 1 (mod 4).

39, 191, 471, 879, 1415, 2079, 2871, 3791, 4839, 6015, 7319, 8751, 10311, 11999, 13815, 15759, 17831, 20031, 22359, 24815, 27399, 30111, 32951, 35919, 39015, 42239, 45591, 49071, 52679, 56415, 60279, 64271, 68391, 72639, 77015, 81519, 86151, 90911, 95799

Offset: 1

Author

Walter Kehowski, Jun 08 2014

Keywords

• nonn
• easy

Comments

If n is of the form 8k+7 such that n=a^2+b^2+c^2+d^2 with gap pattern 112, then [a,b,c,d]=[1,2,3,5]+[4*i,4*i,4*i,4*i], i>=0.

Examples

			a(5)=64*5^2-40*5+15=1415 and m=4*5-3=17, and 1415=17^2+18^2+19^2+21^2.
		

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

I:=[39,191,471]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..60]]; // Vincenzo Librandi, Sep 13 2015

Maple

A243578 := proc(n::posint) return 64*n^3-40*n+15 end;

Mathematica

LinearRecurrence[{3, -3, 1}, {39, 191, 471}, 50] (* Vincenzo Librandi, Sep 13 2015 *)

PARI

Vec(-x*(3*x+13)*(5*x+3)/(x-1)^3 + O(x^100)) \\ Colin Barker, Sep 12 2015

Formula

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

From Colin Barker, Sep 12 2015: (Start)

a(n) = 3*a(n-1)-3*a(n-2)+a(n-3) for n>3.

G.f.: -x*(3*x+13)*(5*x+3) / (x-1)^3.

(End)