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

1,1

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.

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.

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)

Example

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.

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 *)

Prog

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

Crossrefs

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

Sequence in context: A235974 A258095 A193228 * A124619 A290071 A221797

Adjacent sequences: A243575 A243576 A243577 * A243579 A243580 A243581

Keyword

nonn,easy

Author

Walter Kehowski, Jun 08 2014

Status

approved