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] = [2, 3, 5, 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 (web.archive)
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+26*x+87) / (x-1)^3. (End)
EXAMPLE
a(5) = 64*n^2 + 8*5 + 15 = 1655 and m = 4*5 - 2 = 18 so 1655 = 18^2 + 19^2 + 21^2 + 23^2.
MAPLE
A243580 := proc(n::posint) return 64*n^2+8*n+15 end;
MATHEMATICA
Table[64n^2 + 8n + 15, {n, 50}] (* Alonso del Arte, Jun 08 2014 *)
LinearRecurrence[{3, -3, 1}, {87, 287, 615}, 50] (* Harvey P. Dale, Mar 27 2019 *)
PROG
(PARI) Vec(-x*(15*x^2+26*x+87)/(x-1)^3 + O(x^100)) \\ Colin Barker, Sep 13 2015
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Walter Kehowski, Jun 08 2014
STATUS
approved