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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). 6
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified March 29 05:39 EDT 2020. Contains 333105 sequences. (Running on oeis4.)