A243579 - OEIS

%I #38 Feb 16 2025 08:33:22

%S 71,255,567,1007,1575,2271,3095,4047,5127,6335,7671,9135,10727,12447,

%T 14295,16271,18375,20607,22967,25455,28071,30815,33687,36687,39815,

%U 43071,46455,49967,53607,57375,61271,65295,69447,73727,78135,82671,87335,92127,97047,102095,107271,112575,118007,123567,129255,135071,141015,147087

%N 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).

%C 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.

%H Walter Kehowski, <a href="/A243579/b243579.txt">Table of n, a(n) for n = 1..20737</a>

%H J. Owen Sizemore, <a href="http://www.math.wisc.edu/~josizemore/Notes16%284-square%29.pdf">Lagrange's Four Square Theorem</a>

%H R. C. Vaughan, <a href="https://personal.science.psu.edu/rcv4/Foursq.pdf">Lagrange's Four Square Theorem</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LagrangesFour-SquareTheorem.html">Lagrange's Four-Square Theorem</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Lagrange%27s_four-square_theorem">Lagrange's four-square theorem</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

%F a(n) = 64*n^2-8*n+15.

%F From _Colin Barker_, Sep 13 2015: (Start)

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

%F G.f.: x*(15*x^2+42*x+71) / (1-x)^3. (End)

%e 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.

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

%o (PARI) Vec(-x*(15*x^2+42*x+71)/(x-1)^3 + O(x^100)) \\ _Colin Barker_, Sep 13 2015

%o (Magma) [64*n^2-8*n+15 : n in [1..50]]; // _Wesley Ivan Hurt_, Nov 28 2021

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

%K nonn,easy

%O 1,1

%A _Walter Kehowski_, Jun 08 2014