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A243579 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).

%I A243579 #38 Feb 16 2025 08:33:22
%S A243579 71,255,567,1007,1575,2271,3095,4047,5127,6335,7671,9135,10727,12447,
%T A243579 14295,16271,18375,20607,22967,25455,28071,30815,33687,36687,39815,
%U A243579 43071,46455,49967,53607,57375,61271,65295,69447,73727,78135,82671,87335,92127,97047,102095,107271,112575,118007,123567,129255,135071,141015,147087
%N A243579 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 A243579 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 A243579 Walter Kehowski, <a href="/A243579/b243579.txt">Table of n, a(n) for n = 1..20737</a>
%H A243579 J. Owen Sizemore, <a href="http://www.math.wisc.edu/~josizemore/Notes16%284-square%29.pdf">Lagrange's Four Square Theorem</a>
%H A243579 R. C. Vaughan, <a href="https://personal.science.psu.edu/rcv4/Foursq.pdf">Lagrange's Four Square Theorem</a>
%H A243579 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LagrangesFour-SquareTheorem.html">Lagrange's Four-Square Theorem</a>
%H A243579 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lagrange%27s_four-square_theorem">Lagrange's four-square theorem</a>
%H A243579 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A243579 a(n) = 64*n^2-8*n+15.
%F A243579 From _Colin Barker_, Sep 13 2015: (Start)
%F A243579 a(n) = 3*a(n-1)-3*a(n-2)+a(n-3) for n>3.
%F A243579 G.f.: x*(15*x^2+42*x+71) / (1-x)^3. (End)
%e A243579 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 A243579 := proc(n::posint) return 64*n^2-8*n+15 end;
%o A243579 (PARI) Vec(-x*(15*x^2+42*x+71)/(x-1)^3 + O(x^100)) \\ _Colin Barker_, Sep 13 2015
%o A243579 (Magma) [64*n^2-8*n+15 : n in [1..50]]; // _Wesley Ivan Hurt_, Nov 28 2021
%Y A243579 Cf. A008586, A016813, A016825, A004767, A243577, A243578, A243579, A243580, A243581, A243582.
%K A243579 nonn,easy
%O A243579 1,1
%A A243579 _Walter Kehowski_, Jun 08 2014