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A243580 Integers of the form 8k + 7 that can be written as a sum of four distinct squares of the form m, m + 1, m + 3, m + 5, where m == 2 (mod 4).

%I A243580 #24 Feb 16 2025 08:33:22
%S A243580 87,287,615,1071,1655,2367,3207,4175,5271,6495,7847,9327,10935,12671,
%T A243580 14535,16527,18647,20895,23271,25775,28407,31167,34055,37071,40215,
%U A243580 43487,46887,50415,54071,57855,61767,65807,69975,74271,78695,83247,87927,92735,97671,102735,107927,113247,118695,124271,129975,135807,141767,147855
%N A243580 Integers of the form 8k + 7 that can be written as a sum of four distinct squares of the form m, m + 1, m + 3, m + 5, where m == 2 (mod 4).
%C A243580 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.
%H A243580 Walter Kehowski, <a href="/A243580/b243580.txt">Table of n, a(n) for n = 1..20737</a>
%H A243580 J. Owen Sizemore, <a href="http://web.archive.org/web/20130418080049/http://www.math.wisc.edu/~josizemore/Notes16(4-square).pdf">Lagrange's Four Square Theorem</a> (web.archive)
%H A243580 R. C. Vaughan, <a href="https://personal.science.psu.edu/rcv4/Foursq.pdf">Lagrange's Four Square Theorem</a>
%H A243580 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LagrangesFour-SquareTheorem.html">Lagrange's Four-Square Theorem</a>
%H A243580 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lagrange%27s_four-square_theorem">Lagrange's four-square theorem</a>
%H A243580 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A243580 a(n) = 64*n^2 + 8*n + 15.
%F A243580 From _Colin Barker_, Sep 13 2015: (Start)
%F A243580 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>3.
%F A243580 G.f.: -x*(15*x^2+26*x+87) / (x-1)^3. (End)
%e A243580 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.
%p A243580 A243580 := proc(n::posint) return 64*n^2+8*n+15 end;
%t A243580 Table[64n^2 + 8n + 15, {n, 50}] (* _Alonso del Arte_, Jun 08 2014 *)
%t A243580 LinearRecurrence[{3,-3,1},{87,287,615},50] (* _Harvey P. Dale_, Mar 27 2019 *)
%o A243580 (PARI) Vec(-x*(15*x^2+26*x+87)/(x-1)^3 + O(x^100)) \\ _Colin Barker_, Sep 13 2015
%Y A243580 Cf. A008586, A016813, A016825, A004767, A243577, A243578, A243579, A243580, A243581, A243582
%K A243580 nonn,easy
%O A243580 1,1
%A A243580 _Walter Kehowski_, Jun 08 2014