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A243581 Integers of the form 8k + 7 that can be written as a sum of four distinct squares of the form m, m + 2, m + 3, m + 4, where m == 2 (mod 4).

%I A243581 #30 Feb 16 2025 08:33:22
%S A243581 119,351,711,1199,1815,2559,3431,4431,5559,6815,8199,9711,11351,13119,
%T A243581 15015,17039,19191,21471,23879,26415,29079,31871,34791,37839,41015,
%U A243581 44319,47751,51311,54999,58815,62759,66831,71031,75359,79815
%N A243581 Integers of the form 8k + 7 that can be written as a sum of four distinct squares of the form m, m + 2, m + 3, m + 4, where m == 2 (mod 4).
%C A243581 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] = [3, 5, 6, 7] + [4*i, 4*i, 4*i, 4*i], i >= 0.
%H A243581 Walter Kehowski, <a href="/A243581/b243581.txt">Table of n, a(n) for n = 1..20737</a>
%H A243581 J. Owen Sizemore, <a href="http://www.math.wisc.edu/~josizemore/Notes16%284-square%29.pdf">Lagrange's Four Square Theorem</a>
%H A243581 R. C. Vaughan, <a href="https://personal.science.psu.edu/rcv4/Foursq.pdf">Lagrange's Four Square Theorem</a>
%H A243581 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LagrangesFour-SquareTheorem.html">Lagrange's Four-Square Theorem</a>
%H A243581 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lagrange%27s_four-square_theorem">Lagrange's four-square theorem</a>
%H A243581 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A243581 a(n) = 64*n^2 + 40*n + 15.
%F A243581 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - _Colin Barker_, Jun 09 2014
%F A243581 G.f.: -x*(15*x^2-6*x+119) / (x-1)^3. - _Colin Barker_, Jun 09 2014
%e A243581 a(5) = 64*5^2 + 40*5 + 15 = 1815 and 4*5 - 1 = 19 so 1815 = 19^2 + 21^2 + 22^2 + 23^2.
%p A243581 A243581 := proc(n::posint) return 64*n^2+40*n+15 end;
%t A243581 Table[64n^2 + 40n + 15, {n, 50}] (* _Alonso del Arte_, Jun 08 2014 *)
%t A243581 LinearRecurrence[{3,-3,1},{119,351,711},50] (* _Harvey P. Dale_, Jul 23 2014 *)
%o A243581 (PARI) Vec(-x*(15*x^2-6*x+119)/(x-1)^3 + O(x^100)) \\ _Colin Barker_, Jun 09 2014
%o A243581 (Magma) [ 64*n^2 + 40*n + 15 : n in [1..50] ]; // _Wesley Ivan Hurt_, Jun 11 2014
%Y A243581 Cf. A008586, A016813, A016825, A004767, A243577, A243578, A243579, A243580, A243581, A243582.
%K A243581 nonn,easy
%O A243581 1,1
%A A243581 _Walter Kehowski_, Jun 08 2014