Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #37 Dec 18 2023 14:49:28
%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="http://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