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A309779 Squares that can be expressed as the sum of two positive squares but not as the sum of three positive squares.

%I A309779 #53 Oct 29 2023 15:25:08
%S A309779 25,100,400,1600,6400,25600,102400,409600,1638400,6553600,26214400,
%T A309779 104857600,419430400,1677721600,6710886400,26843545600,107374182400,
%U A309779 429496729600,1717986918400,6871947673600,27487790694400,109951162777600,439804651110400,1759218604441600
%N A309779 Squares that can be expressed as the sum of two positive squares but not as the sum of three positive squares.
%C A309779 This sequence comes from the study of A309778, exactly, A309778(n) = 2 iff n^2 belongs to this sequence here.
%C A309779 According to Draxl link, a(n) is a term of this sequence iff a(n) = 5^2 * 4^(n-1) with n >= 1.
%C A309779 This sequence is a subsequence of A219222 whose terms are all of the form b_0 * 4^k with b_0 in A051952, hence, the only primitive term of this sequence here is 25.
%H A309779 H.-P. Baltes, Peter K. J. Draxl, and Eberhard R. Hilf, <a href="http://smallsystems.isn-oldenburg.de/publications/metadocs/ebs.quadratsummen.html">Quadratsummen und gewisse Randwertprobleme der Mathematischen Physik</a>, Publications of the Small Systems Group Oldenburg, preprint, 1973.
%H A309779 H.-P. Baltes, Peter K. J. Draxl, and Eberhard R. Hilf, <a href="https://doi.org/10.1515/crll.1974.268-269.410">Quadratsummen und gewisse Randwertprobleme der Mathematischen Physik</a>, Journ. Reine Angewandte Mathematik, Vol. 268/269, 1974, 410-417.
%H A309779 P. K. J. Draxl, <a href="http://www.numdam.org/item?id=MSMF_1974__37__53_0">Sommes de deux carrés qui ne sont pas sommes de trois carrés.</a>, Mémoires de la SMF, tome 37 (1974), p. 53-53.
%H A309779 <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (4).
%F A309779 a(n) = 5^2 * 4^(n-1) with n >= 1.
%F A309779 a(n) = 4*a(n-1) for n > 1. G.f.: 25*x/(1 - 4*x). - _Chai Wah Wu_, Aug 29 2019
%F A309779 a(n) = 25 * A000302(n-1). - _Alois P. Heinz_, Aug 29 2019
%F A309779 E.g.f.: 25*(exp(4*x) - 1)/4. - _Stefano Spezia_, Oct 28 2023
%e A309779 25 = 5^2 = 3^2 + 4^2,
%e A309779 100 = 10^2 = 6^2 + 8^2,
%e A309779 5^2 * 4^(n-1) = (5 * 2^(n-1))^2 = (3 * 2^(n-1))^2 + (4 * 2^(n-1))^2, but these terms are not the sum of three positive squares.
%t A309779 Array[25*4^(# - 1) &, 24] (* _Michael De Vlieger_, Aug 19 2019 *)
%o A309779 (PARI) a(n) = 25 * 4^(n-1); \\ _Jinyuan Wang_, Aug 18 2019
%Y A309779 Intersection of A000290 and A219222.
%Y A309779 Cf. A000302, A000378, A000408, A051952, A134422, A309778.
%K A309779 nonn,easy
%O A309779 1,1
%A A309779 _Bernard Schott_, Aug 17 2019