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

25, 100, 400, 1600, 6400, 25600, 102400, 409600, 1638400, 6553600, 26214400, 104857600, 419430400, 1677721600, 6710886400, 26843545600, 107374182400, 429496729600, 1717986918400, 6871947673600, 27487790694400, 109951162777600, 439804651110400, 1759218604441600

Offset: 1

Author

Bernard Schott, Aug 17 2019

Keywords

• nonn
• easy

Comments

This sequence comes from the study of A309778, exactly, A309778(n) = 2 iff n^2 belongs to this sequence here.

According to Draxl link, a(n) is a term of this sequence iff a(n) = 5^2 * 4^(n-1) with n >= 1.

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.

Examples

			25 = 5^2 = 3^2 + 4^2,
100 = 10^2 = 6^2 + 8^2,
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.
		

Links

• H.-P. Baltes, Peter K. J. Draxl, and Eberhard R. Hilf, Quadratsummen und gewisse Randwertprobleme der Mathematischen Physik, Publications of the Small Systems Group Oldenburg, preprint, 1973.
• H.-P. Baltes, Peter K. J. Draxl, and Eberhard R. Hilf, Quadratsummen und gewisse Randwertprobleme der Mathematischen Physik, Journ. Reine Angewandte Mathematik, Vol. 268/269, 1974, 410-417.
• P. K. J. Draxl, Sommes de deux carrés qui ne sont pas sommes de trois carrés., Mémoires de la SMF, tome 37 (1974), p. 53-53.
• Index entries for linear recurrences with constant coefficients, signature (4).

Crossrefs

Intersection of A000290 and A219222.

Cf. A000302, A000378, A000408, A051952, A134422, A309778.

Programs

Mathematica

Array[25*4^(# - 1) &, 24] (* Michael De Vlieger, Aug 19 2019 *)

PARI

a(n) = 25 * 4^(n-1); \\ Jinyuan Wang, Aug 18 2019

Formula

a(n) = 5^2 * 4^(n-1) with n >= 1.

a(n) = 4*a(n-1) for n > 1. G.f.: 25*x/(1 - 4*x). - Chai Wah Wu, Aug 29 2019

a(n) = 25 * A000302(n-1). - Alois P. Heinz, Aug 29 2019

E.g.f.: 25*(exp(4*x) - 1)/4. - Stefano Spezia, Oct 28 2023