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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. 1
25, 100, 400, 1600, 6400, 25600, 102400, 409600, 1638400, 6553600, 26214400, 104857600, 419430400, 1677721600, 6710886400, 26843545600, 107374182400, 429496729600, 1717986918400, 6871947673600, 27487790694400, 109951162777600, 439804651110400, 1759218604441600 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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.

LINKS

Table of n, a(n) for n=1..24.

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.

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

EXAMPLE

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.

MATHEMATICA

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

PROG

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

CROSSREFS

Intersection of A000290 and A219222.

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

Sequence in context: A198385 A134422 A016850 * A221274 A042220 A114254

Adjacent sequences:  A309776 A309777 A309778 * A309780 A309781 A309782

KEYWORD

nonn,easy

AUTHOR

Bernard Schott, Aug 17 2019

STATUS

approved

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Last modified August 10 08:39 EDT 2020. Contains 336369 sequences. (Running on oeis4.)