A342903 a(n) is the smallest number that is the sum of n positive squares in two ways.

50, 27, 28, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86

Offset

2,1

Comments

This is r(n,2,2) in Alter's notation.

Links

Paolo Xausa, Table of n, a(n) for n = 2..1000

Ronald Alter, Computations and generalizations on a remark of Ramanujan, pp. 182-196 of "Analytic Number Theory (Philadelphia, 1980)", ed. M. I. Knopp, Lect. Notes Math., Vol. 899, 1981. See Eq. (15), page 188.

Index entries for linear recurrences with constant coefficients, signature (2,-1).

Formula

a(n) = n + 15 for n >= 5.

From Elmo R. Oliveira, Jul 09 2026: (Start)

a(n) = 2*a(n-1) - a(n-2) for n >= 7.

G.f.: x^2*(50 - 73*x + 24*x^2 - 9*x^3 + 9*x^4)/(1 - x)^2.

E.g.f.: exp(x)*(x + 15) + (3*x^4 + 12*x^3 + 132*x^2 - 128*x - 120)/8. (End)

Example

a(2) = 50 = 1+49 = 25+25.
a(3) = 27 = 1+1+25 = 9+9+9.
a(5) = 20 = 1+1+1+1+16 = 4+4+4+4+4.

Mathematica

Join[{50, 27, 28}, Range[20, 100]] (* Paolo Xausa, Feb 09 2026 *)

Crossrefs

Cf. A230563, A342902.

Sequence in context: A033370 A090691 A033980 * A207149 A207142 A244277

Adjacent sequences: A342900 A342901 A342902 * A342904 A342905 A342906

Keyword

nonn,easy,changed

Author

N. J. A. Sloane, Apr 03 2021

Status

approved