A338284 a(n) is the smallest nonsquare m such that the second partial quotient in the continued fraction for sqrt(m) equals n.

7, 2, 23, 5, 47, 10, 79, 17, 119, 26, 167, 37, 223, 50, 287, 65, 359, 82, 439, 101, 527, 122, 623, 145, 727, 170, 839, 197, 959, 226, 1087, 257, 1223, 290, 1367, 325, 1519, 362, 1679, 401, 1847, 442, 2023, 485, 2207, 530, 2399, 577, 2599, 626, 2807, 677, 3023, 730

Offset

1,1

Links

Max Alekseyev, Table of n, a(n) for n = 1..100

Formula

For even n, a(n) = A013945(n) = A002522(n/2) = (n/2)^2 + 1.

For odd n, a(n) = A073577((n+1)/2) = n^2 + 4*n + 2.

O.g.f.: (7 + 2*x + 2*x^2 - x^3 - x^4 + x^5) / ((1-x)^3 * (1+x)^3).

Example

a(3) = 23, since sqrt(23) = [4; 1, 3, ...] and m=23 is the smallest integer such that sqrt(m) has with second partial quotient equal 3.

Mathematica

CoefficientList[Series[(7 + 2*x + 2*x^2 - x^3 - x^4 + x^5) / ((1-x)^3 * (1+x)^3), {x, 0, 20}], x] (* Georg Fischer, Aug 18 2021 *)

Crossrefs

Interweaving of A073577 and A002522.

Cf. A013945 (first partial quotient = n).

Sequence in context: A371214 A213836 A340801 * A269168 A279943 A279999

Adjacent sequences: A338281 A338282 A338283 * A338285 A338286 A338287

Keyword

nonn,easy

Author

Max Alekseyev, Oct 20 2020

Status

approved