A013945 -id:A013945 - cp's OEIS frontend

A065014 Duplicate of A013945.

3, 2, 11, 5, 27, 10, 51, 17, 83, 26, 123, 37, 171, 50, 227, 65, 291, 82, 363, 101, 443, 122, 531, 145, 627, 170, 731, 197, 843, 226, 963, 257, 1091, 290, 1227, 325, 1371, 362, 1523, 401, 1683, 442, 1851, 485, 2027, 530, 2211, 577, 2403, 626, 2603, 677

Offset: 1

Robert G. Wilson v, Nov 01 2001

dead

			a(3) = 11 because the continued fraction for the square root of 11 is 3, {3, 6}.

a = Table[0, {70}]; Do[ b = First[ Last[ ContinuedFraction[ Sqrt[ n]]]]; If[ b < 71 && a[[b]] == 0, a[[b]] = n], {n, 2, 10^4} ]; a

G.f.: (x^5+3x^4-x^3+2x^2+2x+3)/(1-x^2)^3.

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

Original entry on oeis.org

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

Max Alekseyev, Oct 20 2020

			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.

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

Interweaving of A073577 and A002522.

Cf. A013945 (first partial quotient = n).

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 *)

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).

cp's OEIS Frontend

A065014 Duplicate of A013945.

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