A341862 a(n) is the even term in the linear recurrence signature for numerators and denominators of continued fraction convergents to sqrt(n), or 0 if n is a square.

0, 0, 2, 4, 0, 4, 10, 16, 6, 0, 6, 20, 14, 36, 30, 8, 0, 8, 34, 340, 18, 110, 394, 48, 10, 0, 10, 52, 254, 140, 22, 3040, 34, 46, 70, 12, 0, 12, 74, 50, 38, 64, 26, 6964, 398, 322, 48670, 96, 14, 0, 14, 100, 1298, 364, 970, 178, 30, 302, 198, 1060, 62, 59436

Offset

0,3

Comments

The Everest et al. link states that "the continued fraction expansion of a quadratic irrational is eventually periodic, which implies that the numerators px and denominators qx of its convergents satisfy linear recurrence relations".

Let k be the period length minus one of the continued fraction of sqrt(n). Then the linear recurrence signatures with constant coefficients have the form (0, 0, ..., 0, a(n), 0, 0, ..., 0, (-1)^(n+1)), with k zeroes before and behind a(n).

a(n) is twice the numerator of the convergent to sqrt(n) with index k (starting with 0).

These properties result from the mirrored structure of the period of such continued fractions.

The sequence has remarkably many terms in common with A180495 and with 2*A033313.

Links

Jean-François Alcover, Table of n, a(n) for n = 0..10000

Graham Everest, Alf van der Poorten, Igor Shparlinski, and Thomas Ward, Recurrence Sequences, AMS Mathematical Surveys and Monographs, Volume 104 (2003) p. 8, 5th paragraph.

Formula

a(n) = 2*A006702(n) if n is not square, otherwise 0.

Example

The numerators for sqrt(13) begin with 3, 4, 7, 11, 18, 119, ... (A041018) and have the signature (0,0,0,0,36,0,0,0,0,1). The continued fraction has period [1,1,1,1,6], so k=4 and a(13) = 2*A041018(4) = 2*18 = 36. The signature ends with (-1)^4.
The numerators for sqrt(19) begin with 4, 9, 13, 48, 61, 170, 1421, ... (A041028) and have the signature (0,0,0,0,0,340,0,0,0,0,0,-1). The continued fraction has period [2,1,3,1,2,8], so k=5 and a(19) = 2*A041028(5) = 2*170 = 340. The signature ends with (-1)^5.

Crossrefs

Cf. A006702, A033313, A180495.

Sequence in context: A086932 A331762 A221255 * A256487 A079985 A059226

Adjacent sequences: A341859 A341860 A341861 * A341863 A341864 A341865

Keyword

nonn

Author

Georg Fischer, Feb 22 2021

Status

approved