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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.

%I A341862 #22 Feb 24 2021 08:16:49
%S A341862 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,
%T A341862 10,52,254,140,22,3040,34,46,70,12,0,12,74,50,38,64,26,6964,398,322,
%U A341862 48670,96,14,0,14,100,1298,364,970,178,30,302,198,1060,62,59436
%N 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.
%C A341862 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".
%C A341862 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).
%C A341862 a(n) is twice the numerator of the convergent to sqrt(n) with index k (starting with 0).
%C A341862 These properties result from the mirrored structure of the period of such continued fractions.
%C A341862 The sequence has remarkably many terms in common with A180495 and with 2*A033313.
%H A341862 Jean-François Alcover, <a href="/A341862/b341862.txt">Table of n, a(n) for n = 0..10000</a>
%H A341862 Graham Everest, Alf van der Poorten, Igor Shparlinski, and Thomas Ward, <a href="https://bookstore.ams.org/surv-104/16">Recurrence Sequences</a>, AMS Mathematical Surveys and Monographs, Volume 104 (2003) p. 8, 5th paragraph.
%F A341862 a(n) = 2*A006702(n) if n is not square, otherwise 0.
%e A341862 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.
%e A341862 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.
%Y A341862 Cf. A006702, A033313, A180495.
%K A341862 nonn
%O A341862 0,3
%A A341862 _Georg Fischer_, Feb 22 2021