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%I #11 Nov 26 2021 16:57:01
%S 0,1,3,4,7,18,25,43,111,154,265,684,949,1633,4215,5848,10063,25974,
%T 36037,62011,160059,222070,382129,986328,1368457,2354785,6078027,
%U 8432812,14510839,37454490,51965329,89419819,230804967,320224786
%N Numerators of convergents to 3/(1 + sqrt(10)).
%C The partial quotient terms [1 2 1 1 2 1 1 2 1...] are palindromic. The matrix generator for convergents to barover[1 2 1] = [2 3 / 3 4]^n = M^n and is Hermitian (upper right term = lower left). Therefore in any pair of convergents M^n, upper right term = lower left. Example: M^3 = [80 111 / 111 541, where 111 = a(9). Consequently a(3n) = A093612(3n-1), where A093612 = denominators of barover[1 2 1].
%C Denominators give same sequence shifted one place left.
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,6,0,0,1).
%F Partial quotients are [1 2 1 1 2 1...] indicating the operation below a term q. The numerator under q = n = q(n-1) + (n-2), a(1) = 1, a(2) = 2, a(3) = 3 and so on.
%F G.f.: x(1+3x+4x^2+x^3+x^5)/(1-6x^3-x^6). - _Paul Barry_, Apr 12 2010
%e a(5) = 13 = 2*5 + 3.
%t xx = ContinuedFraction[3/(1 + Sqrt[10]), 70]; Table[ Numerator[ FromContinuedFraction[ Take[xx, n]]], {n, 34}] (* _Robert G. Wilson v_, Apr 08 2004 *)
%t LinearRecurrence[{0,0,6,0,0,1},{0,1,3,4,7,18,25},40] (* _Harvey P. Dale_, Nov 26 2021 *)
%K nonn,frac
%O 1,3
%A _Gary W. Adamson_, Apr 04 2004
%E Corrected and extended by _Robert G. Wilson v_, Apr 08 2004
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