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Numerators of continued fraction convergents to sqrt(13)/2 = A295330.
3

%I #15 Feb 27 2019 12:27:39

%S 1,2,9,128,521,649,1819,2468,11691,166142,676259,842401,2361061,

%T 3203462,15174909,215652188,877783661,1093435849,3064655359,

%U 4158091208,19697020191,279916373882,1139362515719,1419278889601,3977920294921,5397199184522,25566717033009,363331237646648,1478891667619601,1842222905266249,5163337478152099,7005560383418348,33185579011825491

%N Numerators of continued fraction convergents to sqrt(13)/2 = A295330.

%C The denominators are given in A295332.

%C The continued fraction expansion of sqrt(13)/2 is [1, repeat(1, 4, 14, 4, 1, 2)] = [b(0), repeat(b(1), b(2), b(3), b(4), b(5), b(6))].

%H Harvey P. Dale, <a href="/A295331/b295331.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (0, 0, 0, 0, 0, 1298, 0, 0, 0, 0, 0, -1).

%F G.f.: (1 + 2*x + 9*x^2 + 128*x^3 + 521*x^4 + 649*x^5 + 521*x^6 - 128*x^7 + 9*x^8 + 2*x^9 + x^10 - x^11) / ((1 - 3*x - x^2)*(1 + 3*x - x^2)*(1 + 3*x + 10*x^2 - 3*x^3 + x^4)*(1 - 3*x + 10*x^2 + 3*x^3 + x^4)). For the derivation see the period 4 example for the denominators of the convergents of sqrt(7)/2 given in A294972. Here the period is 6 and the input for the recurrence a(n) = b(n)*a(n-1) + a(n-2) is a(-1) = 1 = a(0) (a(-2) = 1 - b(0) = 0) with the b(n) modulo 6 given above. Note that a(6*k) = 2*a(6*k-1) + a(6*k-2) is valid only for k >= 1 because a(0) = 1 is not equal to 2*a(-1) + a(-2) = 2. The denominator in the unfactorized form is 1 - 1298*x^6 + x^12.

%F a(n) = 1298*a(n-6) - a(n-12), n >= 12, with inputs a(0)..a(11).

%e The convergents a(n)/A295332 begin: 1, 2, 9/5, 128/71, 521/289, 649/360, 1819/1009, 2468/1369, 11691/6485, 166142/92159, 676259/375121, 842401/467280, 2361061/1309681, 3203462/1776961, 15174909/8417525, 215652188/119622311, 877783661/486906769, 1093435849/606529080, ...

%t Numerator[Convergents[Sqrt[13]/2,40]] (* or *) LinearRecurrence[ {0,0,0,0,0,1298,0,0,0,0,0,-1},{1,2,9,128,521,649,1819,2468,11691,166142,676259,842401},40] (* _Harvey P. Dale_, Jan 23 2019 *)

%Y Cf. A294972/A294973 (sqrt(7/2)), A295330, A295332.

%K nonn,frac,cofr,easy

%O 0,2

%A _Wolfdieter Lang_, Nov 20 2017