A295331 Numerators of continued fraction convergents to sqrt(13)/2 = A295330.
1, 2, 9, 128, 521, 649, 1819, 2468, 11691, 166142, 676259, 842401, 2361061, 3203462, 15174909, 215652188, 877783661, 1093435849, 3064655359, 4158091208, 19697020191, 279916373882, 1139362515719, 1419278889601, 3977920294921, 5397199184522, 25566717033009, 363331237646648, 1478891667619601, 1842222905266249, 5163337478152099, 7005560383418348, 33185579011825491
Offset: 0
Examples
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, ...
Links
- Harvey P. Dale, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 1298, 0, 0, 0, 0, 0, -1).
Programs
-
Mathematica
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 *)
Formula
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.
a(n) = 1298*a(n-6) - a(n-12), n >= 12, with inputs a(0)..a(11).
Comments