A093611 Numerators of convergents to 3/(1 + sqrt(10)).

0, 1, 3, 4, 7, 18, 25, 43, 111, 154, 265, 684, 949, 1633, 4215, 5848, 10063, 25974, 36037, 62011, 160059, 222070, 382129, 986328, 1368457, 2354785, 6078027, 8432812, 14510839, 37454490, 51965329, 89419819, 230804967, 320224786

Offset

1,3

Comments

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].

Denominators give same sequence shifted one place left.

Links

Table of n, a(n) for n=1..34.

Index entries for linear recurrences with constant coefficients, signature (0,0,6,0,0,1).

Formula

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.

G.f.: x(1+3x+4x^2+x^3+x^5)/(1-6x^3-x^6). - Paul Barry, Apr 12 2010

Example

a(5) = 13 = 2*5 + 3.

Mathematica

xx = ContinuedFraction[3/(1 + Sqrt[10]), 70]; Table[ Numerator[ FromContinuedFraction[ Take[xx, n]]], {n, 34}] (* Robert G. Wilson v, Apr 08 2004 *)
LinearRecurrence[{0, 0, 6, 0, 0, 1}, {0, 1, 3, 4, 7, 18, 25}, 40] (* Harvey P. Dale, Nov 26 2021 *)

Crossrefs

Sequence in context: A331115 A327318 A344783 * A042375 A153067 A041593

Adjacent sequences: A093608 A093609 A093610 * A093612 A093613 A093614

Keyword

nonn,frac

Author

Gary W. Adamson, Apr 04 2004

Extensions

Corrected and extended by Robert G. Wilson v, Apr 08 2004

Status

approved