A294972 Numerators of continued fraction convergents to sqrt(7)/2.

1, 4, 41, 127, 295, 1012, 10415, 32257, 74929, 257044, 2645369, 8193151, 19031671, 65288164, 671913311, 2081028097, 4833969505, 16582936612, 170663335625, 528572943487, 1227809222599, 4212000611284, 43347815335439, 134255446617601, 311858708570641, 1069831572329524, 11010174431865881

Offset: 0

Author

Wolfdieter Lang, Nov 18 2017

Keywords

• nonn
• cofr
• frac
• easy

Comments

The denominators are given in A294973.

The continued fraction expansion of sqrt(7)/2 is 1, repeat(3, 10, 3, 2).

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (0,0,0,254,0,0,0,-1).

Crossrefs

Cf. A242703, A294973.

Programs

Mathematica

Numerator[Convergents[Sqrt[7]/2, 30]] (* Vaclav Kotesovec, Nov 19 2017 *)

PARI

Vec((1 + 4*x + 41*x^2 + 127*x^3 + 41*x^4 - 4*x^5 + x^6 - x^7) / ((1 - 16*x^2 + x^4)*(1 + 16*x^2 + x^4)) + O(x^40)) \\ Colin Barker, Nov 21 2017

Formula

From Colin Barker, Nov 19 2017: (Start)

G.f.: (1 + 4*x + 41*x^2 + 127*x^3 + 41*x^4 - 4*x^5 + x^6 - x^7) / ((1 - 16*x^2 + x^4)*(1 + 16*x^2 + x^4)).

a(n) = 254*a(n-4) - a(n-8) for n > 7.

(End)

The proof of the g.f. runs like the one given for the denominators in A294973. The recurrence for a(n) is the same but the input is now a(0) = b(0) = 1 and a(-1) = 1, (a(-2) = 0). - Wolfdieter Lang, Nov 19 2017