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A294972 Numerators of continued fraction convergents to sqrt(7)/2. 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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).

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

MATHEMATICA

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

PROG

(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

CROSSREFS

Cf. A242703, A294973.

Sequence in context: A002677 A119527 A074991 * A213823 A296800 A273418

Adjacent sequences:  A294969 A294970 A294971 * A294973 A294974 A294975

KEYWORD

nonn,cofr,frac,easy

AUTHOR

Wolfdieter Lang, Nov 18 2017

STATUS

approved

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Last modified September 24 02:57 EDT 2020. Contains 337315 sequences. (Running on oeis4.)