A059927 -id:A059927 - cp's OEIS frontend

A059866 Period of the continued fraction for sqrt(2^n-1).

2, 4, 2, 8, 2, 12, 2, 20, 2, 12, 2, 164, 2, 40, 2, 40, 2, 1208, 2, 660, 2, 1304, 2, 3056, 2, 2492, 2, 1080, 2, 13004, 2, 10232, 2, 11296, 2, 148736, 2, 56576, 2, 615482, 2, 44448, 2, 64, 2, 2628524, 2, 28219952, 2, 139558, 2, 3067080, 2, 2683626, 2, 90740360, 2, 103050292, 2

Offset: 2

Labos Elemer, Feb 28 2001

nonn

			For n=7 and n=8 the continued fractions are [[11], [3, 1, 2, 2, 7, 11, 7, 2, 2, 1, 3, 22]] and [[15], [1, 30]] with periods 12 and 2, respectively.

Chai Wah Wu, Table of n, a(n) for n = 2..78

Cf. A003285, A059926, A059927, A064932.

with(numtheory): [seq(nops(cfrac(sqrt(2^k-1),'periodic','quotients')[2]),k=2..30)];

Table[Length@ Last@ ContinuedFraction[Sqrt[2^n - 1]], {n, 2, 56}] (* Michael De Vlieger, Mar 21 2015 *)

Corrected and extended by Naohiro Nomoto, Nov 09 2001

a(57)-a(60) from Daniel Suteu, Jan 25 2019

A064932 Period of the continued fraction for sqrt(3^(2n+1)).

Original entry on oeis.org

2, 10, 30, 98, 270, 818, 2382, 7282, 21818, 65650, 196406, 589982, 1768938, 5309294, 15924930, 47779238, 143322850, 429998586, 1289970842, 3869957114, 11609762666, 34829416842, 104488103446

Offset: 1

Wouter Meeussen, Oct 26 2001

Limit_{n->oo} a(n)/3^n = 1.11... - A.H.M. Smeets, Oct 25 2017

R. K. Guy, personal communication.

Cf. A003285, A013708, A059927.

Table[Length[Last[ContinuedFraction[Sqrt[3^(2n+1)] ]]], {n, 10}]

a(n) = A003285(A013708(n)). - Michel Marcus, Sep 25 2019

a(11)-a(13) from A.H.M. Smeets, Sep 28 2017
a(14) from A.H.M. Smeets, Oct 08 2017
a(15)-a(19) from Daniel Suteu, Jan 24 2019
a(20) from Chai Wah Wu, Sep 23 2019
a(21)-a(23) from Chai Wah Wu, Sep 25 2019

A062328 Length of period of continued fraction expansion of square root of 3^n+1.

Original entry on oeis.org

1, 0, 1, 4, 1, 26, 1, 56, 1, 44, 1, 264, 1, 814, 1, 136, 1, 3730, 1, 20968, 1, 2448, 1, 287980, 1, 397238, 1, 2678, 1, 670896, 1, 8110044, 1, 20696, 1, 1066520, 1, 366601254, 1, 277444, 1, 5903828476, 1, 7701738148, 1, 8208058, 1, 30287795640, 1, 253244432640, 1, 11656644672, 1, 2376211301858, 1, 590009437260, 1

Offset: 0

Labos Elemer, Jul 13 2001

a(n) = 1 iff n is even. In this case, 3^n + 1 = A002522(3^(n/2)) and the continued fraction expansion of sqrt(3^n+1) is {3^(n/2); 2*3^(n/2), 2*3^(n/2), 2*3^(n/2), 2*3^(n/2), ...}. - Bernard Schott, Sep 25 2019

			The period of sqrt(244) contains 26 terms: [1, 1, 1, 1, 1, 2, 1, 5, 1, 1, 9, 1, 6, 1, 9, 1, 1, 5, 1, 2, 1, 1, 1, 1, 1, 30], so a(5) = 26.

Cf. A003285, A034472, A059866, A059926, A059927, A062345, A062682.

with(numtheory): [seq(nops(cfrac(sqrt(3^k+1),'periodic','quotients')[2]),k=2..18)];

Table[Length[Last[ContinuedFraction[Sqrt[3^w+1]]]],{w,1,40}] (* corrected by Harvey P. Dale, Dec 05 2014 *)

a(n) = A003285(A034472(n)). - Bernard Schott, Sep 25 2019

More terms from Harvey P. Dale, Dec 05 2014
a(41)-a(42) from Vaclav Kotesovec, Sep 17 2019
a(0), a(43)-a(48) from Chai Wah Wu, Sep 25 2019
a(49)-a(56) from Chai Wah Wu, Oct 03 2019

A062345 Length of period of continued fraction expansion of square root of 3^n-1.

Original entry on oeis.org

1, 2, 1, 2, 10, 2, 19, 2, 25, 2, 156, 2, 149, 2, 580, 2, 716, 2, 6461, 2, 2485, 2, 123256, 2, 64, 2, 8638, 2, 722190, 2, 3804214, 2, 1783536, 2, 3550696, 2, 86022946, 2, 22119349, 2, 692630166, 2, 8247763078, 2, 43380360, 2, 15150768502, 2, 10229872316, 2, 36580802370, 2, 333495606762, 2, 676122216162, 2

Offset: 1

Labos Elemer, Jul 13 2001

			The period of sqrt(242) contains 10 terms: [1,1,3,1,14,1,3,1,1,30]

Cf. A003285, A024023, A059866, A059926, A059927, A062328.

with(numtheory): [seq(nops(cfrac(sqrt(3^k-1),'periodic','quotients')[2]),k=1..16)];

Table[Length[Last[ContinuedFraction[Sqrt[ -1+3^u]]]], {u, 1, 36}]

a(n) = A003285(A024023(n)). - Michel Marcus, Sep 25 2019

a(37)-a(42) from Vaclav Kotesovec, Aug 28 2019
a(43)-a(44) from Vaclav Kotesovec, Sep 17 2019
a(45)-a(52) from Chai Wah Wu, Sep 25 2019
a(53)-a(56) from Chai Wah Wu, Sep 29 2019

A294226 Length of period of continued fraction expansion of sqrt(3*2^n).

Original entry on oeis.org

2, 2, 2, 2, 2, 4, 4, 8, 8, 12, 16, 32, 36, 60, 72, 128, 136, 244, 292, 508, 576, 972, 1120, 1992, 2272, 3948, 4588, 7924, 9056, 15764, 18132, 31832, 36444, 63216, 72808, 126456, 145332, 253112, 290968, 507096, 581952, 1012312, 1163452, 2026504, 2327844, 4051424, 4656388

Offset: 0

A.H.M. Smeets, Oct 25 2017

nonn

Lim {n->inf} a(2n)/2^n = 0.555...
Lim {n->inf} a(2n+1)/2^n = 0.966...
It seems that Lim {n->inf} a(2n+1)/a(2n) = sqrt(3).
It seems that Lim {n->inf} a(n)/2^n = (Lim {n -> inf} A064932(n)/3^n)/2.

Chai Wah Wu, Table of n, a(n) for n = 0..80 (n = 0..46 from A.H.M. Smeets)

Cf. A003285, A007283, A059927, A064932, A293028.

Array[Length@ Last@ ContinuedFraction@ Sqrt[3*2^#] &, 47, 0] (* Michael De Vlieger, Oct 25 2017 *)

# for odd n
m, p, q = 0, 6, 2
tl, nl, tb, nb = 3, 1, 2, 1
while nl < 10**100000000:
    tl = tl * nb + tb * nl
    nl = 2 * nl * nb
    nb = tl
    tb = p * nl
tl = tl *nb + tb * nl
nl = 2 * nl * nb
tel, noe = tl, nl
while m >= 0:
    tl = tel*q**m
    nl = noe
    a0 = tl//nl
    t = 0
    an = a0
    while an != 2*a0:
        tl = tl - an*nl
        tl, nl = nl, tl
        an = tl//nl
        t = t + 1
    print(2*m+1, t)
    m = m+1

a(n) = A003285(A007283(n)). - Michel Marcus, Oct 02 2019

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A059866 Period of the continued fraction for sqrt(2^n-1).

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A064932 Period of the continued fraction for sqrt(3^(2n+1)).

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A062328 Length of period of continued fraction expansion of square root of 3^n+1.

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A062345 Length of period of continued fraction expansion of square root of 3^n-1.

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A294226 Length of period of continued fraction expansion of sqrt(3*2^n).

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