A036246 -id:A036246 - cp's OEIS frontend

A036245 Numerator of fraction equal to the continued fraction [ 0, 1, 4, ..., n^2 ].

1, 4, 37, 596, 14937, 538328, 26393009, 1689690904, 136891356233, 13690825314204, 1656726754374917, 238582343455302252, 40322072770700455505, 7903364845400744581232, 1778297412287938231232705, 455252040910557587940153712, 131569618120563430852935655473

Offset: 1

Jeff Burch

Seiichi Manyama, Table of n, a(n) for n = 1..253

Cf. A036246, A073824.

Table[Numerator[FromContinuedFraction[Range[0, n]^2]], {n, 1, 20}] (* Vaclav Kotesovec, Aug 14 2021 *)

A036245(n) = my(v=vector(n+1)); for(i=1, n+1, if(i==1, v[i]=0, if(i==2, v[i]=1, v[i]=(i-1)^2*v[i-1]+v[i-2]))); v[n+1] \\ Jianing Song, Nov 30 2019

a(n) = n^2 * a(n-1) + a(n-2) for n > 2. - Seiichi Manyama, Jun 05 2018
Lim_{n->oo} a(n)/A036246(n) = A073824. - Jianing Song, Nov 30 2019
a(n) ~  c * n^(2*n + 1) / exp(2*n), where c = 6.5347337470474831902516177263695578212049901774805425962967688345920604685... - Vaclav Kotesovec, Aug 14 2021

More terms from Seiichi Manyama, Jun 05 2018

A073824 Decimal expansion of number with continued fraction expansion 0, 1, 4, 9, ... (the squares).

Original entry on oeis.org

8, 0, 4, 3, 1, 8, 5, 6, 1, 1, 1, 7, 1, 5, 7, 9, 5, 0, 7, 6, 7, 6, 8, 0, 4, 4, 1, 3, 9, 3, 4, 1, 9, 2, 9, 9, 0, 5, 7, 3, 2, 7, 2, 7, 3, 9, 6, 4, 1, 9, 6, 8, 5, 2, 8, 5, 5, 9, 9, 2, 7, 3, 5, 9, 5, 1, 9, 1, 4, 5, 4, 5, 5, 3, 1, 0, 2, 2, 7, 8, 0, 8, 2, 1, 3, 4, 5, 3, 2, 4, 9, 3, 1, 1, 2, 0, 2, 0, 4, 4, 9, 3, 3, 9

Offset: 0

Rick L. Shepherd, Aug 12 2002

			0.80431856111715795076768044139...

Kerry Mitchell, Table of n, a(n) for n = 0..1494

Cf. A000290 (squares), A052119, A309930, A214070.

The numerators and denominators of convergents to this constant are given by A036245 and A036246 respectively.

RealDigits[FromContinuedFraction[Range[0,100]^2],10,120][[1]] (* Harvey P. Dale, May 07 2018 *)

dec_exp(v)= w=contfracpnqn(v); w[1,1]/w[2,1]+0.
dec_exp(vector(2000,i,(i-1)^2))

A135829 a(n) = F(n)*a(n-1) + a(n-2) with a(0) = 0, a(1) = 1.

Original entry on oeis.org

0, 1, 1, 3, 10, 53, 434, 5695, 120029, 4086681, 224887484, 20019072757, 2882971364492, 671752346999393, 253253517790135653, 154485317604329747723, 152477261728991251138254, 243506341466516632397539361, 629220538826740707106492847078

Offset: 0

Gary W. Adamson, Nov 29 2007

Essentially the same as A071895. [R. J. Mathar, Oct 28 2008]
From Michel Lagneau, Apr 12 2010: (Start)
Determinant of n+1 X n+1 matrix: ((F(0),-1,0,...,0),(1,F(1),-1,0,...,0),(0,1,F(2),-1,0,...,0),...,(0,0,...,1,F(n)). Each determinant is the numerator of the fraction x(n)/y(n) equal to the continued fraction expansion of the diagonal elements [F(0), F(1), ..., F(n)] of the n+1 X n+1 matrix. The value x(n) is obtained by computing the determinant det(n+1 X n+1) from the last column. The value y(n) is obtained by computing this determinant after removal of the first row and the first column (see example below).
The sequence A001040 give the values of each determinant with numerator of continued fraction given by the expansion of the diagonal elements [n,n-1,...,3,2,1]. The same is true for the sequence A084845 with the expansion of the diagonal elements [n,n,...,n], and the sequence A036246 for the elements[ 0, 1, 4, ..., n^2 ].
Examples:
for n = 0, det[0] = 0; for n = 1, det(([[0,-1],[1,1]]) = 1;
for n = 2, det([[0,-1, 0],[1,1,-1],[0,1,1]]))=1;
for n = 3, det([[0,-1, 0,0],[1,1,-1,0],[0,1,1,-1],[0,0,1,2]])) = 3, and the continued fraction expansion is 3/det(([[1,-1, 0],[1,1,-1],[0,1,2]])) = 5/3 = 0 + 1 + 1/(1 + 1/2) => [0,1,1,2]. (End)
a(n) is the denominator of the continued fraction [F(1), F(2), ..., F(n)] for n > 0. - Seung Ju Lee, Aug 23 2020

			a(5) = 53 = F(5)*a(4) + a(3) = 5*10 + 3.

G. C. Greubel, Table of n, a(n) for n = 0..98

Cf. A000045.

Cf. A176232, A176233, A001040, A084845, A036246, A036245, A330638.

a:= proc(n) option remember; `if`(n<2, n,
      combinat[fibonacci](n)*a(n-1)+a(n-2))
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Jan 24 2021

RecurrenceTable[{a[0]==0,a[1]==1,a[n]==Fibonacci[n]*a[n-1]+a[n-2]}, a,{n,0,20}] (* Harvey P. Dale, Apr 26 2012 *)

a(n) = (-a(n-1)*a(n-4)*a(n-2) - a(n-1)*a(n-3)^2 + a(n-1)^2*a(n-3) + a(n-2)^2*a(n-3) + a(n-1)*a(n-2)^2)/(a(n-2)*a(n-3)). - Robert Israel, Dec 04 2016

a(n) ~ c * ((1 + sqrt(5))/2)^(n*(n+1)/2) / 5^(n/2), where c = 2.25240516839867905756631574518868900987391688308922490621152619277084562178... - Vaclav Kotesovec, Dec 29 2019

More terms from Michel Lagneau, Apr 12 2010
Offset changed by N. J. A. Sloane, Apr 21 2010
Replaced n with n+1 where needed. - Seung Ju Lee, Aug 30 2020
Incorrect program removed by Alois P. Heinz, Jan 24 2021

A176232 Determinant of the n X n matrix with rows (1!,-1,0,...,0), (1, 2!,-1,0,...,0), (0,1,3!,-1,0,...,0), ..., (0,0,...,1,n!).

Original entry on oeis.org

1, 1, 3, 19, 459, 55099, 39671739, 199945619659, 8061807424322619, 2925468678338137602379, 10615940739961495538937237819, 423754383328897950597328272711061579, 202979027621555455188781938315330372976764219

Offset: 0

Michel Lagneau, Apr 12 2010

nonn

Each determinant is the numerator of the fraction x(n)/y(n) = [1!, 2!, ..., n!] (the simple continued fraction). The value x(n) is obtained by computing the determinant det(n X n) from the last column. The value y(n) is obtained by computing this determinant after removal of the first row and the first column (see example below).

Also denominator of fraction equal to the continued fraction [ 0; 1!, 2!, ... , n! ]. - Seiichi Manyama, Jun 05 2018

			For n = 1, det[1] = 1.
For n = 2, det([[1,-1],[1,2]]) = 3, and the continued fraction expansion is 3/2 = [1!,2!].
For n = 3, det([[1,-1, 0],[1,2,-1],[0,1,6]]) = 19, and the continued fraction expansion is 19/det([[2,-1],[1,6]]) = 19/13 = [1!,2!,3!].
For n = 4, det([[1,-1,0,0],[1,2,-1,0],[0,1,6,-1],[0,0,1,24]]) = 459, and the continued fraction expansion is 459/det([[2,-1,0],[1,6,-1],[0,1,24]]) = 459/314 = [1!,2!,3!,4!].

J. M. De Koninck, A. Mercier, 1001 problèmes en théorie classique des nombres. Collection ellipses (2004), p.115.

Seiichi Manyama, Table of n, a(n) for n = 0..43

Cf. A001040, A036245, A036246, A084845.

for n from 15 by -1 to 1 do:x0:=n!:for p from n by -1 to 2 do : x0:= (p-1)! + 1/x0 :od:print(x0):od :

a(0) = 1, a(1) = 1, a(n) = n! * a(n-1) + a(n-2). - Daniel Suteu, Dec 20 2016

a(n) ~ c * BarnesG(n+2), where c = 1.5943186620010986362991550255196986158205795892595646967623357407966... - Vaclav Kotesovec, Jun 05 2018

a(0)=1 prepended by Alois P. Heinz, Dec 20 2016

A176233 Determinant of n X n matrix with rows (n^2,-1,0,...,0), (1,n^2,-1, 0,...,0), (0,1,n^2,-1,0,...,0), ...,(0,0,...,1,n^2).

Original entry on oeis.org

1, 17, 747, 66305, 9828200, 2185188193, 679919101029, 281956264747009, 150277722869740455, 100090028003500150001, 81458362232421250207824, 79539026883848399173231873, 91771878445323959814042316673

Offset: 1

Michel Lagneau, Apr 12 2010

nonn

Each determinant is the numerator of the fraction x(n)/y(n) = [n^2, n^2, ..., n^2] (simple continued fraction). The value x(n) is obtained by computing the determinant det(n X n) along the last column. The value y(n) is obtained by computing this determinant after removal of the first row and the first column (see example below).

			For n = 1, det[1] = 1.
For n = 2, det([[4,-1],[1,4]]) = 17, and the continued fraction expansion is 17/4 = [2^2,2^2].
For n = 3, det([[9,-1,0],[1,9,-1],[0,1,9]]) = 747, and the continued fraction expansion is 747/det([[9,-1],[1,9]]) = 747/82 = [3^2,3^2,3^2].

J. M. De Koninck, A. Mercier, 1001 problèmes en théorie classique des nombres. Collection ellipses (2004), p. 115.

Cf. A001040, A036245, A036246, A084845, A176232.

for n from 15 by -1 to 1 do x0:=n^2: for p from n by -1 to 2 do : x0:= n^2 + 1/x0 :od: print(x0): od :

nmax = 20; Do[x0 = n^2; Do[x0 = n^2 + 1/x0, {p, n, 2, -1}]; a[n] = Numerator[x0];, {n, nmax, 1, -1}]; Table[a[n], {n, 1, nmax}] (* Vaclav Kotesovec, Dec 29 2019 *)

a(n) ~ n^(2*n). - Vaclav Kotesovec, Dec 29 2019

A347051 a(0) = 1, a(1) = 2; a(n) = n * (n+1) * a(n-1) + a(n-2).

Original entry on oeis.org

1, 2, 13, 158, 3173, 95348, 4007789, 224531532, 16170278093, 1455549559902, 160126621867313, 21138169636045218, 3297714589844921321, 600205193521411725640, 126046388354086307305721, 30251733410174235165098680, 8228597533955746051214146681, 2517981097123868465906693983066

Offset: 0

Ilya Gutkovskiy, Aug 13 2021

a(n) is the denominator of fraction equal to the continued fraction [0; 2, 6, 12, 20, 30, ..., n*(n+1)].

			a(1) =    2 because 1/(1*2)                               = 1/2.
a(2) =   13 because 1/(1*2 + 1/(2*3))                     = 6/13.
a(3) =  158 because 1/(1*2 + 1/(2*3 + 1/(3*4)))           = 73/158.
a(4) = 3173 because 1/(1*2 + 1/(2*3 + 1/(3*4 + 1/(4*5)))) = 1466/3173.

Cf. A001040, A001046, A002378, A036246, A071896, A102038, A346960, A347052.

a[0] = 1; a[1] = 2; a[n_] := a[n] = n (n + 1) a[n - 1] + a[n - 2]; Table[a[n], {n, 0, 17}]
Table[Denominator[ContinuedFractionK[1, k (k + 1), {k, 1, n}]], {n, 0, 17}]

a(n) ~ c * n^(2*n + 2) / exp(2*n), where c = 6.9478401587876967481571909904361736371398357108358019737901443045685048723... - Vaclav Kotesovec, Aug 14 2021

A329305 Denominators of convergents to A309930, the constant whose continued fraction representation consists of the cubes, [0; 1, 8, 27, 64, ...].

Original entry on oeis.org

1, 1, 9, 244, 15625, 1953369, 421943329, 144728515216, 74101421733921, 54020081172543625, 54020155273965358921, 71900880689729065267476, 124244775852007098747557449, 272965844447740285677448982929, 749018401409375195906018756714625

Offset: 0

Jianing Song, Nov 30 2019

			Convergents to [0; 1, 8, 27, 64, ...]: 0, 1, 8/9, 217/244, 13896/15625, 1737217/1953369, 375252768/421943329, ...

Eric Weisstein's World of Mathematics, Continued Fraction

Cf. A309930, A329304 (numerators), A001040, A036246.

A329305_up_to_n(n) = my(v=vector(n+1)); for(i=1, n+1, if(i==1, v[i]=1, if(i==2, v[i]=1, v[i]=(i-1)^3*v[i-1]+v[i-2]))); v

a(0) = 1, a(1) = 1, a(n) = n^3*a(n-1) + a(n-2) for n >= 2.

A365052 Decimal expansion of continued fraction [1; 4, 9, 16, 25, ... n^2, ... ].

Original entry on oeis.org

1, 2, 4, 3, 2, 8, 8, 4, 7, 8, 3, 9, 9, 7, 1, 5, 6, 4, 4, 0, 8, 2, 4, 9, 6, 5, 4, 5, 3, 9, 4, 4, 2, 9, 4, 9, 9, 2, 3, 1, 2, 0, 0, 2, 6, 1, 1, 9, 7, 4, 4, 6, 8, 8, 5, 0, 6, 6, 4, 9, 7, 4, 5, 9, 8, 8, 1, 6, 3, 0, 3, 2, 2, 3, 3, 8, 2, 5, 3, 4, 2, 1, 4, 5, 9, 6, 4, 9, 8, 1, 5, 6, 1, 2, 1, 8, 5, 5, 9, 5

Offset: 1

Rok Cestnik, Aug 18 2023

			1.243288478399715644...

Cf. A073824 (reciprocal), A036246/A036245 (convergents).

Cf. A226771, A060997, A052119.

A365052 = RealDigits[FromContinuedFraction[Range[1,50]^2],10,#][[1]]&;

p(N) = my(m=contfracpnqn(vector(N, i, i^2))); m[1,1]/m[2,1];
A365052(N) = {my(t=2); while(floor(10^N*p(t)) != floor(10^N*p(t+1)), t++); digits(floor(10^(N-1)*p(t)))};

Equals 1/A073824.

cp's OEIS Frontend

A036245 Numerator of fraction equal to the continued fraction [ 0, 1, 4, ..., n^2 ].

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A073824 Decimal expansion of number with continued fraction expansion 0, 1, 4, 9, ... (the squares).

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A135829 a(n) = F(n)*a(n-1) + a(n-2) with a(0) = 0, a(1) = 1.

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A176232 Determinant of the n X n matrix with rows (1!,-1,0,...,0), (1, 2!,-1,0,...,0), (0,1,3!,-1,0,...,0), ..., (0,0,...,1,n!).

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A176233 Determinant of n X n matrix with rows (n^2,-1,0,...,0), (1,n^2,-1, 0,...,0), (0,1,n^2,-1,0,...,0), ...,(0,0,...,1,n^2).

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A347051 a(0) = 1, a(1) = 2; a(n) = n * (n+1) * a(n-1) + a(n-2).

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A329305 Denominators of convergents to A309930, the constant whose continued fraction representation consists of the cubes, [0; 1, 8, 27, 64, ...].

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