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Numerator of fraction equal to the continued fraction [ 0, 1, 4, ..., n^2 ].
+10
10
1, 4, 37, 596, 14937, 538328, 26393009, 1689690904, 136891356233, 13690825314204, 1656726754374917, 238582343455302252, 40322072770700455505, 7903364845400744581232, 1778297412287938231232705, 455252040910557587940153712, 131569618120563430852935655473
FORMULA
a(n) ~ c * n^(2*n + 1) / exp(2*n), where c = 6.5347337470474831902516177263695578212049901774805425962967688345920604685... - Vaclav Kotesovec, Aug 14 2021
MATHEMATICA
Table[Numerator[FromContinuedFraction[Range[0, n]^2]], {n, 1, 20}] (* Vaclav Kotesovec, Aug 14 2021 *)
PROG
(PARI) 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
Decimal expansion of number with continued fraction expansion 0, 1, 4, 9, ... (the squares).
+10
9
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
EXAMPLE
0.80431856111715795076768044139...
MATHEMATICA
RealDigits[FromContinuedFraction[Range[0, 100]^2], 10, 120][[1]] (* Harvey P. Dale, May 07 2018 *)
PROG
(PARI) dec_exp(v)= w=contfracpnqn(v); w[1, 1]/w[2, 1]+0.
dec_exp(vector(2000, i, (i-1)^2))
CROSSREFS
The numerators and denominators of convergents to this constant are given by A036245 and A036246 respectively.
a(n) = F(n)*a(n-1) + a(n-2) with a(0) = 0, a(1) = 1.
+10
4
0, 1, 1, 3, 10, 53, 434, 5695, 120029, 4086681, 224887484, 20019072757, 2882971364492, 671752346999393, 253253517790135653, 154485317604329747723, 152477261728991251138254, 243506341466516632397539361, 629220538826740707106492847078
COMMENTS
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
FORMULA
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
EXAMPLE
a(5) = 53 = F(5)*a(4) + a(3) = 5*10 + 3.
MAPLE
a:= proc(n) option remember; `if`(n<2, n,
combinat[fibonacci](n)*a(n-1)+a(n-2))
end:
MATHEMATICA
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 *)
EXTENSIONS
Replaced n with n+1 where needed. - Seung Ju Lee, Aug 30 2020
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!).
+10
4
1, 1, 3, 19, 459, 55099, 39671739, 199945619659, 8061807424322619, 2925468678338137602379, 10615940739961495538937237819, 423754383328897950597328272711061579, 202979027621555455188781938315330372976764219
COMMENTS
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
REFERENCES
J. M. De Koninck, A. Mercier, 1001 problèmes en théorie classique des nombres. Collection ellipses (2004), p.115.
FORMULA
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
EXAMPLE
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!].
MAPLE
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 :
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).
+10
3
1, 17, 747, 66305, 9828200, 2185188193, 679919101029, 281956264747009, 150277722869740455, 100090028003500150001, 81458362232421250207824, 79539026883848399173231873, 91771878445323959814042316673
COMMENTS
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).
REFERENCES
J. M. De Koninck, A. Mercier, 1001 problèmes en théorie classique des nombres. Collection ellipses (2004), p. 115.
EXAMPLE
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].
MAPLE
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 :
MATHEMATICA
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(0) = 1, a(1) = 2; a(n) = n * (n+1) * a(n-1) + a(n-2).
+10
2
1, 2, 13, 158, 3173, 95348, 4007789, 224531532, 16170278093, 1455549559902, 160126621867313, 21138169636045218, 3297714589844921321, 600205193521411725640, 126046388354086307305721, 30251733410174235165098680, 8228597533955746051214146681, 2517981097123868465906693983066
COMMENTS
a(n) is the denominator of fraction equal to the continued fraction [0; 2, 6, 12, 20, 30, ..., n*(n+1)].
FORMULA
a(n) ~ c * n^(2*n + 2) / exp(2*n), where c = 6.9478401587876967481571909904361736371398357108358019737901443045685048723... - Vaclav Kotesovec, Aug 14 2021
EXAMPLE
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.
MATHEMATICA
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}]
Denominators of convergents to A309930, the constant whose continued fraction representation consists of the cubes, [0; 1, 8, 27, 64, ...].
+10
1
1, 1, 9, 244, 15625, 1953369, 421943329, 144728515216, 74101421733921, 54020081172543625, 54020155273965358921, 71900880689729065267476, 124244775852007098747557449, 272965844447740285677448982929, 749018401409375195906018756714625
FORMULA
a(0) = 1, a(1) = 1, a(n) = n^3*a(n-1) + a(n-2) for n >= 2.
EXAMPLE
Convergents to [0; 1, 8, 27, 64, ...]: 0, 1, 8/9, 217/244, 13896/15625, 1737217/1953369, 375252768/421943329, ...
PROG
(PARI) 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
Decimal expansion of continued fraction [1; 4, 9, 16, 25, ... n^2, ... ].
+10
1
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
MATHEMATICA
A365052 = RealDigits[FromContinuedFraction[Range[1, 50]^2], 10, #][[1]]&;
PROG
(PARI)
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)))};
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