A010853 -id:A010853 - cp's OEIS frontend

A040042 Continued fraction for sqrt(50) = 5*sqrt(2).

7, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14

Offset: 0

N. J. A. Sloane

			7.07106781186547524400844... = 7 + 1/(14 + 1/(14 + 1/(14 + 1/(14 + ...)))). - _Harry J. Smith_, Jun 01 2009

James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 276.

Harry J. Smith, Table of n, a(n) for n = 0..20000
G. Xiao, Contfrac
Index entries for continued fractions for constants
Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A010503 (decimal expansion), A041084/A041085 (convergents), A248275 (Egyptian fraction).

Cf. A040000.

Digits := 100: convert(evalf(sqrt(N)),confrac,90,'cvgts'):

ContinuedFraction[Sqrt[50],300] (* Vladimir Joseph Stephan Orlovsky, Mar 07 2011 *)

{ allocatemem(932245000); default(realprecision, 47000); x=contfrac(sqrt(50)); for (n=0, 20000, write("b040042.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 01 2009

From Elmo R. Oliveira, Feb 07 2024: (Start)
a(n) = 14 = A010853(n) for n >= 1.
G.f.: 7*(1+x)/(1-x).
E.g.f.: 14*exp(x) - 7.
a(n) = 7*A040000(n). (End)

A023012 Number of partitions of n into parts of 14 kinds.

Original entry on oeis.org

1, 14, 119, 770, 4165, 19754, 84602, 333608, 1228080, 4263770, 14071827, 44420796, 134793918, 394805110, 1119974875, 3086034350, 8280022023, 21678277754, 55486209625, 139065013640, 341779759755, 824753397814, 1956347387428

Offset: 0

David W. Wilson

nonn

a(n) is Euler transform of A010853. - Alois P. Heinz, Oct 17 2008

Seiichi Manyama, Table of n, a(n) for n = 0..1000
N. J. A. Sloane, Transforms
Index entries for expansions of Product_{k >= 1} (1-x^k)^m

14th column of A144064. - Alois P. Heinz, Oct 17 2008

with(numtheory): a:= proc(n) option remember; `if`(n=0, 1, add(add(d*14, d=divisors(j)) *a(n-j), j=1..n)/n) end: seq(a(n), n=0..40); # Alois P. Heinz, Oct 17 2008

CoefficientList[Series[1/QPochhammer[x]^14, {x, 0, 30}], x] (* Indranil Ghosh, Mar 27 2017 *)

Vec(1/eta(x)^14 + O(x^30)) \\ Indranil Ghosh, Mar 27 2017

a(0) = 1, a(n) = (14/n)*Sum_{k=1..n} A000203(k)*a(n-k) for n > 0. - Seiichi Manyama, Mar 27 2017

a(n) ~ m^((m+1)/4) * exp(Pi*sqrt(2*m*n/3)) / (2^((3*m+5)/4) * 3^((m+1)/4) * n^((m+3)/4)) * (1 - ((9+Pi^2)*m^2+36*m+27) / (24*Pi*sqrt(6*m*n))), set m = 14. - Vaclav Kotesovec, Jun 28 2025

cp's OEIS Frontend

A040042 Continued fraction for sqrt(50) = 5*sqrt(2).

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