A010859 -id:A010859 - cp's OEIS frontend

A040090 Continued fraction for sqrt(101).

10, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20

Offset: 0

N. J. A. Sloane

			10 + 1/(20 + 1/(20 + 1/(20 + 1/(20 + ...)))) = sqrt(101).

Cf. A248803 (decimal expansion), A041180/A041181 (convergents).

Cf. A040000, A040002, A040020.

Digits := 200: convert(evalf(sqrt(101)),confrac,150,'cvgts'):

ContinuedFraction[Sqrt[101],300] (* Vladimir Joseph Stephan Orlovsky, Mar 10 2011*)
PadRight[{10},100,{20}] (* Harvey P. Dale, Apr 22 2022 *)

From Elmo R. Oliveira, Feb 11 2024: (Start)
a(n) = 20 = A010859(n) for n >= 1.
G.f.: 10*(1+x)/(1-x).
E.g.f.: 20*exp(x) - 10.
a(n) = 10*A040000(n) = 5*A040002(n) = 2*A040020(n). (End)

A023018 Number of partitions of n into parts of 20 kinds.

Original entry on oeis.org

1, 20, 230, 1960, 13685, 82524, 443870, 2175800, 9869990, 41907380, 168012824, 640438680, 2334121995, 8171039800, 27580783270, 90058003200, 285253928790, 878572253720, 2636748302650, 7725084195240, 22130265931900, 62079251390180

Offset: 0

David W. Wilson

nonn

a(n) is Euler transform of A010859. - 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

20th column of A144064. - Alois P. Heinz, Oct 17 2008

with(numtheory): a:= proc(n) option remember; `if`(n=0, 1, add(add(d*20, 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]^20, {x, 0, 30}], x] (* Indranil Ghosh, Mar 27 2017 *)

Vec(1/eta(x)^20 + O(x^30)) \\ Indranil Ghosh, Mar 27 2017

G.f.: Product_{m>=1} 1/(1-x^m)^20.
a(0) = 1, a(n) = (20/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 = 20. - Vaclav Kotesovec, Jun 28 2025

A166577 Inverse binomial transform of A166517.

Original entry on oeis.org

1, 4, -5, 10, -20, 40, -80, 160, -320, 640, -1280, 2560, -5120, 10240, -20480, 40960, -81920, 163840, -327680, 655360, -1310720, 2621440, -5242880, 10485760, -20971520, 41943040, -83886080, 167772160, -335544320, 671088640, -1342177280, 2684354560, -5368709120

Offset: 0

Paul Curtz, Oct 17 2009

The definition assumes that the offset of A166517 is changed to 0.
A166517 mod 9 yields a periodic sequence with period 1, 5, 4, 8, 7, 2.
This set of numbers in the period appears also in A153130, A146501, and A166304.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-2).

Cf. A010716, A010692, A010859.

Join[{1,4},NestList[-2#&,-5,40]] (* Harvey P. Dale, Aug 02 2012 *)
Join[{1, 4}, LinearRecurrence[{-2}, {-5}, 48]] (* G. C. Greubel, May 17 2016 *)

a(n) = -2*a(n-1), n>2.
a(n) = (-1)^(n+1)*A020714(n-2), n>1.
From Colin Barker, Jan 07 2013: (Start)
a(n) = -5*(-1)^n*2^(n-2) for n>1.
G.f.: (3*x^2+6*x+1)/(2*x+1). (End)
E.g.f.: (9/4) + (3/2)*x - (5/4)*exp(-2*x). - Alejandro J. Becerra Jr., Feb 15 2021

Edited, comments not concerning this sequence removed, and extended by R. J. Mathar, Oct 21 2009

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A040090 Continued fraction for sqrt(101).

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A023018 Number of partitions of n into parts of 20 kinds.

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A166577 Inverse binomial transform of A166517.

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