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A206737 G.f.: 1/(1 - x/(1 - x^4/(1 - x^7/(1 - x^10/(1 - x^13/(1 - x^16/(1 -...- x^(3*n-2)/(1 -...)))))))), a continued fraction. 2
1, 1, 1, 1, 1, 2, 3, 4, 5, 7, 10, 14, 20, 28, 39, 54, 76, 107, 150, 210, 294, 412, 578, 811, 1137, 1593, 2233, 3131, 4390, 6155, 8629, 12097, 16959, 23777, 33336, 46737, 65524, 91863, 128790, 180563, 253149, 354912, 497581, 697602, 978031, 1371190, 1922395 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,6

COMMENTS

From Peter Bala, 15 Dec 2015: (Start)

We have the simple continued fraction expansions (A(x) is the sequence o.g.f.): A(1/n) = [1; n - 2, 1, n^3 - 2, 1, n^4 - 2, 1, n^6 - 2, 1, n^7 - 2, 1, n^9 - 2, 1, n^10 - 2, 1, ...] for n >= 3 and A(-1/n) = [0; 1, n - 1, 1, n^3 - 1, n^4 - 1, 1, n^6 - 1, n^7 - 1, 1, n^9 - 1, n^10 - 1, 1, ...] for n >= 2. Cf. A005169, A111317 and A143951. (End)

LINKS

Robert Israel, Table of n, a(n) for n = 0..2000

P. Bala, Some simple continued fraction expansions associated with A206737

FORMULA

a(n) ~ c * d^n, where d = 1.40198938377739909105003523518827... and c = 0.34165269320144328278000954698... - Vaclav Kotesovec, Aug 25 2017

EXAMPLE

G.f.: A(x) = 1 + x + x^2 + x^3 + x^4 + 2*x^5 + 3*x^6 + 4*x^7 + 5*x^8 + ...

Simple continued fraction expansions: A(1/10) = 1.11112345816325284441923227158 ... = [1, 8, 1, 998, 1, 9998, 1, 999998, 1, 9999998, 1, 999999998, 1, 9999999998, 1, ...]; A(-1/10) = 0.909082643877542661578687284018 ... = [0, 1, 9, 1, 999, 9999, 1, 999999, 9999999, 1, 999999999, 9999999999, 1, ...]. - Peter Bala, Dec 15 2015

MAPLE

N:= 100: # to get a(0) .. a(N)

C:= [0, [1, 1], seq([-x^i, 1], i=1..N, 3)]:

S:= series(numtheory:-cfrac(C), x, N+1):

seq(coeff(S, x, j), j=0..N); # Robert Israel, Dec 28 2015

MATHEMATICA

max = 15; CF = 1+x*O[x]^max; M = Sqrt[max+1]//Floor; For[k=0, k <= M, k++, CF = 1/(1-x^(3M-3k+1)*CF)]; CoefficientList[CF, x] (* Jean-Fran├žois Alcover, Dec 29 2015, adapted from PARI *)

nmax = 50; CoefficientList[Series[1/Fold[(1 - #2/#1) &, 1, Reverse[x^(3*Range[nmax + 1]-2)]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 25 2017 *)

PROG

(PARI) {a(n)=local(CF=1+x*O(x^n), M=sqrtint(n+1)); for(k=0, M, CF=1/(1-x^(3*M-3*k+1)*CF)); polcoeff(CF, n, x)}

for(n=0, 55, print1(a(n), ", "))

CROSSREFS

Cf. A206738, A143951, A005169, A111317.

Sequence in context: A295072 A206739 A107586 * A275174 A282582 A282502

Adjacent sequences:  A206734 A206735 A206736 * A206738 A206739 A206740

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Feb 12 2012

STATUS

approved

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Last modified February 20 07:15 EST 2018. Contains 299359 sequences. (Running on oeis4.)