|
| |
|
|
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
|
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. - Peter Bala, Dec 15 2015
|
|
|
LINKS
|
|
|
|
FORMULA
|
a(n) ~ c * d^n, where d = 1.40198938377739909105003523518827... and c = 0.34165269320144328278000954698... - Vaclav Kotesovec, Aug 25 2017
O.g.f. as a ratio of q-series: N(q)/D(q), where N(q) = Sum_{n >= 0} (-1)^n*q^(3*n^2+n)/((1-q^3)*(1-q^6)*...*(1-q^(3*n))) and D(q) = Sum_{n >= 0} (-1)^n*q^(3*n^2-2*n)/((1-q^3)*(1-q^6)*...*(1-q^(3*n))). Cf. A143951, A224704 and A206738.
D(q) has a simple real zero at x = 0.7132721628.... The constants c and d quoted in the above asymptotic approximation for a(n) are given by d = 1/x and c = - N(x)/(x*D'(x)), where the prime indicates differentiation w.r.t. q. (End)
|
|
|
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):
|
|
|
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-François 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
|
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
