A226728 -id:A226728 - cp's OEIS frontend

A227309 G.f.: 1/G(0) where G(k) = 1 - q^(k+1) / (1 - q^(k+2)/G(k+1) ).

1, 1, 1, 2, 3, 6, 10, 19, 34, 63, 115, 213, 391, 723, 1333, 2463, 4547, 8403, 15522, 28686, 53006, 97963, 181042, 334606, 618415, 1142994, 2112545, 3904592, 7216810, 13338856, 24654268, 45568784, 84225393, 155675230, 287737327, 531830605, 982993368, 1816887637, 3358192905

Offset: 0

Joerg Arndt, Jul 06 2013

nonn

Sums along falling diagonals of A161492 (skew Ferrers diagrams by area and number of columns). [Joerg Arndt, Mar 23 2014]

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
M. P. Delest, J. M. Fedou, Enumeration of skew Ferrers diagrams, Discrete Mathematics. vol.112, no.1-3, pp.65-79, (1993)

Cf. A049346 (g.f.: 1-1/G(0), G(k)= 1 + q^(k+1) / (1 - q^(k+1)/G(k+1) ) ).
Cf. A227310 (g.f.: 1/G(0), G(k) = 1 + (-q)^(k+1) / (1 - (-q)^(k+1)/G(k+1) ) ).
Cf. A226728 (g.f.: 1/G(0), G(k) = 1 + q^(k+1) / (1 - q^(k+1)/G(k+2) ) ).
Cf. A226729 (g.f.: 1/G(0), G(k) = 1 - q^(k+1) / (1 - q^(k+1)/G(k+2) ) ).
Cf. A006958 (g.f.: 1/G(0), G(k) = 1 - q^(k+1) / (1 - q^(k+1)/G(k+1) ) ).

nmax = 40; CoefficientList[Series[1/Fold[(1 - #2/#1) &, 1, Reverse[x^(Range[2, nmax] - Floor[Range[2, nmax]/2])]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 05 2017 *)

N = 66;  q = 'q + O('q^N);
G(k) = if(k>N, 1, 1 - q^(k+1) / (1 - q^(k+2) / G(k+1) ) );
Vec( 1 / G(0) )

/* formula from the Delest/Fedou reference with t=q: */
N=66;  q='q+O('q^N);  t=q;
qn(n) = prod(k=1, n, 1-q^k );
nm = sum(n=0, N, (-1)^n* q^(n*(n+1)/2) / ( qn(n) * qn(n+1) ) * (t*q)^(n+1) );
dn = sum(n=0, N, (-1)^n* q^(n*(n-1)/2) / ( qn(n)^2 ) * (t*q)^n );
v=Vec(nm/dn)

G.f.: 1/(1-q /(1-q^2/(1-q^2/(1-q^3/(1-q^3/(1-q^4/(1-q^4/(1-q^5/(1-q^5/(1-...) )) )) )) )) ).
G.f.: 1/x - Q(0)/(2*x), where Q(k)= 1 + 1/(1 - 1/(1 - 1/(2*x^(k+1)) + 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 09 2013
G.f.: 1/x - U(0)/x, where U(k)= 1 - x^(k+1)/(1 - x^(k+1)/U(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Aug 15 2013
G.f.: -W(0)/x, where W(k)= 1 - x^(k+1) - x^k - x^(2*k+2)/W(k+1); (continued fraction). - Sergei N. Gladkovskii, Aug 15 2013
G.f.: G(0) where G(k) = 1 - q/(q^(k+2) - 1 / G(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Jan 18 2016
a(n) ~ c * d^n, where d = 1.84832326133106924642685135202616091890310896530577301386219207630312784... and c = 0.244648950328338656997216931963422920467577616734159139510762093105072... - Vaclav Kotesovec, Sep 05 2017

A227310 G.f.: 1/G(0) where G(k) = 1 + (-q)^(k+1) / (1 - (-q)^(k+1)/G(k+1) ).

Original entry on oeis.org

1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 2, 1, 1, 3, 2, 3, 4, 4, 6, 7, 8, 11, 13, 16, 20, 24, 31, 37, 46, 58, 70, 88, 108, 133, 167, 204, 252, 315, 386, 479, 594, 731, 909, 1122, 1386, 1720, 2124, 2628, 3254, 4022, 4980, 6160, 7618, 9432, 11665, 14433, 17860, 22093, 27341, 33824, 41847, 51785, 64065, 79267

Offset: 0

Joerg Arndt, Jul 06 2013

nonn

Number of rough sandpiles: 1-dimensional sandpiles (see A186085) with n grains without flat steps (no two successive parts of the corresponding composition equal), see example. - Joerg Arndt, Mar 08 2014
The sequence of such sandpiles by base length starts (n>=0) 1, 1, 0, 1, 0, 2, 0, 5, 0, 14, 0, 42, 0, ... (A097331, essentially A000108 with interlaced zeros).  This is a consequence of the obvious connection to Dyck paths, see example. - Joerg Arndt, Mar 09 2014
a(n>=1) are the Dyck paths with area n between the x-axis and the path which return to the x-axis only once (at their end), whereas A143951 includes paths with intercalated touches of the x-axis. - R. J. Mathar, Aug 22 2018

			From _Joerg Arndt_, Mar 08 2014: (Start)
The a(21) = 7 rough sandpiles are:
:
:   1:      [ 1 2 1 2 1 2 1 2 1 2 3 2 1 ]  (composition)
:
:           o
:  o o o o ooo
: ooooooooooooo  (rendering of sandpile)
:
:
:   2:      [ 1 2 1 2 1 2 1 2 3 2 1 2 1 ]
:
:         o
:  o o o ooo o
: ooooooooooooo
:
:
:   3:      [ 1 2 1 2 1 2 3 2 1 2 1 2 1 ]
:
:       o
:  o o ooo o o
: ooooooooooooo
:
:
:   4:      [ 1 2 1 2 3 2 1 2 1 2 1 2 1 ]
:
:     o
:  o ooo o o o
: ooooooooooooo
:
:
:   5:      [ 1 2 3 2 1 2 1 2 1 2 1 2 1 ]
:
:   o
:  ooo o o o o
: ooooooooooooo
:
:
:   6:      [ 1 2 3 2 3 4 3 2 1 ]
:
:      o
:   o ooo
:  ooooooo
: ooooooooo
:
:
:   7:      [ 1 2 3 4 3 2 3 2 1 ]
:
:    o
:   ooo o
:  ooooooo
: ooooooooo
(End)
From _Joerg Arndt_, Mar 09 2014: (Start)
The A097331(9) = 14 such sandpiles with base length 9 are:
01:  [ 1 2 1 2 1 2 1 2 1 ]
02:  [ 1 2 1 2 1 2 3 2 1 ]
03:  [ 1 2 1 2 3 2 3 2 1 ]
04:  [ 1 2 1 2 3 2 1 2 1 ]
05:  [ 1 2 1 2 3 4 3 2 1 ]
06:  [ 1 2 3 2 1 2 3 2 1 ]
07:  [ 1 2 3 2 1 2 1 2 1 ]
08:  [ 1 2 3 2 3 2 1 2 1 ]
09:  [ 1 2 3 2 3 2 3 2 1 ]
10:  [ 1 2 3 4 3 2 1 2 1 ]
11:  [ 1 2 3 2 3 4 3 2 1 ]
12:  [ 1 2 3 4 3 2 3 2 1 ]
13:  [ 1 2 3 4 3 4 3 2 1 ]
14:  [ 1 2 3 4 5 4 3 2 1 ]
(End)

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Joerg Arndt)
A. M. Odlyzko and H. S. Wilf, The editor's corner: n coins in a fountain, Amer. Math. Monthly, 95 (1988), 840-843.

Cf. A049346 (g.f.: 1 - 1/G(0), where G(k)= 1 + q^(k+1) / (1 - q^(k+1)/G(k+1) ) ).
Cf. A226728 (g.f.: 1/G(0), where G(k) = 1 + q^(k+1) / (1 - q^(k+1)/G(k+2) ) ).
Cf. A226729 (g.f.: 1/G(0), where G(k) = 1 - q^(k+1) / (1 - q^(k+1)/G(k+2) ) ).
Cf. A006958 (g.f.: 1/G(0), where G(k) = 1 - q^(k+1) / (1 - q^(k+1)/G(k+1) ) ).
Cf. A227309 (g.f.: 1/G(0), where G(k) = 1 - q^(k+1) / (1 - q^(k+2)/G(k+1) ) ).
Cf. A173258, A291874.

N = 66;  q = 'q + O('q^N);
G(k) = if(k>N, 1, 1 + (-q)^(k+1) / (1 - (-q)^(k+1) / G(k+1) ) );
gf = 1 / G(0);
Vec(gf)

N = 66;  q = 'q + O('q^N);
F(q,y,k) = if(k>N, 1, 1/(1 - y*q^2 * F(q, q^2*y, k+1) ) );
Vec( 1 + q * F(q,q,0) ) \\ Joerg Arndt, Mar 09 2014

a(0) = 1 and a(n) = abs(A049346(n)) for n>=1.
G.f.: 1/ (1-q/(1+q/ (1+q^2/(1-q^2/ (1-q^3/(1+q^3/ (1+q^4/(1-q^4/ (1-q^5/(1+q^5/ (1+-...) )) )) )) )) )).
G.f.: 1 + q * F(q,q) where F(q,y) = 1/(1 - y * q^2 * F(q, q^2*y) ); cf. A005169 and p. 841 of the Odlyzko/Wilf reference; 1/(1 - q * F(q,q)) is the g.f. of A143951. - Joerg Arndt, Mar 09 2014
G.f.: 1 + q/(1 - q^3/(1 - q^5/(1 - q^7/ (...)))) (from formulas above). - Joerg Arndt, Mar 09 2014
G.f.: F(x, x^2) where F(x,y) is the g.f. of A239927. - Joerg Arndt, Mar 29 2014
a(n) ~ c * d^n, where d = 1.23729141259673487395949649334678514763130846902468... and c = 0.0773368373684184197215007198148835507944051447907... - Vaclav Kotesovec, Sep 05 2017
G.f.: A(x) = 2 -1/A143951(x). - R. J. Mathar, Aug 23 2018

A226729 G.f.: 1 / G(0), where G(k) = 1 - q^(k+1) / (1 - q^(k+1)/G(k+2) ).

Original entry on oeis.org

1, 1, 2, 4, 8, 17, 36, 76, 162, 345, 734, 1564, 3332, 7098, 15124, 32224, 68658, 146291, 311704, 664152, 1415124, 3015237, 6424636, 13689132, 29167776, 62148513, 132421414, 282153672, 601192008, 1280975135, 2729406380, 5815615784, 12391480916, 26402844538, 56257214530, 119868682488

Offset: 0

Joerg Arndt, Jun 29 2013

nonn

What does this sequence count?
Conjectures from John Tyler Rascoe, Nov 04 2023: (Start)
a(n) is the number of integer compositions of n into two kinds of odd parts with the following restrictions. Each composition has first part 1a. For all a parts pa_i >= px_{i+1} and for all b parts pb_i >= px_{i+1} or pb_i = (p+2)a_{i+1}.
In general if B(i) = b_1, b_2, ..., b_i is an infinite sequence where b_1 > 0 and b_i <= b_{i+1} for all i, let A(q) = 1/(1-q^b_1/(1-q^b_2/(1-q^b_3/(1-...)))) be a generating function where the exponents of q are the sequence B(i).
Then A(q) counts integer compositions into parts b_i with the following restrictions. Every composition has first part p_1 = b_1 and for every pair of parts (p_j,p_{j+1}), B^-1(p_j) + 1 >= B^-1(p_{j+1}). Where j is the position of the part p_j within the composition itself and B^-1(p_j) is the index of p_j in B(i). (End)

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A226728 (g.f.: 1/G(0), G(k) = 1 + q^(k+1) / (1 - q^(k+1)/G(k+2) ) ).
Cf. A227309 (g.f.: 1/G(0), G(k) = 1 - q^(k+1) / (1 - q^(k+2)/G(k+1) ) ).
Cf. A129183.

nmax = 50; CoefficientList[Series[1/Fold[(1 - #2/#1) &, 1, Reverse[x^(2*Range[nmax + 1] - 2*Floor[Range[nmax + 1]/2] - 1)]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 05 2017 *)

N = 66;  q = 'q + O('q^N);
G(k) = if(k>N, 1, 1 - q^(k+1) / (1 - q^(k+1) / G(k+2) ) );
gf = 1 / G(0)
Vec(gf)

G.f.: 1/(1-q/(1-q/(1-q^3/(1-q^3/(1-q^5/(1-q^5/(1-q^7/(1-q^7/(1-...))))))))).
G.f.: 1/W(0), where W(k)= 1 - x^(2*k+1)/(1 - x^(2*k+1)/W(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Aug 16 2013
a(n) ~ c * d^n, where d = 2.13072551790181698200128321720925945740967671226348407873633962907725871... and c = 0.38040216799237980431596440625527448705929594287571043849218282414099437... - Vaclav Kotesovec, Sep 05 2017
Conjecture: a(n) = Sum_{i=0..floor((n-sqrt(2*n-1))/2)} A129183(n-(2*i),n-i). - John Tyler Rascoe, Nov 04 2023

A227045 G.f.: 1/(1 - q/G(0)) where G(k) = 1 - q^(k+1) / (1 - q^(k+1) / G(k+1) ).

Original entry on oeis.org

1, 1, 2, 5, 13, 35, 95, 260, 713, 1959, 5386, 14815, 40759, 112151, 308609, 849240, 2337009, 6431246, 17698332, 48704714, 134032593, 368850417, 1015056867, 2793383746, 7687248186, 21154913043, 58217239536, 160210872557, 440892153268, 1213312738702, 3338974845151, 9188688696438

Offset: 0

Joerg Arndt, Jul 06 2013

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A006958 (g.f.: 1/G(0), where G(k) = 1 - q^(k+1) / (1 - q^(k+1)/G(k+1) ) ).
Cf. A226729 (g.f.: 1/G(0), where G(k) = 1 - q^(k+1) / (1 - q^(k+1)/G(k+2) ) ).
Cf. A226728 (g.f.: 1/G(0), where G(k) = 1 + q^(k+1) / (1 - q^(k+1)/G(k+2) ) ).
Cf. A227309 (g.f.: 1/G(0), where G(k) = 1 - q^(k+1) / (1 - q^(k+2)/G(k+1) ) ).
Cf. A227310 (g.f.: 1/G(0), where G(k) = 1 + (-q)^(k+1) / (1 - (-q)^(k+1)/G(k+1) ) ).

nmax = 40; CoefficientList[Series[1/(1 - x/Fold[(1 - #2/#1) &, 1, Reverse[x^(Range[nmax + 2] - Floor[Range[nmax + 2]/2])]]), {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 05 2017 *)

N = 66;  q = 'q + O('q^N);
G(k) = if(k>N, 1, 1 - q^(k+1) / (1 - q^(k+1) / G(k+1) ) );
gf = 1 /(1- q/G(0));
Vec(gf)

G.f.: 1/(1-q/ (1-q/(1-q/ (1-q^2/(1-q^2/ (1-q^3/(1-q^3/ (1-q^4/(1-q^4/ (1-q^5/(1-q^5/ (1-...))))))))))) ).
G.f. A(x) = 1/(1 - B(x)) where B(x) is the g.f. of A006958.
a(n) ~ c * d^n, where d = 2.751949072495748078279227332764623096815571855905843246297955690122791154... and c = 0.215973947378529032758849789768859077066690378163074586384819930605436492... - Vaclav Kotesovec, Sep 05 2017

cp's OEIS Frontend

A227309 G.f.: 1/G(0) where G(k) = 1 - q^(k+1) / (1 - q^(k+2)/G(k+1) ).

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A227310 G.f.: 1/G(0) where G(k) = 1 + (-q)^(k+1) / (1 - (-q)^(k+1)/G(k+1) ).

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A226729 G.f.: 1 / G(0), where G(k) = 1 - q^(k+1) / (1 - q^(k+1)/G(k+2) ).

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A227045 G.f.: 1/(1 - q/G(0)) where G(k) = 1 - q^(k+1) / (1 - q^(k+1) / G(k+1) ).

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