A347901 -id:A347901 - cp's OEIS frontend

A005169 Number of fountains of n coins.

1, 1, 1, 2, 3, 5, 9, 15, 26, 45, 78, 135, 234, 406, 704, 1222, 2120, 3679, 6385, 11081, 19232, 33379, 57933, 100550, 174519, 302903, 525734, 912493, 1583775, 2748893, 4771144, 8281088, 14373165, 24946955, 43299485, 75153286, 130440740, 226401112, 392955956, 682038999, 1183789679, 2054659669, 3566196321, 6189714276

Offset: 0

N. J. A. Sloane

A fountain is formed by starting with a row of coins, then stacking additional coins on top so that each new coin touches two in the previous row.
Also the number of Dyck paths for which the sum of the heights of the vertices that terminate an upstep (i.e., peaks and doublerises) is n. Example: a(4)=3 because we have UDUUDD, UUDDUD and UDUDUDUD. - Emeric Deutsch, Mar 22 2008
Also the number of ordered trees with path length n (follows from previous comment via a standard bijection). - Emeric Deutsch, Mar 22 2008
Probably first studied by Jim Propp (unpublished).
Number of compositions of n with c(1) = 1 and c(i+1) <= c(i) + 1. (Slide each row right 1/2 step relative to the row below, and count the columns.) - Franklin T. Adams-Watters, Nov 24 2009
With the additional requirement for weak unimodality one obtains A001524. - Joerg Arndt, Dec 09 2012

			An example of a fountain with 19 coins:
... O . O O
.. O O O O O O . O
. O O O O O O O O O
From _Peter Bala_, Dec 26 2012: (Start)
F(1/10) = Sum_{n >= 0} a(n)/10^n has the simple continued fraction expansion 1 + 1/(8 + 1/(1 + 1/(8 + 1/(1 + 1/(98 + 1/(1 + 1/(98 + 1/(1 + 1/(998 + 1/(1 + 1/(998 + 1/(1 + ...)))))))))))).
F(-1/10) = Sum_{n >= 0} (-1)^n*a(n)/10^n has the simple continued fraction expansion 1/(1 + 1/(9 + 1/(1 + 1/(9 + 1/(99 + 1/(1 + 1/(99 + 1/(999 + 1/(1 + 1/(999 + 1/(9999 + 1/(1 + ...)))))))))))).
(End)

S. R. Finch, Mathematical Constants, Cambridge, 2003, p. 381.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..4178 (first 501 terms from T. D. Noe)
Peter Bala, Some simple continued fraction expansions
P. Flajolet, Combinatorial aspects of continued fractions, Discrete Mathematics 32 (1980), pp. 125-161.
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 331.
Atli Fannar Franklín, Pattern avoidance enumerated by inversions, arXiv:2410.07467 [math.CO], 2024. See pp. 2, 4.
Atli Fannar Franklín, Anders Claesson, Christian Bean, Henning Úlfarsson, and Jay Pantone, Restricted Permutations Enumerated by Inversions, arXiv:2406.16403 [cs.DM], 2024. See p. 2.
M. L. Glasser, V. Privman, N. M. Svrakic, Temperley's triangular lattice compact cluster model: exact solution in terms of the q series. J. Phys. A 20 (1987), no. 18, L1275-L1280.
H. W. Gould, R. K. Guy, and N. J. A. Sloane, Correspondence, 1987.
R. K. Guy, Letter to N. J. A. Sloane, Sep 25 1986.
R. K. Guy, Letter to N. J. A. Sloane, 1987
R. K. Guy, The strong law of small numbers, Amer. Math. Monthly 95 (1988), no. 8, 697-712.
R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy]
R. K. Guy and N. J. A. Sloane, Correspondence, 1988.
Kival Ngaokrajang, Illustration for initial terms
A. M. Odlyzko and H. S. Wilf, The editor's corner: n coins in a fountain, Amer. Math. Monthly, 95 (1988), 840-843.
A. M. Odlyzko, Asymptotic enumeration methods, pp. 1063-1229 of R. L. Graham et al., eds., Handbook of Combinatorics, 1995; see Example 10.7 (pdf, ps)
Eric Weisstein's World of Mathematics, Rogers-Ramanujan Continued Fraction.

Cf. A001524, A192728, A192729, A192730, A111317, A143951, A285903, A226999 (inverse Euler transform), A291148 (convolution inverse).
First column of A168396. - Franklin T. Adams-Watters, Nov 24 2009
Diagonal of A185646.
Row sums of A047998. Column sums of A138158. - Emeric Deutsch, Mar 22 2008

a005169 0 = 1
a005169 n = a168396 n 1  -- Reinhard Zumkeller, Sep 13 2013; corrected by R. J. Mathar, Sep 16 2013

P[0]:=1: for n to 40 do P[n]:=sort(expand(t*(sum(P[j]*P[n-j-1]*t^(n-j-1),j= 0..n-1)))) end do: F:=sort(sum(P[k],k=0..40)): seq(coeff(F,t,j),j=0..36); # Emeric Deutsch, Mar 22 2008
# second Maple program:
A005169_G:= proc(x,NK); Digits:=250; Q2:=1;
        for k from NK by -1 to 0 do  Q1:=1-x^k/Q2; Q2:=Q1; od;
        Q3:=Q2; S:=1-Q3;
end:
series(A005169_G(x, 20), x, 21); # Sergei N. Gladkovskii, Dec 18 2011

m = 36; p[0] = 1; p[n_] := p[n] = Expand[t*Sum[p[j]*p[n-j-1]*t^(n-j-1), {j, 0, n-1}]]; f[t_] = Sum[p[k], {k, 0, m}]; CoefficientList[Series[f[t], {t, 0, m}], t] (* Jean-François Alcover, Jun 21 2011, after Emeric Deutsch *)
max = 43; Series[1-Fold[Function[1-x^#2/#1], 1, Range[max, 0, -1]], {x, 0, max}] // CoefficientList[#, x]& (* Jean-François Alcover, Sep 16 2014 *)
b[n_, i_] := b[n, i] = If[n==0, 1, Sum[b[n-j, j], {j, 1, Min[i+1, n]}]];
c[n_] :=  b[n, 0] - b[n-1, 0];
c /@ Range[0, 50] // Accumulate  (* Jean-François Alcover, Nov 14 2020, after Alois P. Heinz in A289080 *)

/* using the g.f. from p. L1278 of the Glasser, Privman, Svrakic paper */
N=30;  x='x+O('x^N);
P(k)=sum(n=0,N, (-1)^n*x^(n*(n+1+k))/prod(j=1,n,1-x^j));
G=1+x*P(1)/( (1-x)*P(1)-x^2*P(2) );
Vec(G) /* Joerg Arndt, Feb 10 2011 */

/* As a continued fraction: */
{a(n)=local(A=1+x,CF);CF=1+x;for(k=0,n,CF=1/(1-x^(n-k+1)*CF+x*O(x^n));A=CF);polcoeff(A,n)} /* Paul D. Hanna */

/* By the Rogers-Ramanujan continued fraction identity: */
{a(n)=local(A=1+x,P,Q);
P=sum(m=0,sqrtint(n),(-1)^m*x^(m*(m+1))/prod(k=1,m,1-x^k));
Q=sum(m=0,sqrtint(n),(-1)^m*x^(m^2)/prod(k=1,m,1-x^k));
A=P/(Q+x*O(x^n));polcoeff(A,n)}  /* Paul D. Hanna */

A005169(n) = f(n, 1), where f(n, p) = 0 if p > n, 1 if p = n, Sum(1 <= q <= p+1; f(n-p, q)) if p < n. f=A168396.
G.f.: F(t) = Sum_{k>=0} P[k], where P[0]=1, P[n] = t*Sum_{j= 0..n-1} P[j]*P[n-j-1]*t^(n-j-1) for n >= 1. - Emeric Deutsch, Mar 22 2008
G.f.: 1/(1-x/(1-x^2/(1-x^3/(1-x^4/(1-x^5/(...)))))) [given on the first page of the Odlyzko/Wilf reference]. - Joerg Arndt, Mar 08 2011
G.f.: 1/G(0), where G(k)= 1 - x^(k+1)/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Jun 29 2013
G.f.: A(x) = P(x)/Q(x) where
  P(x) = Sum_{n>=0} (-1)^n* x^(n*(n+1)) / Product_{k=1..n} (1-x^k),
  Q(x) = Sum_{n>=0} (-1)^n* x^(n^2) / Product_{k=1..n} (1-x^k),
due to the Rogers-Ramanujan continued fraction identity. - Paul D. Hanna, Jul 08 2011
From Peter Bala, Dec 26 2012: (Start)
Let F(x) denote the o.g.f. of this sequence. For positive integer n >= 3, the real number F(1/n) has the simple continued fraction expansion 1 + 1/(n-2 + 1/(1 + 1/(n-2 + 1/(1 + 1/(n^2-2 + 1/(1 + 1/(n^2-2 + 1/(1 + ...)))))))), while for n >= 2, F(-1/n) has the simple continued fraction expansion 1/(1 + 1/(n-1 + 1/(1 + 1/(n-1 + 1/(n^2-1 + 1/(1 + 1/(n^2-1 + 1/(n^3-1 + 1/(1 + ...))))))))). Examples are given below. Cf. A111317 and A143951.
(End)
a(n) = c * x^(-n) + O((5/3)^n), where c = 0.312363324596741... and x = A347901 = 0.576148769142756... is the lowest root of the equation Q(x) = 0, Q(x) see above (Odlyzko & Wilf 1988). - Vaclav Kotesovec, Jul 18 2013, updated Sep 24 2020
G.f.: G(0), where G(k)= 1 - x^(k+1)/(x^(k+1) - 1/G(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Aug 06 2013
G.f.: 1 - 1/x + 1/(x*W(0)), where W(k)= 1 - x^(2*k+2)/(1 - x^(2*k+1)/W(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Aug 16 2013

More terms from David W. Wilson, Apr 30 2001

A003116 Expansion of the reciprocal of the g.f. defining A039924.

Original entry on oeis.org

1, 1, 2, 4, 7, 13, 23, 41, 72, 127, 222, 388, 677, 1179, 2052, 3569, 6203, 10778, 18722, 32513, 56455, 98017, 170161, 295389, 512755, 890043, 1544907, 2681554, 4654417, 8078679, 14022089, 24337897, 42242732, 73319574, 127258596, 220878683

Offset: 0

N. J. A. Sloane, Herman P. Robinson

Conjecture: a(n) is the number of compositions p(1) + p(2) + ... + p(m) = n with p(i)-p(i-1) <= 1, see example; cf. A034297. - Vladeta Jovovic, Feb 09 2004
Row sums and central terms of the triangle in A168396: a(n) = A168396(2*n+1,n) and for n > 0: a(n) = Sum_{k=1..n} A168396(n,k). - Reinhard Zumkeller, Sep 13 2013
Former definition was "Expansion of reciprocal of a determinant." - N. J. A. Sloane, Aug 10 2018
From Doron Zeilberger, Aug 10 2018: (Start)
Jovovic's conjecture can be proved as follows. There is a sign-changing involution defined on pairs (L1,L2) where L1 is a partition with difference >= 2 between consecutive parts and L2 is the number of compositions described by Jovovic, with the sign (-1)^(Number of parts of L1).
Let a be the largest part of L1 and b the largest part of L2. If b-a>=2 then move b from L2 to the top of L1, otherwise move a to the top of L2.
Since this is an involution and it changes the sign (the number of parts of L1 changes parity) this proves it, since the g.f. of A039924 is exactly the signed-enumeration of the set given by L1. (End)

			From _Joerg Arndt_, Dec 29 2012: (Start)
There are a(6)=23 compositions p(1)+p(2)+...+p(m)=6 such that p(k)-p(k-1) <= 1:
[ 1]  [ 1 1 1 1 1 1 ]
[ 2]  [ 1 1 1 1 2 ]
[ 3]  [ 1 1 1 2 1 ]
[ 4]  [ 1 1 2 1 1 ]
[ 5]  [ 1 1 2 2 ]
[ 6]  [ 1 2 1 1 1 ]
[ 7]  [ 1 2 1 2 ]
[ 8]  [ 1 2 2 1 ]
[ 9]  [ 1 2 3 ]
[10]  [ 2 1 1 1 1 ]
[11]  [ 2 1 1 2 ]
[12]  [ 2 1 2 1 ]
[13]  [ 2 2 1 1 ]
[14]  [ 2 2 2 ]
[15]  [ 2 3 1 ]
[16]  [ 3 1 1 1 ]
[17]  [ 3 1 2 ]
[18]  [ 3 2 1 ]
[19]  [ 3 3 ]
[20]  [ 4 1 1 ]
[21]  [ 4 2 ]
[22]  [ 5 1 ]
[23]  [ 6 ]
Replacing the condition with p(k)-p(k-1) <= 0 gives integer partitions.
(End)

D. H. Lehmer, Combinatorial and cyclotomic properties of certain tridiagonal matrices. Proceedings of the Fifth Southeastern Conference on Combinatorics, Graph Theory, and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1974), pp. 53-74. Congressus Numerantium, No. X, Utilitas Math., Winnipeg, Man., 1974. MR0441852.
H. P. Robinson, Letter to N. J. A. Sloane, Nov 19 1973.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..4176 (first 401 terms from T. D. Noe)
Roland Bacher, Generic numerical semigroups, hal-03221466 [math.CO], 2021.
George Beck and Shane Chern, Reciprocity between partitions and compositions, arXiv:2108.04363 [math.CO], 2021.
Shalosh B. Ekhad and Doron Zeilberger, D.H. Lehmer's Tridiagonal determinant: An Etude in (Andrews-Inspired) Experimental Mathematics, arXiv:1808.06730 [math.CO], 2018.
Miguel Mendez, Shift-plethysm, Hydra continued fractions, and m-distinct partitions, arXiv:2009.04623 [math.CO], 2020.
Herman P. Robinson, Letter to N. J. A. Sloane, Nov 13 1973.
Herman P. Robinson, Letter to N. J. A. Sloane, Nov 19 1973.

Cf. A003114, A039924, A034297, A168443, A224959.

a003116 n = a168396 (2 * n + 1) n  -- Reinhard Zumkeller, Sep 13 2013

max = 35; f[x_] := 1/Sum[x^k^2*((-1)^k/Product[1 - x^i, {i, 1, k}]), {k, 0, Floor[Sqrt[max]]}]; CoefficientList[ Series[f[x], {x, 0, max}], x](* Jean-François Alcover, Jun 12 2012, after PARI *)
b[n_, k_] := b[n, k] = Expand[If[n == 0, 1, x*
     Sum[b[n - j, j], {j, 1, Min[n, k + 1]}]]];
a[n_] := Total@CoefficientList[b[n, n], x];
Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Apr 14 2022, after Alois P. Heinz in A168443 *)

a(n)=if(n<0,0,polcoeff(1/sum(k=0,sqrtint(n),x^k^2/prod(i=1,k,x^i-1,1+x*O(x^n))),n))

G.f.: 1/(Sum_{k>=0} x^(k^2)(-1)^k/(Product_{i=1..k} 1-x^i)).

a(n) ~ c * d^n, where d = 1/A347901 = 1.73566282453034742565826074971966853... and c = 0.9180565304926754125870866477349969555868577236908640010903420353... - Vaclav Kotesovec, Nov 01 2021

Definition revised by N. J. A. Sloane, Aug 10 2018 at the suggestion of Doron Zeilberger

A226999 Inverse Euler transform of A005169 (fountains of coins).

Original entry on oeis.org

1, 0, 1, 1, 2, 3, 5, 8, 13, 21, 35, 55, 93, 149, 248, 403, 671, 1098, 1827, 3013, 5013, 8313, 13859, 23063, 38534, 64341, 107715, 180355, 302565, 507784, 853507, 1435415, 2416941, 4072272, 6868062, 11590807, 19577555, 33088481, 55964327, 94712212

Offset: 1

R. J. Mathar, Jun 26 2013

nonn

If G005169(x) = Sum_{i>=0} A005169(n)*x^n is the generating function of A005169, the a(n) are defined through G005169(x) = Product_{n>=1} 1/(1-x^n)^a(n), the inverse Euler transform of A005169.

Steven R. Finch, Mathematical Constants, Cambridge, 2003, p. 381.

Alois P. Heinz, Table of n, a(n) for n = 1..4178
R. K. Guy, Letter to N. J. A. Sloane, Sep 25 1986.
R. K. Guy, Letter to N. J. A. Sloane, 1987
R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712.
R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy]

Cf. A005169, A005170.

max = 100;
A005169 = Series[1 - Fold[Function[1 - x^#2/#1], 1, Range[max, 0, -1]], {x, 0, max}] // CoefficientList[#, x]&;
mob[m_, n_] := If[Mod[m, n] == 0, MoebiusMu[m/n], 0];
EULERi[b_] := Module[{a, c, i, d}, c = {}; For[i = 1, i <= Length[b], i++, c = Append[c, i*b[[i]] - Sum[c[[d]]*b[[i - d]], {d, 1, i - 1}]]]; a = {}; For[i = 1, i <= Length[b], i++, a = Append[a, (1/i)*Sum[mob[i, d]*c[[d]], {d, 1, i}]]]; Return[a]];
EULERi[A005169 // Rest] (* Jean-François Alcover, Jan 06 2020 *)

a(n) ~ 1 / (n * r^n), where r = A347901 = 0.57614876914275660229786857371993878235472466311897446868515653431946822937499... - Vaclav Kotesovec, Oct 09 2019

A285636 G.f.: (1 + x/(1 + x^2/(1 + x^3/(1 + x^4/(1 + ...))))) / (1 - x/(1 - x^2/(1 - x^3/(1 - x^4/(1 - ...))))), a continued fraction.

Original entry on oeis.org

1, 2, 2, 2, 4, 8, 14, 22, 36, 64, 114, 198, 340, 586, 1018, 1772, 3076, 5332, 9248, 16054, 27872, 48376, 83952, 145700, 252888, 438938, 761846, 1322286, 2295022, 3983384, 6913822, 12000054, 20828006, 36150354, 62744812, 108903838, 189020310, 328075444, 569428264, 988335418, 1715417004

Offset: 0

Ilya Gutkovskiy, Apr 23 2017

nonn

			G.f.: A(x) = 1 + 2*x + 2*x^2 + 2*x^3 + 4*x^4 + 8*x^5 + 14*x^6 + 22*x^7 + 36*x^8 + 64*x^9 + ...

Steven R. Finch, Mathematical Constants, Cambridge, 2003, p. 381.

Robert Israel, Table of n, a(n) for n = 0..600
Eric Weisstein's World of Mathematics, Rogers-Ramanujan Continued Fraction

Cf. A003823, A005169, A007325, A055101, A285635, A285637, A285638.

A10:= [1, seq([x^i,1],i=1..10)]: B10:= [1, seq([-x^i,1],i=1..10)]:
S:= series(numtheory:-nthconver(A10,10)/numtheory:-nthconver(B10,10),x,51):
A:= [seq(coeff(S,x,i),i=0..50)]; # Robert Israel, Dec 15 2024

nmax = 40; CoefficientList[Series[(1/(1 + ContinuedFractionK[-x^k, 1, {k, 1, nmax}]))/(1/(1 + ContinuedFractionK[x^k, 1, {k, 1, nmax}])), {x, 0, nmax}], x]
nmax = 40; CoefficientList[Series[Sum[(-1)^k x^(k (k + 1))/Product[(1 - x^m), {m, 1, k}], {k, 0, nmax}] / (Product[(1 - x^(5 k - 1)) (1 - x^(5 k - 4))/((1 - x^(5 k - 2)) (1 - x^(5 k - 3))), {k, 1, nmax}]  Sum[(-1)^k x^(k^2)/Product[(1 - x^m), {m, 1, k}], {k, 0, nmax}]), {x, 0, nmax}], x]

G.f.: A(x) = P(x)/(R(x)*Q(x)), where P(x) = Sum_{k>=0} (-1)^k*x^(k*(k+1)) / Product_{m=1..k} (1 - x^m), R(x) = Product_{k>=1} (1 - x^(5*k-1))*(1 - x^(5*k-4)) / ((1 - x^(5*k-2))*(1 - x^(5*k-3))) and Q(x) = Sum_{k>=0} (-1)^k*x^(k^2) / Product_{m=1..k} (1 - x^m).

a(n) ~ c / r^n, where r = A347901 = 0.576148769142756602297868573719938782354724663118974... is the lowest root of the equation Sum_{k>=0} (-1)^k * r^(k^2) / QPochhammer(r, r, k) = 0 and c = 0.452642356466453742995961374156022446123012... - Vaclav Kotesovec, Aug 26 2017, updated Sep 24 2020

A285637 G.f.: 1/( (1 - x/(1 - x^2/(1 - x^3/(1 - x^4/(1 - ...))))) * (1 + x/(1 + x^2/(1 + x^3/(1 + x^4/(1 + ...))))) ), a continued fraction.

Original entry on oeis.org

1, 0, 1, 2, 1, 4, 6, 10, 19, 30, 55, 92, 161, 282, 483, 846, 1462, 2538, 4409, 7642, 13276, 23032, 39977, 69394, 120426, 209036, 362800, 629698, 1092952, 1896968, 3292522, 5714678, 9918752, 17215620, 29880461, 51862438, 90015657, 156236814, 271174435, 470667300, 816919764, 1417897172, 2460991365

Offset: 0

Ilya Gutkovskiy, Apr 23 2017

nonn

			G.f.: A(x) = 1 + x^2 + 2*x^3 + x^4 + 4*x^5 + 6*x^6 + 10*x^7 + 19*x^8 + 30*x^9 + 55*x^10 + ...

Eric Weisstein's World of Mathematics, Rogers-Ramanujan Continued Fraction

Cf. A003823, A005169, A007325, A055101, A285635, A285636, A285638.

nmax = 42; CoefficientList[Series[(1/(1 + ContinuedFractionK[-x^k, 1, {k, 1, nmax}])) (1/(1 + ContinuedFractionK[x^k, 1, {k, 1, nmax}])), {x, 0, nmax}], x]
nmax = 42; CoefficientList[Series[Product[(1 - x^(5 k - 1)) (1 - x^(5 k - 4))/((1 - x^(5 k - 2)) (1 - x^(5 k - 3))), {k, 1, nmax}] Sum[(-1)^k x^(k (k + 1))/Product[(1 - x^m), {m, 1, k}], {k, 0, nmax}] / Sum[(-1)^k x^(k^2)/Product[(1 - x^m), {m, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]

G.f.: A(x) = R(x)*P(x)/Q(x), where R(x) = Product_{k>=1} (1 - x^(5*k-1))*(1 - x^(5*k-4)) / ((1 - x^(5*k-2))*(1 - x^(5*k-3))), P(x) = Sum_{k>=0} (-1)^k*x^(k*(k+1)) / Product_{m=1..k} (1 - x^m) and Q(x) = Sum_{k>=0} (-1)^k*x^(k^2) / Product_{m=1..k} (1 - x^m).

a(n) ~ c * d^n, where d = 1/A347901 = 1.7356628245303474256582607497196685302546528472903927546099... and c = 0.215558365582078354136603033062960103377669... - Vaclav Kotesovec, Aug 26 2017

A285903 G.f.: 1 + Sum_{n>=1} a(n)*x^n/(1 - x^n) = 1/(1 - x/(1 - x^2/(1 - x^3/(1 - x^4/(1 - ...))))).

Original entry on oeis.org

1, 0, 1, 2, 4, 7, 14, 23, 43, 73, 134, 223, 405, 689, 1216, 2094, 3678, 6333, 11080, 19152, 33363, 57798, 100549, 174262, 302898, 525328, 912448, 1583069, 2748892, 4769842, 8281087, 14371045, 24946819, 43295806, 75153267, 130434130, 226401111, 392944875, 682038592, 1183770424, 2054659668, 3566162246

Offset: 1

Ilya Gutkovskiy, Apr 28 2017

nonn

			G.f.: 1 + x/(1 - x) + x^3/(1 - x^3) + 2*x^4/(1 - x^4) + 4*x^5/(1 - x^5) + ... = 1/(1 - x/(1 - x^2/(1 - x^3/(1 - x^4/(1 - ...))))).

Vaclav Kotesovec, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Rogers-Ramanujan Continued Fraction
N. J. A. Sloane, Transforms

Cf. A005169, A192728.

nn = 42; f[x_] := 1 + Sum[a[n] x^n/(1 - x^n), {n, 1, nn}]; sol = SolveAlways[0 == Series[f[x] - 1/(1 + ContinuedFractionK[-x^n, 1, {n, 1, nn}]), {x, 0, nn}], x]; Table[a[n], {n, 1, nn}] /. sol // Flatten

G.f.: 1 + Sum_{n>=1} a(n)*x^n/(1 - x^n) = (Sum_{n>=0} (-1)^n*x^(n*(n+1)) /Product_{k=1..n} (1 - x^k)) / (Sum_{n>=0} (-1)^n*x^(n^2)/Product_{k=1..n} (1 - x^k)).
Sum_{d|n} a(d) = A005169(n) for n > 0.
From Vaclav Kotesovec, Apr 30 2017: (Start)
a(n) ~ c * d^n, where
d = 1/A347901 = 1.735662824530347425658260749719668530254652847290392754609934...
c = 0.31236332459674145306627970724066492149823012868471473538681348971946...
(End)

A168445 Number of compositions a(1),...,a(k) of n, for some k, such that a(i+1) <= a(i) + 1 for 1 <= i < k and a(1) <= a(k) + 1.

Original entry on oeis.org

1, 2, 4, 6, 11, 18, 31, 52, 91, 155, 268, 464, 802, 1390, 2411, 4178, 7249, 12578, 21823, 37870, 65724, 114061, 197960, 343578, 596317, 1034983, 1796359, 3117837, 5411478, 9392460, 16302081, 28294850, 49110242, 85238716, 147945552, 256783448, 445689300

Offset: 1

Vladeta Jovovic, Nov 25 2009

nonn

Steven R. Finch, Mathematical Constants, Cambridge, 2003, p. 381.

Alois P. Heinz, Table of n, a(n) for n = 1..400

Cf. A003116, A005169.

b:= proc(n,r,f) option remember; `if`(n=0, `if`(f-1<=r, 1, 0),
      add(b(n-i, i, f), i=1..min(r+1, n)))
    end:
a:= n-> add(b(n-i, i, i), i=1..n):
seq(a(n), n=1..40);  # Alois P. Heinz, Dec 15 2009

b[n_, r_, f_] := b[n, r, f] = If[n == 0, If[f - 1 <= r, 1, 0], Sum[b[n - i, i, f], {i, 1, Min [r + 1, n]}]];
a[n_] := Sum[b[n - i, i, i], {i, 1, n}];
Array[a, 40] (* Jean-François Alcover, Nov 10 2020, after Alois P. Heinz *)

a(n) ~ c / r^n, where r = A347901 = 0.576148769142756602297868573719938782354724663118974... is the lowest root of the equation Sum_{k>=0} (-1)^k * r^(k^2) / QPochhammer(r, r, k) = 0 and c = 0.6149126319329581124890112676009720339906790088212712130894... - Vaclav Kotesovec, May 01 2014, updated Sep 09 2020

More terms from Alois P. Heinz, Dec 15 2009

cp's OEIS Frontend

A379091 Decimal expansion of 1/A347901.

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A005169 Number of fountains of n coins.

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A003116 Expansion of the reciprocal of the g.f. defining A039924.

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A226999 Inverse Euler transform of A005169 (fountains of coins).

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A285636 G.f.: (1 + x/(1 + x^2/(1 + x^3/(1 + x^4/(1 + ...))))) / (1 - x/(1 - x^2/(1 - x^3/(1 - x^4/(1 - ...))))), a continued fraction.

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A285637 G.f.: 1/( (1 - x/(1 - x^2/(1 - x^3/(1 - x^4/(1 - ...))))) * (1 + x/(1 + x^2/(1 + x^3/(1 + x^4/(1 + ...))))) ), a continued fraction.

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A285903 G.f.: 1 + Sum_{n>=1} a(n)*x^n/(1 - x^n) = 1/(1 - x/(1 - x^2/(1 - x^3/(1 - x^4/(1 - ...))))).

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