A185646 -id:A185646 - 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Probably first studied by Jim Propp (unpublished).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

A173173 a(n) = ceiling(Fibonacci(n)/2).

Original entry on oeis.org

0, 1, 1, 1, 2, 3, 4, 7, 11, 17, 28, 45, 72, 117, 189, 305, 494, 799, 1292, 2091, 3383, 5473, 8856, 14329, 23184, 37513, 60697, 98209, 158906, 257115, 416020, 673135, 1089155, 1762289, 2851444, 4613733, 7465176, 12078909, 19544085, 31622993, 51167078, 82790071

Offset: 0

Roger L. Bagula, Nov 22 2010

Also the independence number of the n-Fibonacci cube graph. - Eric W. Weisstein, Sep 06 2017
Also the edge cover number of the (n-2)-Fibonacci cube graph. - Eric W. Weisstein, Dec 26 2017
Also the calque covering number of the (n-2)-Fibonacci cube graph. - Eric W. Weisstein, Apr 20 2019

Vincenzo Librandi, Table of n, a(n) for n = 0..280
Eric Weisstein's World of Mathematics, Clique Covering Number
Eric Weisstein's World of Mathematics, Edge Cover Number
Eric Weisstein's World of Mathematics, Fibonacci Cube Graph
Eric Weisstein's World of Mathematics, Independence Number
Index entries for linear recurrences with constant coefficients, signature (1,1,1,-1,-1).

Column m=3 of A185646.

[Fibonacci(n) - Floor(Fibonacci(n)/2): n in [0..50]]; // Vincenzo Librandi, Apr 24 2011

with(combinat,fibonacci): seq(ceil(fibonacci(n)/2),n=0..33) # Mircea Merca, Jan 04 2010

Table[Fibonacci[n] - Floor[Fibonacci[n]/2], {n, 0, 40}] (* Harvey P. Dale, Jun 09 2013 *)
(* Start from Eric W. Weisstein, Sep 06 2017 *)
Table[Ceiling[Fibonacci[n]/2], {n, 0, 20}]
Ceiling[Fibonacci[Range[0, 20]]/2]
LinearRecurrence[{1, 1, 1, -1, -1}, {1, 2, 3, 4, 7}, 20]
CoefficientList[Series[(1 + x - 2 x^3 - x^4)/(1 - x - x^2 - x^3 + x^4 + x^5), {x, 0, 20}], x]
(* End *)

/* Continued Fraction: */
{a(n)=my(CF); CF=1+x; for(k=0, n, CF=1/(1 - x^(n-k+1)*(1 - x^(n-k+4)) *CF +x*O(x^n) )); polcoeff(x*CF, n)} \\ Paul D. Hanna, Jul 08 2013

{a(n)=polcoeff( x*(1 - x^2 - x^3) / ((1-x^3)*(1 - x - x^2 +x*O(x^n))),n)} \\ Paul D. Hanna, Jul 18 2013

a(n)=(fibonacci(n)+1)\2 \\ Charles R Greathouse IV, Jun 11 2015

a(n) = ceiling(Fibonacci(n)/2). - Mircea Merca, Jan 04 2010
a(n) = a(n-1) +a(n-2) +a(n-3) -a(n-4) -a(n-5) - Joerg Arndt, Apr 24 2011
G.f.: x/(1 - x*(1-x^4)/(1 - x^2*(1-x^5)/(1 - x^3*(1-x^6)/(1 - x^4*(1-x^7)/(1 - x^5*(1-x^8)/(1 - x^6*(1-x^9)/(1 - x^7*(1-x^10)/(1 - x^8*(1-x^11)/(1 - ...))))))))), (continued fraction) - Paul D. Hanna, Jul 08 2013
G.f.: x*(1 - x^2 - x^3) / ((1-x^3)*(1 - x - x^2)). [Paul D. Hanna, Jul 18 2013, from Joerg Arndt's formula]
a(n) = A061347(n)/6 +1/3 +A000045(n)/2. - R. J. Mathar, Jul 19 2013
For n > 1, if n == 0 (mod 3) then a(n) = a(n-1) + a(n-2) - 1; otherwise a(n) = a(n-1) + a(n-2). - Franklin T. Adams-Watters, Jun 11 2018

Name simplified using Mircea Merca's formula by Eric W. Weisstein, Sep 06 2017

A143064 Expansion of a Ramanujan false theta series variation of A089801 in powers of x.

Original entry on oeis.org

1, 1, 0, 0, 0, -1, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Michael Somos, Jul 21 2008

sign

			G.f. = 1 + x - x^5 - x^8 + x^16 + x^21 - x^33 - x^40 + x^56 + x^65 - x^85 - x^96 + ...
G.f. = q + q^4 - q^16 - q^25 + q^49 + q^64 - q^100 - q^121 + q^169 + q^196 + ...

Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 41, 13th equation.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
G. E. Andrews, An introduction to Ramanujan's "lost" notebook, Amer. Math. Monthly 86 (1979), no. 2, 89-108. See page 89, Equation (1.2), page 100, Equation (5.3)
G. E. Andrews, The Bhargava-Adiga Summation and Partitions, 2016. See page 3, Equation (1.5)

Cf. A089801, A010054, A143062, A143065.

Column m=0 of A185646.

a[ n_] := With[ {m = Sqrt[3 n + 1]}, If[ IntegerQ @ m, (-1)^Quotient[ m, 3], 0]]; (* Michael Somos, Jun 30 2015 *)
a[ n_] := SeriesCoefficient[ Sum[ (-1)^k x^(3 k^2 + 2 k) (1 + x^(2 k + 1)), {k, 0, n}], {x, 0, n}]; (* Michael Somos, Nov 04 2013 *)
a[ n_] := SeriesCoefficient[ Sum[ x^k QPochhammer[ x, x^2, k], {k, 0, n}], {x, 0, n}]; (* Michael Somos, Jun 30 2015 *)
a[ n_] := SeriesCoefficient[ Sum[ x^k / QPochhammer[ -x, x^2, k + 1], {k, 0, 2 n}], {x, 0, 2 n}]; (* Michael Somos, Jun 30 2015 *)

{a(n) = my(m); if( issquare( 3*n + 1, &m), (-1)^(m \ 3) )};

{a(n) = my(A, p, e); if( n<0, 0, n = 3*n + 1; A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if( e%2, 0, p==2, -(-1)^(e/2), p == 3, 0, p%6 == 1, 1, (-1)^(e/2))))}; /* Michael Somos, Jul 19 2013 */

/* Continued Fraction: */
{a(n)=local(CF); CF=1+x; for(k=0, n, CF=1/(1 - x^(n-k+1)*(1 - x^(n-k+1))*CF+x*O(x^n))); polcoeff(CF, n)} \\ Paul D. Hanna, Jul 18 2013

Expansion of Sum_{k>=0} x^k / (Product_{j=0..k} ( 1 + x^(2*k + 1) ) ) in powers of x^2. - Michael Somos, Nov 04 2013
a(n) = b(3*n + 1) where b() is multiplicative with b(p^(2*e)) = -(-1)^e if p = 2, b(p^(2*e)) = (-1)^e if p = 5 (mod 6), b(p^(2*e)) = 1 if p = 1 (mod 6), and b(p^(2*e-1)) = b(3^e) = 0 if e>0. - Michael Somos, Jul 19 2013
a(4*n + 2) = a(4*n + 3) = a(8*n + 4) = 0.
a(8*n) = A143062(n). Convolution of A010054 with A143065. - Michael Somos, Jul 19 2013
G.f.: Sum_{k>=0} (-1)^k * x^(3*k^2 + 2*k) * ( 1 + x^(2*k + 1) ).
G.f.: 1/(1 - x*(1-x)/(1 - x^2*(1-x^2)/(1 - x^3*(1-x^3)/(1 - x^4*(1-x^4)/(1 - ...))))), a continued fraction. - Paul D. Hanna, Jul 18 2013
abs(a(n)) = A089801(n). - Michael Somos, Jun 30 2015
G.f.: 1 + x*(1-x) + x^2*(1-x)*(1-x^3) + x^3*(1-x)*(1-x^3)*(1-x^5) + ... . - Michael Somos, Aug 03 2017

A227375 G.f.: 1/(1 - x(1-x^6)/(1 - x^2(1-x^7)/(1 - x^3(1-x^8)/(1 - x^4(1-x^9)/(1 - x^5*(1-x^10)/(1 - ...)))))), a continued fraction.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 9, 14, 24, 41, 69, 118, 200, 340, 579, 985, 1677, 2854, 4858, 8270, 14078, 23966, 40798, 69453, 118235, 201280, 342655, 583328, 993046, 1690543, 2877949, 4899369, 8340598, 14198887, 24171937, 41149884, 70052848, 119256753, 203020631, 345618810, 588375486, 1001640259

Offset: 0

Paul D. Hanna, Jul 09 2013

nonn

Radius of convergence r is a root of 1 - r - r^2 - r^3 + r^5 + r^6 + r^7 = 0,
where r = Limit a(n)/a(n+1) = 0.587411973105598587998520092901249815195963...
Compare to sequence A227376, generated by 1/(1-x-x^2-x^3+x^5+x^6+x^7).

			G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 3*x^4 + 5*x^5 + 9*x^6 + 14*x^7 + 24*x^8 +...

Index entries for linear recurrences with constant coefficients, signature (1, 1, 1, 0, 0, -2, -2, -1, 0, 1, 1, 1).

Cf. A173173, A227374, A227360, A227376, A228644, A228645.

Column m=5 of A185646.

nMax = 42; col[m_ /; 0 <= m <= nMax] := 1/(1 + ContinuedFractionK[-x^k (1 - x^(m + k)), 1, {k, 1, Ceiling[nMax/2]}]) + O[x]^(2 nMax) // CoefficientList[#, x]&; A227375 = col[5][[1 ;; nMax]] (* Jean-François Alcover, Nov 03 2016 *)
LinearRecurrence[{1,1,1,0,0,-2,-2,-1,0,1,1,1},{1,1,1,2,3,5,9,14,24,41,69,118},50] (* Harvey P. Dale, Jul 08 2023 *)

a(n)=local(CF); CF=1+x; for(k=0, n, CF=1/(1 - x^(n-k+1)*(1 - x^(n-k+6))*CF+x*O(x^n))); polcoeff(CF, n)
for(n=0,50,print1(a(n),", "))

/* From R. J. Mathar's g.f. formula: */
{a(n)=polcoeff((1-x-x^4)*(1+x-x^3-x^4-x^5)/((1-x^5)*(1-x-x^2-x^3+x^5+x^6+x^7) +x*O(x^n)),n)}
for(n=0,50,print1(a(n),", ")) \\ Paul D. Hanna, Jul 18 2013

Conjecture: G.f. -(x^4+x-1)*(x^5+x^4+x^3-x-1) / ( (x-1)*(x^4+x^3+x^2+x+1)*(x^7+x^6+x^5-x^3-x^2-x+1) ). - R. J. Mathar, Jul 17 2013

A227360 G.f.: 1/(1 - x(1-x^3)/(1 - x^2(1-x^4)/(1 - x^3(1-x^5)/(1 - x^4(1-x^6)/(1 - ...))))), a continued fraction.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 5, 6, 10, 14, 21, 32, 46, 71, 104, 157, 235, 350, 527, 785, 1179, 1763, 2639, 3954, 5915, 8861, 13262, 19857, 29731, 44507, 66640, 99765, 149366, 223625, 334795, 501247, 750434, 1123518, 1682076, 2518314, 3770306, 5644701, 8450977, 12652376

Offset: 0

Paul D. Hanna, Jul 08 2013

nonn

Compare to the continued fraction representation for the g.f. of A173173, where A173173(n) = ceiling(Fibonacci(n)/2).

Limit a(n)/a(n+1) = 0.6679357039724580760720733281356826861233293827578332775311...

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

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A173173, A227374, A227375, A228644, A228645.

Column m=2 of A185646.

nMax = 44; col[m_ /; 0 <= m <= nMax] := 1/(1 + ContinuedFractionK[-x^k (1 - x^(m + k)), 1, {k, 1, Ceiling[nMax/2]}]) + O[x]^(2 nMax) // CoefficientList[#, x] &; A227360 = col[2][[1 ;; nMax]] (* Jean-François Alcover, Nov 03 2016 *)

{a(n)=local(CF); CF=1+x; for(k=0, n, CF=1/(1 - x^(n-k+1)*(1 - x^(n-k+3))*CF+x*O(x^n))); polcoeff(CF, n)}
for(n=0,50,print1(a(n),", "))

A227374 G.f.: 1/(1 - x(1-x^5)/(1 - x^2(1-x^6)/(1 - x^3(1-x^7)/(1 - x^4(1-x^8)/(1 - x^5*(1-x^9)/(1 - ...)))))), a continued fraction.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 8, 13, 22, 36, 61, 101, 169, 283, 473, 793, 1325, 2220, 3715, 6220, 10413, 17431, 29185, 48856, 81797, 136937, 229257, 383813, 642564, 1075762, 1800995, 3015171, 5047886, 8451001, 14148368, 23686705, 39655467, 66389797, 111147511, 186079299, 311527531, 521548600

Offset: 0

Paul D. Hanna, Jul 09 2013

nonn

Limit a(n)/a(n+1) = 0.597312551712707899432116871133154503665320273329853...

			G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 3*x^4 + 5*x^5 + 8*x^6 + 13*x^7 + 22*x^8 +...

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A173173, A227360, A227375, A228644, A228645.

Column m=4 of A185646.

nMax = 42; col[m_ /; 0 <= m <= nMax] := 1/(1 + ContinuedFractionK[-x^k (1 - x^(m + k)), 1, {k, 1, Ceiling[nMax/2]}]) + O[x]^(2 nMax) // CoefficientList[#, x]&; A227374 = col[4][[1 ;; nMax]] (* Jean-François Alcover, Nov 03 2016 *)

{a(n)=local(CF); CF=1+x; for(k=0, n, CF=1/(1 - x^(n-k+1)*(1 - x^(n-k+5))*CF+x*O(x^n))); polcoeff(CF, n)}
for(n=0,50,print1(a(n),", "))

G.f.: T(0), where T(k) = 1 - x^(k+1)*(1-x^(k+5))/(x^(k+1)*(1-x^(k+5)) - 1/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 18 2013

A228644 Expansion of g.f. 1/ (1-x^1(1-x^(m+1))/ (1-x^2(1-x^(m+2))/ (1- ... ))) for m=7.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 9, 15, 26, 44, 76, 131, 225, 389, 670, 1156, 1994, 3439, 5934, 10236, 17661, 30470, 52569, 90699, 156483, 269985, 465811, 803677, 1386609, 2392357, 4127611, 7121498, 12286951, 21199078, 36575462, 63104849, 108876873, 187848862, 324101847

Offset: 0

Alois P. Heinz, Aug 28 2013

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Paul D. Hanna et al., Formula Needed for a Family of Continued Fractions and follow-up messages on the SeqFan list, Jul 28 2013
Index entries for linear recurrences with constant coefficients, signature (1,1,1,0,0,-1,-1,-3,-2,-1,0,2,2,3,3,1,0,0,-2,-1,-1,-1).

Cf. A143064 (m=0), A227360 (m=2), A173173 (m=3), A227374 (m=4), A227375 (m=5), A228646(m=6), A228645 (m=9).

Column m=7 of A185646.

a:= n-> coeff(series(-(x^18 +x^17 +x^16 +2*x^15 +x^14 -2*x^11 -2*x^10 -2*x^9 -2*x^8 +x^5 +x^4 +x^3 +x^2-1) / ((x-1)*(x^6 +x^5 +x^4 +x^3 +x^2 +x +1)*(x^15 +x^14 +x^13 +2*x^12 -x^9 -2*x^8 -2*x^7 -x^6 +x^3 +x^2 +x-1)), x, n+1), x, n): seq(a(n), n=0..50);

nMax = 39; col[m_ /; 0 <= m <= nMax] := 1/(1 + ContinuedFractionK[-x^k (1 - x^(m + k)), 1, {k, 1, Ceiling[nMax/2]}]) + O[x]^(2 nMax) // CoefficientList[#, x]&; A228644 = col[7][[1 ;; nMax]] (* Jean-François Alcover, Nov 03 2016 *)

G.f.: -(x^18 +x^17 +x^16 +2*x^15 +x^14 -2*x^11 -2*x^10 -2*x^9 -2*x^8 +x^5 +x^4 +x^3 +x^2-1) / ((x-1)*(x^6 +x^5 +x^4 +x^3 +x^2 +x +1)*(x^15 +x^14 +x^13 +2*x^12 -x^9 -2*x^8 -2*x^7 -x^6 +x^3 +x^2 +x-1)).

A228645 Expansion of g.f. 1/ (1-x^1(1-x^(m+1))/ (1-x^2(1-x^(m+2))/ (1- ... ))) for m=9.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 9, 15, 26, 45, 78, 134, 232, 402, 695, 1205, 2086, 3613, 6259, 10841, 18780, 32531, 56354, 97621, 169111, 292954, 507488, 879136, 1522947, 2638242, 4570298, 7917253, 13715281, 23759370, 41159039, 71300984, 123516755, 213971647, 370669282

Offset: 0

Alois P. Heinz, Aug 28 2013

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Paul D. Hanna et al., Formula Needed for a Family of Continued Fractions and follow-up messages on the SeqFan list, Jul 28 2013
Index entries for linear recurrences with constant coefficients, signature (1, 1, 1, 0, 0, -1, -1, -2, -1, -3, -2, 0, 1, 3, 4, 4, 4, 4, 2, 0, -2, -3, -5, -4, -4, -3, -2, 0, 1, 1, 2, 2, 1, 1, 1).

Cf. A143064 (m=0), A227360 (m=2), A173173 (m=3), A227374 (m=4), A227375 (m=5), A228646 (m=6), A228644 (m=7).

Column m=9 of A185646.

a:= n-> coeff(series(-(x^30 +x^29 +x^28 +2*x^27 +2*x^26 +2*x^25 +x^24 +x^23 -x^22 -2*x^21 -2*x^20 -4*x^19 -4*x^18 -3*x^17 -2*x^16 -x^15 +2*x^13 +2*x^12 +3*x^11 +3*x^10 +x^9 +x^8 -x^5 -x^4 -x^3 -x^2+1) / ((x-1)*(x^2 +x+1)*(x^6 +x^3+1)*(x^26 +x^25 +x^24 +2*x^23 +2*x^22 +x^21 +x^20 -2*x^18 -2*x^17 -3*x^16 -3*x^15 -3*x^14 -x^13 -x^12 +x^11 +2*x^10 +2*x^9 +2*x^8 +x^7 +x^6 -x^3 -x^2 -x+1)), x, n+1), x, n): seq(a(n), n=0..50);

nMax = 39; col[m_ /; 0 <= m <= nMax] := 1/(1 + ContinuedFractionK[-x^k (1 - x^(m + k)), 1, {k, 1, Ceiling[nMax/2]}]) + O[x]^(2 nMax) // CoefficientList[#, x] &; A228645 = col[9][[1 ;; nMax]] (* Jean-François Alcover, Nov 03 2016 *)

G.f.: -(x^30 +x^29 +x^28 +2*x^27 +2*x^26 +2*x^25 +x^24 +x^23 -x^22 -2*x^21 -2*x^20 -4*x^19 -4*x^18 -3*x^17 -2*x^16 -x^15 +2*x^13 +2*x^12 +3*x^11 +3*x^10 +x^9 +x^8 -x^5 -x^4 -x^3 -x^2+1) / ((x-1)*(x^2 +x+1)*(x^6 +x^3+1)*(x^26 +x^25 +x^24 +2*x^23 +2*x^22 +x^21 +x^20 -2*x^18 -2*x^17 -3*x^16 -3*x^15 -3*x^14 -x^13 -x^12 +x^11 +2*x^10 +2*x^9 +2*x^8 +x^7 +x^6 -x^3 -x^2 -x+1)).

cp's OEIS Frontend

A005169 Number of fountains of n coins.

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A173173 a(n) = ceiling(Fibonacci(n)/2).

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A227375 G.f.: 1/(1 - x*(1-x^6)/(1 - x^2*(1-x^7)/(1 - x^3*(1-x^8)/(1 - x^4*(1-x^9)/(1 - x^5*(1-x^10)/(1 - ...)))))), a continued fraction.

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A227360 G.f.: 1/(1 - x*(1-x^3)/(1 - x^2*(1-x^4)/(1 - x^3*(1-x^5)/(1 - x^4*(1-x^6)/(1 - ...))))), a continued fraction.

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A227374 G.f.: 1/(1 - x*(1-x^5)/(1 - x^2*(1-x^6)/(1 - x^3*(1-x^7)/(1 - x^4*(1-x^8)/(1 - x^5*(1-x^9)/(1 - ...)))))), a continued fraction.

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A227375 G.f.: 1/(1 - x(1-x^6)/(1 - x^2(1-x^7)/(1 - x^3(1-x^8)/(1 - x^4(1-x^9)/(1 - x^5*(1-x^10)/(1 - ...)))))), a continued fraction.

A227360 G.f.: 1/(1 - x(1-x^3)/(1 - x^2(1-x^4)/(1 - x^3(1-x^5)/(1 - x^4(1-x^6)/(1 - ...))))), a continued fraction.

A227374 G.f.: 1/(1 - x(1-x^5)/(1 - x^2(1-x^6)/(1 - x^3(1-x^7)/(1 - x^4(1-x^8)/(1 - x^5*(1-x^9)/(1 - ...)))))), a continued fraction.

A228644 Expansion of g.f. 1/ (1-x^1(1-x^(m+1))/ (1-x^2(1-x^(m+2))/ (1- ... ))) for m=7.

A228645 Expansion of g.f. 1/ (1-x^1(1-x^(m+1))/ (1-x^2(1-x^(m+2))/ (1- ... ))) for m=9.

A228646 Expansion of g.f. 1/ (1-x^1(1-x^(m+1))/ (1-x^2(1-x^(m+2))/ (1- ... ))) for m=6.

A185648 Expansion of g.f. 1/ (1-x^1(1-x^(m+1))/ (1-x^2(1-x^(m+2))/ (1- ... ))) for m=8.