A005169 -id:A005169 - cp's OEIS frontend

A206738 G.f.: 1/(1 - x^2/(1 - x^5/(1 - x^8/(1 - x^11/(1 - x^14/(1 - x^17/(1 -...- x^(3*n-1)/(1 -...)))))))), a continued fraction.

1, 0, 1, 0, 1, 0, 1, 1, 1, 2, 1, 3, 2, 4, 4, 6, 7, 9, 11, 14, 18, 22, 29, 35, 46, 56, 73, 90, 116, 144, 184, 231, 292, 370, 465, 591, 742, 942, 1185, 1502, 1893, 2395, 3023, 3819, 4826, 6093, 7702, 9724, 12290, 15519, 19611, 24767, 31294, 39527, 49937, 63082

Offset: 0

Paul D. Hanna, Feb 12 2012

We have the simple continued fraction expansions (A(x) is the sequence o.g.f.): A(1/n) = [1; n^2 - 2, 1, n^3 - 2, 1, n^5 - 2, 1, n^6 - 2, 1, n^8 - 2, 1, n^9 - 2, 1, n^11 - 2, 1, n^12 - 2, 1, ...] for n >= 2 and A(-1/n) = [ 1, n^2 - 1, n^3 - 1, 1, n^5 - 1, n^6 - 1, 1, n^8 - 1, n^9 - 1, 1, n^11 - 1, n^12 - 1, 1, ...] for n >= 2. Cf. A005169, A111317 and A143951. - Peter Bala, Dec 15 2015

			G.f.: A(x) = 1 + x^2 + x^4 + x^6 + x^7 + x^8 + 2*x^9 + x^10 + 3*x^11 + ...
Simple continued fraction expansions: A(1/2) = 1.34788543155288690684 ... = [1; 2, 1, 6, 1, 30, 1, 62, 1, 254, 1, 510, 1, 2046, 1, 4094, 1, ...] and A(-1/2) = 1.3199498363818812865 ... = [1; 3, 7, 1, 31, 63, 1, 255, 511, 1, 2047, 4095, 1, ...]. - _Peter Bala_, Dec 15 2015

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

Cf. A206737, A143951, A005169, A111317, A143951, A224704.

N:= 100:
C:= [0,[1,1],seq([-x^i,1],i=2..N,3)]:
S:= series(numtheory:-cfrac(C),x,N+1):
seq(coeff(S,x,j),j=0..N); # Robert Israel, Feb 18 2024

nmax = 60; CoefficientList[Series[1/Fold[(1 - #2/#1) &, 1, Reverse[x^(3*Range[nmax + 1]-1)]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 25 2017 *)

{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+2)*CF)); polcoeff(CF, n, x)}
for(n=0,55,print1(a(n),", "))

a(n) ~ c * d^n, where d = 1.26326802855134275222... and c = 0.16506173508242936... - Vaclav Kotesovec, Aug 25 2017
From Peter Bala, Jul 03 2019: (Start)
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+2*n)/((1-q^3)*(1-q^6)*...*(1-q^(3*n))) and D(q) = Sum_{n >= 0} (-1)^n*q^(3*n^2-n)/((1-q^3)*(1-q^6)*...*(1-q^(3*n))). Cf. A143951, A224704 and A206737.
D(q) has a simple real zero at x = 0.79159764784576529644 .... The constants c and d quoted in the above asymptotic approximation are given by d = 1/x and c = - N(x)/(x*D'(x)), where the prime indicates differentiation w.r.t. q. (End)

A238871 Number of weakly unimodal compositions of n with absolute difference of successive parts <= 1.

Original entry on oeis.org

1, 1, 2, 4, 6, 10, 14, 21, 27, 40, 52, 70, 92, 124, 156, 206, 264, 335, 425, 539, 673, 847, 1052, 1300, 1611, 1990, 2433, 2977, 3638, 4420, 5367, 6496, 7829, 9439, 11341, 13590, 16270, 19425, 23135, 27525, 32697, 38745, 45844, 54168, 63875, 75247, 88493, 103892

Offset: 0

Joerg Arndt, Mar 21 2014

nonn

			The a(8) = 27 such compositions are:
01:  [ 1 1 1 1 1 1 1 1 ]
02:  [ 1 1 1 1 1 1 2 ]
03:  [ 1 1 1 1 1 2 1 ]
04:  [ 1 1 1 1 2 1 1 ]
05:  [ 1 1 1 1 2 2 ]
06:  [ 1 1 1 2 1 1 1 ]
07:  [ 1 1 1 2 2 1 ]
08:  [ 1 1 1 2 3 ]
09:  [ 1 1 2 1 1 1 1 ]
10:  [ 1 1 2 2 1 1 ]
11:  [ 1 1 2 2 2 ]
12:  [ 1 2 1 1 1 1 1 ]
13:  [ 1 2 2 1 1 1 ]
14:  [ 1 2 2 2 1 ]
15:  [ 1 2 2 3 ]
16:  [ 1 2 3 2 ]
17:  [ 2 1 1 1 1 1 1 ]
18:  [ 2 2 1 1 1 1 ]
19:  [ 2 2 2 1 1 ]
20:  [ 2 2 2 2 ]
21:  [ 2 3 2 1 ]
22:  [ 2 3 3 ]
23:  [ 3 2 1 1 1 ]
24:  [ 3 2 2 1 ]
25:  [ 3 3 2 ]
26:  [ 4 4 ]
27:  [ 8 ]

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

Cf. A001522, A001523, A005169, A034297, A238870, A238872.

A239928 Expansion of F(x^2, x) where F(x,y) is the g.f. of A239927.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 2, 0, 1, 3, 1, 1, 4, 3, 2, 5, 6, 4, 6, 10, 8, 9, 15, 15, 15, 22, 26, 26, 33, 43, 45, 52, 69, 76, 85, 109, 127, 141, 173, 209, 235, 278, 340, 390, 452, 550, 643, 742, 890, 1054, 1221, 1445, 1720, 2007, 2356, 2803, 3291, 3853, 4568, 5385, 6309, 7450, 8800, 10330, 12164, 14372, 16905, 19879

Offset: 0

Joerg Arndt, Mar 29 2014

nonn

What does this sequence count?

Joerg Arndt, Table of n, a(n) for n = 0..1000

Cf. A000108 (F(1, x)), A143951 (F(x, 1)), A005169 (F(x, x), with interlaced zeros), A227310 (F(x, x^2)).

N=66; x='x+O('x^N);
F(x, y, d=0)=if (d>N, 1, 1 / (1-x*y * F(x, x^2*y, d+1) ) );
Vec( F(x^2, x) )

G.f.: 1/(1 - x^3/(1 - x^7/(1 - x^11/(1 - x^15/(1 - x^19/(1 - x^23/( ... ))))))).

A286933 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of continued fraction 1/(1 - kx/(1 - kx^2/(1 - kx^3/(1 - kx^4/(1 - k*x^5/(1 - ...)))))).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 4, 2, 0, 1, 4, 9, 12, 3, 0, 1, 5, 16, 36, 32, 5, 0, 1, 6, 25, 80, 135, 88, 9, 0, 1, 7, 36, 150, 384, 513, 248, 15, 0, 1, 8, 49, 252, 875, 1856, 1971, 688, 26, 0, 1, 9, 64, 392, 1728, 5125, 9024, 7533, 1920, 45, 0, 1, 10, 81, 576, 3087, 11880, 30125, 43776, 28836, 5360, 78, 0

Offset: 0

Ilya Gutkovskiy, May 16 2017

			G.f. of column k: A(x) = 1 + k*x + k^2*x^2 + k^2*(k + 1)*x^3 + k^3*(k + 2)*x^4 + k^3*(k^2 + 3*k + 1)*x^5 + ...
Square array begins:
  1,  1,   1,    1,     1,     1,  ...
  0,  1,   2,    3,     4,     5,  ...
  0,  1,   4,    9,    16,    25,  ...
  0,  2,  12,   36,    80,   150,  ...
  0,  3,  32,  135,   384,   875,  ...
  0,  5,  88,  513,  1856,  5125,  ...

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

Columns k=0..1 give: A000007, A005169.
Rows n=0..3 give: A000012, A001477, A000290, A011379.
Main diagonal gives A291274.
Cf. A286932.

Table[Function[k, SeriesCoefficient[1/(1 + ContinuedFractionK[-k x^i, 1, {i, 1, n}]), {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten

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

G.f. of column k (for k > 0): (Sum_{j>=0} (-k)^j*x^(j*(j+1))/Product_{i=1..j} (1 - x^i)) / (Sum_{j>=0} (-k)^j*x^(j^2)/Product_{i=1..j} (1 - x^i)).

A288005 Number of symmetrical fountains of n coins.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 3, 5, 4, 9, 8, 15, 14, 26, 24, 46, 42, 79, 73, 137, 128, 239, 221, 414, 385, 719, 668, 1249, 1161, 2167, 2016, 3762, 3499, 6531, 6075, 11336, 10546, 19676, 18306, 34153, 31775, 59279, 55155, 102890, 95733, 178587, 166165, 309968, 288412

Offset: 0

Seiichi Manyama, Sep 01 2017

nonn

			a(7) = 5:
.. O O O ....... O O ....... O ... O ......... O ........................
. O O O O ... O O O O O ... O O O O O ... O O O O O O ... O O O O O O O .

Alois P. Heinz, Table of n, a(n) for n = 0..8354

Cf. A005169, A288006.

b:= proc(n, i, p) option remember; `if`(n<0, 0, `if`(n=0,
      `if`(p<0 and i=1, 1, 0), `if`(n=i or n=i+p, 1, 0)+
      `if`(i<1 and p=1, 0, b(n-2*i, i, -p))+b(n-2*(i+p), i+p, -p)))
    end:
a:= n-> `if`(n=0, 1, b(n, 0, 1)):
seq(a(n), n=0..60);  # Alois P. Heinz, Sep 02 2017

b[n_, i_, p_] := b[n, i, p] = If[n < 0, 0, If[n == 0, If[p < 0 && i == 1, 1, 0], If[n == i || n == i + p, 1, 0] + If[i < 1 && p == 1, 0, b[n - 2i, i, -p]] + b[n - 2(i + p), i + p, -p]]];
a[n_] := If[n == 0, 1, b[n, 0, 1]];
a /@ Range[0, 60] (* Jean-François Alcover, Nov 14 2020, after Alois P. Heinz *)

a(33)-a(48) from Alois P. Heinz, Sep 02 2017

A288006 Number of distinct fountains of n coins.

Original entry on oeis.org

1, 1, 1, 2, 2, 4, 6, 10, 15, 27, 43, 75, 124, 216, 364, 634, 1081, 1879, 3229, 5609, 9680, 16809, 29077, 50482, 87452, 151811, 263201, 456871, 792468, 1375530, 2386580, 4142425, 7188332, 12476743, 21652780, 37582311, 65225643, 113210394, 196487131, 341036576

Offset: 0

Seiichi Manyama, Sep 01 2017

nonn

We regard fountains as equivalent if one can be transformed into another by symmetries.

Alois P. Heinz, Table of n, a(n) for n = 0..4179

Cf. A005169, A288005.

g:= proc(n, i) option remember; `if`(n=0, 1,
      add(g(n-j, j), j=1..min(i+1, n)))
    end:
b:= proc(n, i, p) option remember; `if`(n<0, 0, `if`(n=0,
      `if`(p<0 and i=1, 1, 0), `if`(n=i or n=i+p, 1, 0)+
      `if`(i<1 and p=1, 0, b(n-2*i, i, -p))+b(n-2*(i+p), i+p, -p)))
    end:
a:= n-> (g(n, 0)+`if`(n=0, 1, b(n, 0, 1)))/2:
seq(a(n), n=0..60);  # Alois P. Heinz, Sep 02 2017

g[n_, i_] := g[n, i] = If[n == 0, 1,
     Sum[g[n-j, j], {j, 1, Min[i+1, n]}]];
b[n_, i_, p_] := b[n, i, p] = If[n < 0, 0, If[n == 0,
     If[p < 0 && i == 1, 1, 0], If[n == i || n == i+p, 1, 0] +
     If[i < 1 && p == 1, 0, b[n - 2i, i, -p]] + b[n - 2(i+p), i+p, -p]]];
a[n_] := (g[n, 0] + If[n == 0, 1, b[n, 0, 1]])/2;
Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Aug 01 2021, after Alois P. Heinz *)

a(n) = (A005169(n) + A288005(n)) / 2.

a(33)-a(39) from Alois P. Heinz, Sep 02 2017

A291146 Expansion of 1/(1 - x/(1 - x^8/(1 - x^27/(1 - x^64/(1 - x^125/(1 - x^216/(1 - ... - x^(n^3)/(1 - ...)))))))), a continued fraction.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 14, 18, 23, 29, 36, 44, 53, 64, 78, 96, 119, 148, 184, 228, 281, 345, 423, 519, 639, 788, 973, 1202, 1484, 1830, 2254, 2774, 3415, 4206, 5183, 6390, 7880, 9717, 11979, 14762, 18188, 22408, 27609, 34022, 41931

Offset: 0

Seiichi Manyama, Aug 18 2017

nonn

			G.f. = 1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + 2*x^9 + 3*x^10 + 4*x^11 + 5*x^12 + ...

Column k=3 of A290771.

Cf. A005169, A206739.

A291878 Triangle read by rows: T(n,k) = number of fountains of n coins and height k.

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 2, 0, 1, 4, 0, 1, 7, 1, 0, 1, 12, 2, 0, 1, 20, 5, 0, 1, 33, 11, 0, 1, 54, 22, 1, 0, 1, 88, 44, 2, 0, 1, 143, 85, 5, 0, 1, 232, 161, 12, 0, 1, 376, 302, 25, 0, 1, 609, 559, 52, 1, 0, 1, 986, 1026, 105, 2, 0, 1, 1596, 1870, 207, 5, 0, 1

Offset: 0

Seiichi Manyama, Sep 05 2017

Same as A187080, with trailing zeros omitted.

			T(6, 1) = 1;
. O O O O O O .
-------------------------------------------------------
T(6, 2) = 7;
.. O O ......... O O ..... O . O ..
. O O O O ... O O O O ... O O O O .
.......................................................
.. O ............. O ............. O ............. O ..
. O O O O O ... O O O O O ... O O O O O ... O O O O O .
-------------------------------------------------------
T(6, 3) = 1;
... O ...
.. O O ..
. O O O .
-------------------------------------------------------
First few rows are:
  1;
  0,  1;
  0,  1;
  0,  1,   1;
  0,  1,   2;
  0,  1,   4;
  0,  1,   7,   1;
  0,  1,  12,   2;
  0,  1,  20,   5;
  0,  1,  33,  11;
  0,  1,  54,  22,  1;
  0,  1,  88,  44,  2;

Alois P. Heinz, Rows n = 0..1000, flattened

Row sums give A005169.
Columns 0-2 give A000007, A000012, A000071.
Cf. A047998, A187080.

b:= proc(n, i, h) option remember; `if`(n=0, x^h,
      add(b(n-j, j, max(h, j)), j=1..min(i+1, n)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0$2)):
seq(T(n), n=0..30);  # Alois P. Heinz, Sep 05 2017

b[n_, i_, h_] := b[n, i, h] = If[n == 0, x^h, Sum[b[n - j, j, Max[h, j]], {j, 1, Min[i + 1, n]}]];
T[n_] := Table[Coefficient[#, x, i], {i, 0, Exponent[#, x]}]& @ b[n, 0, 0];
Table[T[n], {n, 0, 30}] // Flatten (* Jean-François Alcover, May 31 2019, after Alois P. Heinz *)

from sympy.core.cache import cacheit
from sympy import Symbol, Poly, flatten
x=Symbol('x')
@cacheit
def b(n, i, h): return x**h if n==0 else sum([b(n - j, j, max(h, j)) for j in range(1, min(i + 1, n) + 1)])
def T(n): return 1 if n==0 else Poly(b(n, 0, 0)).all_coeffs()[::-1]
print(flatten(map(T, range(31)))) # Indranil Ghosh, Sep 06 2017

A295944 Expansion of e.g.f. 1/(1 - x/(1 - x^2/(2 - x^3/(3 - x^4/(4 - x^5/(5 - x^6/(6 - x^7/(7 - ...)))))))), a continued fraction.

Original entry on oeis.org

1, 1, 2, 9, 48, 330, 2760, 26670, 295680, 3686760, 51080400, 778516200, 12944131200, 233156523600, 4522777459200, 94000269963600, 2083918752115200, 49086474041404800, 1224240044169542400, 32229413145084355200, 893129953569780326400, 25987602379142314310400, 792175050968260985625600

Offset: 0

Ilya Gutkovskiy, Nov 30 2017

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..300

Cf. A005169, A181167, A257544, A295945.

nmax = 22; CoefficientList[Series[1/(1 + ContinuedFractionK[-x^k, k, {k, 1, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!

a(n) ~ c * d^n * n!, where d = 1.38558212161941692858602713469062337279193542118277136584639901149123656221... and c = 0.53969028910223464320214486945875671476165137860949073877514057198146... - Vaclav Kotesovec, Sep 24 2020

A005170 Erroneous version of A226999.

Original entry on oeis.org

1, 0, 1, 1, 2, 3, 5, 8, 13, 21, 35, 55, 93, 149, 248, 403, 670, 1082

Offset: 1

dead

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

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]

cp's OEIS Frontend

A206738 G.f.: 1/(1 - x^2/(1 - x^5/(1 - x^8/(1 - x^11/(1 - x^14/(1 - x^17/(1 -...- x^(3*n-1)/(1 -...)))))))), a continued fraction.

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A238871 Number of weakly unimodal compositions of n with absolute difference of successive parts <= 1.

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A239928 Expansion of F(x^2, x) where F(x,y) is the g.f. of A239927.

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A286933 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of continued fraction 1/(1 - k*x/(1 - k*x^2/(1 - k*x^3/(1 - k*x^4/(1 - k*x^5/(1 - ...)))))).

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A288005 Number of symmetrical fountains of n coins.

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A288006 Number of distinct fountains of n coins.

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A291146 Expansion of 1/(1 - x/(1 - x^8/(1 - x^27/(1 - x^64/(1 - x^125/(1 - x^216/(1 - ... - x^(n^3)/(1 - ...)))))))), a continued fraction.

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A291878 Triangle read by rows: T(n,k) = number of fountains of n coins and height k.

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A286933 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of continued fraction 1/(1 - kx/(1 - kx^2/(1 - kx^3/(1 - kx^4/(1 - k*x^5/(1 - ...)))))).