A005169 -id:A005169 - cp's OEIS frontend

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.

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

A167750 Row sums of triangle A167749.

Original entry on oeis.org

1, 1, 2, 4, 9, 20, 45, 102, 231, 524, 1189, 2699, 6127, 13910, 31581, 71702, 162796, 369622, 839216, 1905419, 4326211, 9822571, 22301948, 50636130, 114968339, 261033373, 592671205, 1345648489, 3055268857, 6936928864, 15750162908

Offset: 0

Paul Barry, Nov 10 2009

Hankel transform is A167752.

			G.f. = 1 + x + 2*x^2 + 4*x^3 + 9*x^4 + 20*x^5 + 45*x^6 + 102*x^7 + ...

G.f.: 1/(1-x/(1-x/(1-x^2/(1-x^3/(1-x^4/(1-...)))))) (continued fraction);

G.f.: 1/(1-x*f(x)), f(x) the g.f. of A005169.

A168368 Number of stable connected piles of n bricks.

Original entry on oeis.org

0, 1, 1, 2, 4, 7, 12, 21

Offset: 0

Franklin T. Adams-Watters, Nov 24 2009

This sequence is similar to various sequences with rows of coins (often pennies). Each brick must be offset by 1/2 brick from any bricks under it. However, a brick might only have a brick under one half, provided the pile is stable. A definition of stability is provided in the Paterson paper.
Since the main goal of the Paterson paper is the search for stable piles with maximum overhang (and not to find the number of stable piles), it is likely that some stable piles configurations not leading to significant overhangs have been passed over. So it is not easy to determine whether or not all the stable piles configurations have been accounted for! So even though I added graphic depictions for n=8 and n=9 (cf. A168368a), I will abstain for now from appending a(8) and a(9) to this sequence until it has been ascertained that none are missing (please verify A168368a).
Connected piles only (allowing piles with disconnected subpiles would produce a different sequence).
For this sequence I assumed that the bricks on any given row need not be contiguous. A different sequence would be produced if we required the bricks on any given row to be contiguous (fewer piles of bricks since the piles would constitute a subset of the ones obtained from the current piles).
a(0) is set to 0 (no pile) instead of 1 (empty pile) since the number a(n) of stable connected piles of n bricks with height <= 2 is F_n (n-th Fibonacci number) and at least 4 bricks are needed to make a stable connected pile of height greater than 2 and outgrow the Fibonacci sequence {0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...}.
The number of stable connected piles of n bricks is T_n (n-th tribonacci numbers, with a(0)=0, a(1)=a(2)=1, T_n = {0, 1, 1, 2, 4, 7, 13, 24, 44, 81, ...}) up to n=5, after which it falls behind (for n >= 6).
The following few comments describe only one of many kinds of stacks known to be self stable or stabilizable via a balancing set on top, e.g. the n-diamonds stacks.
The n-diamonds  (of width n, with n^2 bricks) are self stable (no balancing set needed) for n = 1 to 4 only (<1>, <2>, <3>, <4> only):
... & ..... & ....... & ...|=|...
... & ..... & ....... & ..|=|=|..
... & ..... & ..|=|.. & .|=|=|=|.
... & ..... & .|=|=|. & |=|=|=|=|
... & .|=|. & |=|=|=| & .|=|=|=|.
... & |=|=| & .|=|=|. & ..|=|=|..
|=| & .|=|. & ..|=|.. & ...|=|...
For n >= 5, the n-diamonds  need a balancing set with at least 2^n-n^2-1 bricks on top of the n^2 bricks of the n-diamond for a combined total of 2^n-1 bricks.
For n=5 we need a balancing set of at least 2^5-5^2-1 = 6 bricks |B|, e.g., a 2-diamond <2> on top of |B|B| to stabilize <5>:
.<2>.
|B|B|
.<5>.
-----
= 6 = balancing bricks.
For n=6 we need a balancing set of at least 2^6-6^2-1 = 27 bricks |B| to stabilize <6>, e.g.:
.|B|.
|B|B|
.<2>. or .<2>.
|B|B| or |B|B|
.<2>. or .<2>. or .<2>. or .|B|.
|B|B| or |B|B| or |B|B| or |B|B|
.<2>. or .<2>. or .<3>. or .<2>. or .<3>.
|B|B| or |B|B| or |B|B| or |B|B| or |B|B|
.<2>. or .<3>. or .<3>. or .<4>. or .<4>.
|B|B| or |B|B| or |B|B| or |B|B| or |B|B|
.<6>. or .<6>. or .<6>. or .<6>. or .<6>.
----- .. ----- .. ----- .. ----- .. -----
= 27= .. = 29= .. = 28= .. = 27= .. = 29= balancing bricks.
For n=7 we need a balancing set of at least 2^7-7^2-1 = 78 bricks |B| to stabilize <7>, where the balancing set itself might need its own balancing subset as in the following example:
.|B|.
|B|B|
.<3>.
|B|B|
.<5>.
|B|B|
.<6>.
|B|B|
.<7>.
-----
= 79 = balancing bricks (where <5> itself needs 6 balancing bricks).

			Following is a graphic depiction of the stable connected piles of bricks for n = 0 to 4 ordered by increasing height (all piles of a given height within curly braces) and each variant of a given pattern within square brackets, where C(k, i) is k choose i (binomial coefficient), F_n is n-th Fibonacci number [F_n = Sum_{k+i = n-1, i <= k} C(k, i)]. Also, the piles of heights 1 and 2 are grouped within parentheses (since they give the n-th Fibonacci number).
For n = 0, the following 0 [F_0] piles:
( { } )
For n = 1, the following 1 [F_1 = C(0, 0) = 1] pile:
( { |=| } )
For n = 2, the following 1 [F_2 = C(1, 0) = 1] pile:
( { |=|=| } )
For n = 3, the following 2 [F_3 = C(2, 0) + C(1, 1) = 2] piles:
( { ....... } & { .|=|. } )
( { |=|=|=| } & { |=|=| } )
For n = 4, the following 4 [F_4 + 1 = (C(3, 0) + C(2, 1)) + 1 = 3 + 1] piles (where the brick on the third level is necessary for stability):
( { ......... } & { ....... & ....... } ) & { .|=|. }
( { ......... } & { .|=|... & ...|=|. } ) & { |=|=| }
( { |=|=|=|=| } & { |=|=|=| & |=|=|=| } ) & { .|=|. }

Daniel Forgues, Graphic depictions for larger n
Mike Paterson et al., Maximum Overhang
Eric Weisstein's World of Mathematics, Fibonacci Number
Eric Weisstein's World of Mathematics, Tribonacci Number
Eric Weisstein's World of Mathematics, Fibonacci n-Step Number

Cf. A001524, A005169.

Edited by Daniel Forgues, Nov 29 2009, Dec 13 2009

A168396 Triangle, T(n,k) = number of compositions a(1),...,a(j) of n with a(1) = k, such that a(i+1) <= a(i) + 1 for 1 <= i < j.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 5, 4, 2, 1, 1, 9, 6, 4, 2, 1, 1, 15, 11, 7, 4, 2, 1, 1, 26, 19, 12, 7, 4, 2, 1, 1, 45, 33, 21, 13, 7, 4, 2, 1, 1, 78, 57, 37, 22, 13, 7, 4, 2, 1, 1, 135, 99, 64, 39, 23, 13, 7, 4, 2, 1, 1, 234, 172, 112, 68, 40, 23, 13, 7, 4, 2, 1, 1, 406, 298, 194, 119, 70, 41, 23, 13, 7, 4, 2, 1, 1

Offset: 1

Franklin T. Adams-Watters, Nov 24 2009

The definition is a replica of the recursion formula in A005169: T(n,1) = A005169(n). Row sums, central terms and A003116 coincide: sum(T(n,k): k=1..n) = A003116(n); T(2*n-1,n) = A003116(n-1). - Reinhard Zumkeller, Sep 13 2013

			First 16 rows of triangle:
.   1:     1
.   2:     1    1
.   3:     2    1    1
.   4:     3    2    1   1
.   5:     5    4    2   1   1
.   6:     9    6    4   2   1   1
.   7:    15   11    7   4   2   1   1
.   8:    26   19   12   7   4   2   1  1
.   9:    45   33   21  13   7   4   2  1  1
.  10:    78   57   37  22  13   7   4  2  1  1
.  11:   135   99   64  39  23  13   7  4  2  1  1
.  12:   234  172  112  68  40  23  13  7  4  2  1 1
.  13:   406  298  194 119  70  41  23 13  7  4  2 1 1
.  14:   704  518  337 207 123  71  41 23 13  7  4 2 1 1
.  15:  1222  898  586 360 214 125  72 41 23 13  7 4 2 1 1
.  16:  2120 1559 1017 626 373 218 126 72 41 23 13 7 4 2 1 1

Reinhard Zumkeller, Rows n=1..120 of triangle, flattened

Cf. A005169 (first column), A003116 (apparently row sums).

a168396 n k = a168396_tabl !! (n-1) !! (k-1)
a168396_row n = a168396_tabl !! (n-1)
a168396_tabl = [1] : f [[1]] where
   f xss = ys : f (ys : xss) where
     ys = (map sum $ zipWith take [2..] xss) ++ [1] -- Reinhard Zumkeller, Sep 13 2013

b:= proc(n, k) option remember; `if`(n=0, 1,
      add(b(n-j, j+1), j=1..min(n, k)))
    end:
T:= (n, k)-> b(n-k, k+1):
seq(seq(T(n, k), k=1..n), n=1..14); # Alois P. Heinz, Sep 19 2013

t[n_, k_] /; k > n = 0; t[n_, n_] = 1; t[n_, k_] := t[n, k] = Sum[ t[n-k, j], {j, 1, k+1}]; Flatten[ Table[ t[n, k], {n, 1, 13}, {k, 1, n}] ](* Jean-François Alcover, Feb 17 2012, after Pari *)

T(n,k)=if(k>=n,k==n,sum(j=1,k+1,T(n-k,j)))

Tm(n)=local(m);m=matrix(n,n);for(i=1,n,for(j=1,i,m[i,j]=if(i==j,1,sum(k=1,j+1,m[i-j,k]))));m

A238872 Number of strongly unimodal compositions of n with absolute difference of successive parts = 1.

Original entry on oeis.org

1, 1, 1, 3, 2, 3, 3, 4, 3, 6, 4, 3, 5, 6, 4, 9, 5, 3, 7, 7, 5, 9, 6, 6, 8, 9, 5, 9, 8, 6, 10, 6, 5, 15, 8, 9, 10, 7, 7, 12, 10, 3, 11, 15, 7, 15, 8, 6, 13, 12, 9, 12, 9, 9, 14, 12, 7, 15, 12, 6, 15, 13, 6, 21, 12, 12, 13, 6, 11, 15, 15, 9, 14, 12, 8, 24, 10, 9

Offset: 0

Joerg Arndt, Mar 21 2014

nonn

			The a(33) = 15 such compositions of 33 are:
01:  [ 1 2 3 4 5 6 5 4 3 ]
02:  [ 2 3 4 5 6 7 6 ]
03:  [ 3 4 5 6 5 4 3 2 1 ]
04:  [ 3 4 5 6 7 8 ]
05:  [ 4 5 6 7 6 5 ]
06:  [ 5 6 7 6 5 4 ]
07:  [ 5 6 7 8 7 ]
08:  [ 6 7 6 5 4 3 2 ]
09:  [ 7 8 7 6 5 ]
10:  [ 8 7 6 5 4 3 ]
11:  [ 10 11 12 ]
12:  [ 12 11 10 ]
13:  [ 16 17 ]
14:  [ 17 16 ]
15:  [ 33 ]
G.f. = 1 + x + x^2 + 3*x^3 + 2*x^4 + 3*x^5 + 3*x^6 + 4*x^7 + 3*x^8 + 6*x^9 + ...

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Dandan Chen and Rong Chen, Generating Functions of the Hurwitz Class Numbers Associated with Certain Mock Theta Functions, arXiv:2107.04809 [math.NT], 2021.

Cf. A001522, A001523, A005169, A034297, A059618, A238870, A238871, A130695.

a[ n_] := If[ n < 1, Boole[n == 0], If[ OddQ[n], 1, 1/3] Length @ FindInstance[ {x >= 0, y >= 0, z >= 0, x y + y z + z x + x + y + z + 1 == n}, {x, y, z}, Integers, 10^9]]; (* Michael Somos, Jul 04 2015 *)
a[ n_] := If[ n < 1, Boole[n == 0], Length @ FindInstance[ {1 <= y <= n, 1 <= x <= y, 1 <= z <= y, y^2 + (x - x^2 + z - z^2) / 2 == n}, {x, y, z}, Integers, 10^9]]; (* Michael Somos, Jul 04 2015 *)

\\ generate the compositions
a(n)=
{
    if ( n==0, return(1) );
    my( ret=0 );
    my( as, ts );
    for (f=1, n,  \\ first part
        as = 0;
        for (p=f, n, \\ numper of parts in rising half
            as += p; \\ ascending sum
            if ( as > n, break() );
            if ( as == n,  ret+=1;  break() );
            ts = as;  \\ total sum
            forstep (q=p-1, 1, -1,
                ts += q;  \\ descending sum
                if ( ts > n, break() );
                if ( ts == n,  ret+=1;  break() );
            );
        );
    );
    return( ret );
}
v=vector(100,n,a(n-1))

a(2*n) = A130695(2*n) / 3 if n>0. a(2*n + 1) = A130695(2*n + 1) = 3 * H(8*n + 3), where H is the Hurwitz class number, if n>0. - Michael Somos, Jul 04 2015

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

A238870 Number of compositions of n with c(1) = 1, c(i+1) - c(i) <= 1, and c(i+1) - c(i) != 0.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 2, 2, 1, 4, 4, 4, 9, 10, 11, 21, 25, 30, 51, 62, 80, 125, 157, 208, 309, 399, 536, 772, 1013, 1373, 1938, 2574, 3503, 4882, 6540, 8918, 12329, 16611, 22672, 31183, 42182, 57588, 78952, 107092, 146202, 200037, 271831, 371057, 507053, 689885, 941558, 1285655, 1750672, 2388951, 3260459, 4442179, 6060948

Offset: 0

Joerg Arndt, Mar 09 2014

nonn

Number of fountains of n coins with at most two successive coins on the same level.

			The a(10) = 4 such compositions are:
:
:   1:  [ 1 2 1 2 1 2 1 ]  (composition)
:
:  o o o
: ooooooo   (rendering as composition)
:
:     O   O   O
:    O O O O O O O  (rendering as fountain of coins)
:
:
:   2:  [ 1 2 1 2 3 1 ]
:
:     o
:  o oo
: oooooo
:
:           O
:      O   O O
:     O O O O O O
:
:
:   3:  [ 1 2 3 1 2 1 ]
:
:   o
:  oo o
: oooooo
:
:       O
:      O O   O
:     O O O O O O
:
:
:   4:  [ 1 2 3 4 ]
:
:    o
:   oo
:  ooo
: oooo
:
:         O
:        O O
:       O O O
:      O O O O
:

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

Cf. A005169 (fountains of coins), A001524 (weakly unimodal fountains of coins).
Cf. A186085 (1-dimensional sandpiles), A227310 (rough sandpiles).
Cf. A023361 (fountains of coins with all valleys at lowest level).

b:= proc(n, i) option remember; `if`(n=0, 1, add(
      `if`(i=j, 0, b(n-j, j)), j=1..min(n, i+1)))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..60);  # Alois P. Heinz, Mar 11 2014

b[n_, i_] := b[n, i] = If[n == 0, 1, Sum[If[i == j, 0, b[n-j, j]], {j, 1, Min[n, i+1]}]];
a[n_] := b[n, 0];
a /@ Range[0, 60] (* Jean-François Alcover, Nov 07 2020, after Alois P. Heinz *)

# translation of the Maple program by Alois P. Heinz
@CachedFunction
def F(n, i):
    if n == 0: return 1
    return sum( (i!=j) * F(n-j, j) for j in [1..min(n,i+1)] ) # A238870
#    return sum( F(n-j, j) for j in [1, min(n,i+1)] ) # A005169
def a(n): return F(n, 0)
print([a(n) for n in [0..50]])
# Joerg Arndt, Mar 20 2014

a(n) ~ c / r^n, where r = 0.733216317061133379740342579187365700397652443391231594... and c = 0.172010618097928709454463097802313209201440229976513439... . - Vaclav Kotesovec, Feb 17 2017

A285635 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, 0, 0, -2, 2, -4, 0, -2, -6, 0, -10, -6, -12, -20, -20, -40, -46, -68, -104, -132, -204, -280, -394, -578, -790, -1154, -1616, -2294, -3286, -4614, -6610, -9340, -13278, -18878, -26748, -38060, -53978, -76684, -108912, -154600, -219622, -311812, -442818, -628866, -892962, -1268168

Offset: 0

Ilya Gutkovskiy, Apr 23 2017

sign

			G.f.: A(x) = 1 - 2*x + 2*x^2 - 2*x^3 - 2*x^6 + 2*x^7 - 4*x^8 - 2*x^10 - 6*x^11 - ...

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

Cf. A003823, A005169, A007325, A055101, A285636, A285637, A285638.

nmax = 48; 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 = 48; 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^2)/Product[(1 -
x^m), {m, 1, k}], {k, 0, nmax}] / Sum[(-1)^k x^(k (k + 1))/Product[(1 - x^m), {m, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]

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

A285638 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, 0, -1, -2, 0, 0, -1, -4, -4, -2, -3, -6, -13, -16, -19, -24, -38, -60, -82, -110, -150, -224, -324, -458, -637, -898, -1289, -1838, -2609, -3680, -5223, -7430, -10571, -15004, -21272, -30202, -42903, -60960, -86543, -122860, -174450, -247762, -351883, -499668, -709521, -1007532

Offset: 0

Ilya Gutkovskiy, Apr 23 2017

sign

			G.f.: A(x) = 1 - x^2 - 2*x^3 - x^6 - 4*x^7 - 4*x^8 - 2*x^9 - 3*x^10 - 6*x^11 - 13*x^12 - ...

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

Cf. A003823, A005169, A007325, A055101, A285635, A285636, A285637.

nmax = 45; CoefficientList[Series[1/((1/(1 + ContinuedFractionK[-x^k, 1, {k, 1, nmax}])) (1/(1 + ContinuedFractionK[x^k, 1, {k, 1, nmax}]))), {x, 0, nmax}], x]
nmax = 45; CoefficientList[Series[Sum[(-1)^k x^(k^2)/Product[(1 - x^m), {m, 1, k}], {k, 0, nmax}] / (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}]), {x, 0, nmax}], x]

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

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)

cp's OEIS Frontend

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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A167750 Row sums of triangle A167749.

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A168368 Number of stable connected piles of n bricks.

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A168396 Triangle, T(n,k) = number of compositions a(1),...,a(j) of n with a(1) = k, such that a(i+1) <= a(i) + 1 for 1 <= i < j.

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A238872 Number of strongly unimodal compositions of n with absolute difference of successive parts = 1.

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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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A238870 Number of compositions of n with c(1) = 1, c(i+1) - c(i) <= 1, and c(i+1) - c(i) != 0.

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A285635 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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