A224704 -id:A224704 - cp's OEIS frontend

A111330 Let qf(a,q) = Product_{j >= 0} (1-a*q^j); g.f. is qf(q,q^4)/qf(q^3,q^4).

1, -1, 0, 1, -1, -1, 2, 0, -2, 1, 1, -1, -1, 1, 2, -2, -2, 3, 1, -4, 0, 5, -1, -5, 2, 5, -4, -5, 6, 4, -6, -4, 7, 4, -10, -2, 12, 0, -13, 2, 13, -4, -14, 6, 17, -10, -17, 14, 15, -17, -15, 21, 15, -26, -13, 31, 9, -35, -5, 39, 2, -44, 3, 49, -12, -52, 21, 53, -27, -55, 35, 57, -47, -57, 59, 55, -69, -52, 80, 49, -95, -43, 110, 34, -122

Offset: 0

N. J. A. Sloane, Nov 09 2005

Cf. A111317, A111335, A111374, A224704, A284313, A284316.

From Peter Bala, Nov 28 2020: (Start)
O.g.f.: A(x) = F(x)/G(x) where F(x) = Product_{k >= 0} 1 - x^(4*k+1) (see A284313) and G(x) = Product_{k >= 0} 1 - x^(4*k+3) (see A284316).
Continued fraction representations: A(x) =  1 - x/(1 + x^2 - x^3/(1 + x^4 - x^5/(1 + x^6 - ... ))).
A(x) = 1 - x/(1 - x^2*(x - 1)/(1 - x^5/(1 - x^4*(x^3 - 1)/(1 - x^9/(1 - x^6*(x^5 - 1)/(1 - ... )))))). Cf. A224704. (End)

A206737 G.f.: 1/(1 - x/(1 - x^4/(1 - x^7/(1 - x^10/(1 - x^13/(1 - x^16/(1 -...- x^(3*n-2)/(1 -...)))))))), a continued fraction.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 3, 4, 5, 7, 10, 14, 20, 28, 39, 54, 76, 107, 150, 210, 294, 412, 578, 811, 1137, 1593, 2233, 3131, 4390, 6155, 8629, 12097, 16959, 23777, 33336, 46737, 65524, 91863, 128790, 180563, 253149, 354912, 497581, 697602, 978031, 1371190, 1922395

Offset: 0

Paul D. Hanna, Feb 12 2012

nonn

We have the simple continued fraction expansions (A(x) is the sequence o.g.f.): A(1/n) = [1; n - 2, 1, n^3 - 2, 1, n^4 - 2, 1, n^6 - 2, 1, n^7 - 2, 1, n^9 - 2, 1, n^10 - 2, 1, ...] for n >= 3 and A(-1/n) = [0; 1, n - 1, 1, n^3 - 1, n^4 - 1, 1, n^6 - 1, n^7 - 1, 1, n^9 - 1, n^10 - 1, 1, ...] for n >= 2. Cf. A005169, A111317 and A143951. - Peter Bala, Dec 15 2015

			G.f.: A(x) = 1 + x + x^2 + x^3 + x^4 + 2*x^5 + 3*x^6 + 4*x^7 + 5*x^8 + ...
Simple continued fraction expansions: A(1/10) = 1.11112345816325284441923227158 ... = [1, 8, 1, 998, 1, 9998, 1, 999998, 1, 9999998, 1, 999999998, 1, 9999999998, 1, ...]; A(-1/10) = 0.909082643877542661578687284018 ... = [0, 1, 9, 1, 999, 9999, 1, 999999, 9999999, 1, 999999999, 9999999999, 1, ...]. - _Peter Bala_, Dec 15 2015

Robert Israel, Table of n, a(n) for n = 0..2000
Peter Bala, Some simple continued fraction expansions associated with A206737

Cf. A206738, A143951, A005169, A111317, A143951, A224704.

N:= 100: # to get a(0) .. a(N)
C:= [0,[1,1],seq([-x^i,1],i=1..N,3)]:
S:= series(numtheory:-cfrac(C),x,N+1):
seq(coeff(S,x,j),j=0..N); # Robert Israel, Dec 28 2015

max = 15; CF = 1+x*O[x]^max; M = Sqrt[max+1]//Floor; For[k=0, k <= M, k++, CF = 1/(1-x^(3M-3k+1)*CF)]; CoefficientList[CF, x] (* Jean-François Alcover, Dec 29 2015, adapted from PARI *)
nmax = 50; CoefficientList[Series[1/Fold[(1 - #2/#1) &, 1, Reverse[x^(3*Range[nmax + 1]-2)]], {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+1)*CF)); polcoeff(CF, n, x)}
for(n=0,55,print1(a(n),", "))

a(n) ~ c * d^n, where d = 1.40198938377739909105003523518827... and c = 0.34165269320144328278000954698... - 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+n)/((1-q^3)*(1-q^6)*...*(1-q^(3*n))) and D(q) =  Sum_{n >= 0} (-1)^n*q^(3*n^2-2*n)/((1-q^3)*(1-q^6)*...*(1-q^(3*n))). Cf. A143951, A224704 and A206738.
D(q) has a simple real zero at x = 0.7132721628.... The constants c and d quoted in the above asymptotic approximation for a(n) are given by d = 1/x and c = - N(x)/(x*D'(x)), where the prime indicates differentiation w.r.t. q. (End)

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.

Original entry on oeis.org

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)

A326792 Triangular array: T(n,k) equals the number of small Schröder paths such that the area between the path and the x-axis contains n up-triangles and k down-triangles; n >= 1, k >= 0.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 5, 5, 1, 1, 7, 13, 8, 1, 1, 9, 25, 28, 11, 1, 1, 11, 41, 68, 51, 15, 1, 1, 13, 61, 136, 155, 86, 19, 1, 1, 15, 85, 240, 371, 314, 135, 24, 1, 1, 17, 113, 388, 763, 882, 585, 202, 29, 1, 1, 19, 145, 588, 1411, 2086, 1899, 1019, 290, 35, 1

Offset: 1

Peter Bala, Jul 25 2019

Equivalent definition: T(n,k) equals the number of triangle stacks, as defined in A224704, containing n up-triangles and k down-triangles.
We define two types of plane triangles - up-triangles with vertices at the integer lattice points (x, y), (x+1, y+1) and (x+2, y) and down-triangles with vertices at the integer lattice points (x, y), (x-1, y+1) and (x+1, y+1). The area beneath a small Schröder path may be decomposed in a unique manner into a collection of up- and down-triangles.
To construct a triangle stack (of small Schröder type) we start with a horizontal row of k contiguous up-triangles forming the base row of the stack. Subsequent rows of the stack are formed by placing down-triangles in some, all or none of the spaces between the up-triangles of the previous row. Further up-triangles may be then be placed on these down-triangles and the process repeated. For an example, see the illustration in the Links section. There is an obvious bijective correspondence between triangle stacks with a base of m up-triangles and small Schröder paths of semilength m.

			Triangle begins
  n\k| 0   1   2   3   4   5   6   7  8   9
  - - - - - - - - - - - - - - - - - - - - -
   1 | 1
   2 | 1   1
   3 | 1   3   1
   4 | 1   5   5   1
   5 | 1   7  13   8   1
   6 | 1   9  25  28  11   1
   7 | 1  11  41  68  51  15   1
   8 | 1  13  61 136 155  86  19   1
   9 | 1  15  85 240 371 314 135  24  1
  10 | 1  17 113 388 763 882 585 202 29  1
  ...

P. Bala, Illustration for terms of row 4
P. Bala, Notes on A326792

Row sums A326793. Cf. A224704.

O.g.f. as a continued fraction including initial term 1: (u marks up-triangles and d marks down-triangles)
A(u,d) = 1/(1 - u/(1 - u*d - u^2*d/(1 - u^2*d^2 - u^3*d^2/(1 - u^3*d^3 - u^4*d^3/(1 - u^4*d^4 - (...) ))))) = 1 + u + (1 + d)*u^2 + (1 + 3*d + d^2)*u^3 + ....
A(u,d) = 1/(2 - (1 + u)/(2 - (1 + u^2*d)/(2 - (1 + u^3*d^2)/(2 - (...) )))).
O.g.f. as a ratio of q-series: N(u,d)/D(u,d), where N(u,d) = Sum_{n >= 0} (-1)^n*u^(n^2+n)*d^(n^2)/( Product_{k = 1..n} ( 1 - (u*d)^k )^2 ) and D(u,d) = Sum_{n >= 0} (-1)^n*u^(n^2)*d^(n^2-n)/( Product_{k = 1..n} ( 1 - (u*d)^k )^2 )
Row sums = A326793.

A326793 The number of small Schröder paths such that the area between the path and the x-axis contains n up-triangles.

Original entry on oeis.org

1, 1, 2, 5, 12, 30, 75, 188, 472, 1186, 2981, 7494, 18842, 47376, 119126, 299545, 753220, 1894018, 4762640, 11976010, 30114592, 75725485, 190417684, 478820320, 1204031670, 3027633300, 7613224740, 19144059492, 48139261637, 121050006438

Offset: 0

Peter Bala, Jul 25 2019

We define two types of plane triangles - up-triangles with vertices at the integer lattice points (x, y), (x+1, y+1) and (x+2, y) and down-triangles with vertices at the integer lattice points (x, y), (x-1, y+1) and (x+1, y+1). The area beneath a small Schröder path may be decomposed in a unique manner into a collection of up- and down-triangles. This decomposition produces a triangle stack in the sense of A224704. Here we are counting triangle stacks containing n up-triangles. See the Links section for an illustration.

P. Bala, Illustration for a(3) = 5

Cf. A224704, A326792.

O.g.f. as a continued fraction: (u marks up-triangles)
A(u) = 1/(1 - u/(1 - u - u^2/(1 - u^2 - u^3/(1 - u^3 - u^4/(1 - u^4 - (...) ))))) = 1 + u + 2*u^2 + 5*u^3 + 12*u^4 + ....
A(u) = 1/(1 - u/(1 - (u + u^2)/(1 - u^3/(1 - (u^2 + u^4)/(1 - u^5/(1 - (u^3 + u^6)/(1 - u^7/( (...) )))))))).
A(u) = 1/(2 - (1 + u)/(2 - (1 + u^2)/(2 - (1 + u^3)/(2 - (...) )))).
A(u) = N(u)/D(u), where N(u) = Sum_{n >= 0} u^(n^2+n)/ Product_{k = 1..n} ((1 - u^k)^2) and D(u) = Sum_{n >= 0} u^(n^2)/ Product_{k = 1..n} ((1 - u^k)^2).
a(n) ~ c*d^n, where c = 0.29475 98606 22204 98206 41002 ..., d = 2.51457 96438 78729 18851 04371 ....
Row sums of A326792.

A316585 Number of ways to stack n triangles symmetrically (pointing upwards or downwards depending on row parity).

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 3, 6, 7, 12, 12, 21, 23, 39, 43, 74, 81, 138, 151, 257, 281, 479, 525, 895, 981, 1671, 1830, 3116, 3414, 5813, 6370, 10847, 11887, 20239, 22177, 37758, 41375, 70442, 77193, 131425, 144020, 245197, 268693, 457451, 501288, 853446, 935235, 1592242, 1744834, 2970580, 3255261

Offset: 0

Seiichi Manyama, Jul 07 2018

nonn

			a(9) = 12.
    *   *   *   *   *   *   *   *   *
   / \ / \ / \ / \ / \ / \ / \ / \ / \
  *---*---*---*---*---*---*---*---*---*
.
    *   *   *   *---*   *   *   *
   / \ / \ / \ / \ / \ / \ / \ / \
  *---*---*---*---*---*---*---*---*
.
    *   *   *---*---*   *   *
   / \ / \ / \ / \ / \ / \ / \
  *---*---*---*---*---*---*---*
.
    *   *---*   *   *---*   *
   / \ / \ / \ / \ / \ / \ / \
  *---*---*---*---*---*---*---*
.
    *---*   *   *   *   *---*
   / \ / \ / \ / \ / \ / \ / \
  *---*---*---*---*---*---*---*
.
    *   *---*---*---*   *
   / \ / \ / \ / \ / \ / \
  *---*---*---*---*---*---*
.
    *---*   *---*   *---*
   / \ / \ / \ / \ / \ / \
  *---*---*---*---*---*---*
.
    *---*---*---*---*
   / \ / \ / \ / \ / \
  *---*---*---*---*---*
.
          *   *
         / \ / \
    *   *---*---*   *
   / \ / \ / \ / \ / \
  *---*---*---*---*---*
.
      *           *
     / \         / \
    *---*   *   *---*
   / \ / \ / \ / \ / \
  *---*---*---*---*---*
.
      *       *
     / \     / \
    *---*---*---*
   / \ / \ / \ / \
  *---*---*---*---*
.
        *
       / \
      *---*
     / \ / \
    *---*---*
   / \ / \ / \
  *---*---*---*
.

Cf. A224704, A316384.

Motzk := proc(x,y,twoar)
    option remember;
    if x =0 then
        if y <> 0 or twoar <>0 then
            return 0;
        else
            return 1;
        end if;
    elif y < 0 or y > x or twoar A316585 := proc(twoar)
    local a,x,y ;
    a:= 0 ;
    for x from 0 to twoar do
    for y from 0 to x do
        a := a+Motzk(x,y,twoar) ;
    end do:
    end do:
    a ;
end proc:
seq(A316585(n),n=0..50) ; # R. J. Mathar, Aug 23 2018

Motzk[x_, y_, twoar_] := Motzk[x, y, twoar] = Which[
x == 0, If[y != 0 || twoar != 0, 0, 1],
y < 0 || y > x || twoar < x, 0,
y == 0 , If[Mod[x, 2] == 0, Motzk[x - 1, y + 1, twoar - 2y - 1], 0],
Mod[y, 2] == Mod[x, 2], Motzk[x - 1, y + 1, twoar - 2y - 1] + Motzk[x - 1, y, twoar - 2y] + Motzk[x - 1, y - 1, twoar - 2y + 1],
True, Motzk[x - 1, y, twoar - 2y]];
A316585[twoar_] := Module[{a, x, y}, a = 0; For[x = 0, x <= twoar , x++, For[y = 0, y <= x, y++, a = a + Motzk[x, y, twoar]]]; a];
Table[A316585[n], {n, 0, 50}] (* Jean-François Alcover, Nov 08 2023, after R. J. Mathar *)

a(36)-a(50) from R. J. Mathar, Aug 23 2018

cp's OEIS Frontend

A111330 Let qf(a,q) = Product_{j >= 0} (1-a*q^j); g.f. is qf(q,q^4)/qf(q^3,q^4).

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A206737 G.f.: 1/(1 - x/(1 - x^4/(1 - x^7/(1 - x^10/(1 - x^13/(1 - x^16/(1 -...- x^(3*n-2)/(1 -...)))))))), a continued fraction.

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Mathematica

PARI

Formula

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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Maple

Mathematica

PARI

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A326792 Triangular array: T(n,k) equals the number of small Schröder paths such that the area between the path and the x-axis contains n up-triangles and k down-triangles; n >= 1, k >= 0.

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Author

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Comments

Examples

Links

Crossrefs

Formula

A326793 The number of small Schröder paths such that the area between the path and the x-axis contains n up-triangles.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

A316585 Number of ways to stack n triangles symmetrically (pointing upwards or downwards depending on row parity).

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Mathematica

Extensions