A088352 -id:A088352 - cp's OEIS frontend

A088354 G.f. = continued fraction: A(x)=1/(1-x-x/(1-x^2-x^2/(1-x^3-x^3/(1-x^4-x^4/(...))))).

1, 2, 4, 10, 24, 60, 150, 376, 944, 2372, 5962, 14988, 37684, 94752, 238252, 599090, 1506440, 3788036, 9525280, 23952020, 60229184, 151450970, 380835368, 957640640, 2408063340, 6055266600, 15226449480, 38288118984, 96278523274, 242100012876, 608779761460, 1530825191912

Offset: 0

Paul D. Hanna, Sep 26 2003

From Peter Bala, Jul 29 2019: (Start)
a(n) is the number of triangle stacks of large Schröder type containing n down-triangles. See Links for a definition and an illustration.
Cf. A088352 for triangle stacks of large Schröder type on n triangles. Cf. A224704, which enumerates triangle stacks (of small Schröder type) on n triangles. (End)

P. Bala, Illustration for a(3) = 10.
P. Bala, Triangle Stacks of large Schröder type

Cf. A088352, A224704.

nmax = 40; CoefficientList[Series[1/(1 - x + ContinuedFractionK[-x^k, 1 - x^(k + 1), {k, 1, nmax}]), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 01 2019 *)

N = 66; x = 'x + O('x^N);
Q(k) = if(k>N, 1,  1 - x^(k+1)*( 1 + 1/Q(k+1) ) );
gf = 1/Q(0);
Vec(gf)
/* Joerg Arndt, May 01 2013 */

G.f.: 1/Q(0), where Q(k)= 1 - x^(k+1) - x^(k+1)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Apr 30 2013
G.f.: T(0)/(1-x), where T(k) = 1 - x^(k+1)/(x^(k+1) - (1-x^(k+1))*(1-x^(k+2))/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 14 2013
a(n) ~ c * d^n, where d = 2.514579643878729188510437194343099820141030855900783271935495710723840992... and c = 0.589519721244409964128200577034763735132770782513329859477444288778116... - Vaclav Kotesovec, Jul 01 2019
From Peter Bala, Jul 29 2019: (Start)
O.g.f. as a continued fraction:
1/(1 - 2*d/(1 - d^2/(1 - (d^2 + d^3)/(1 - d^4/(1 - (d^3 + d^5)/(1 - d^6/( (...) ))))))).
O.g.f. as a ratio of q-series: A(q) = N(q)/D(q), where N(q) = Sum_{n >= 0} (-1)^n*d^(n^2+n)/( (1 - d^(n+1))*Product_{k = 1..n} (1 - d^k)^2 ) and D(q) = Sum_{n >= 0} (-1)^n*d^(n^2)/( Product_{k = 1..n} (1 - d^k)^2 ).
In the above asymptotic formula of Kotesovec, the constant 1/d = 0.3976807823... is the minimal positive real zero of D(q), and is the dominant singularity of N(q)/D(q). (End)

Added more terms, Joerg Arndt, May 01 2013

A326676 Triangular array: T(n,k) equals the number of n triangle stacks of large Schröder type with k down-triangles in the bottom row of the stack.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 0, 2, 1, 0, 0, 1, 3, 1, 0, 0, 1, 3, 4, 1, 0, 0, 1, 3, 6, 5, 1, 0, 0, 0, 4, 7, 10, 6, 1, 0, 0, 0, 3, 10, 14, 15, 7, 1, 0, 0, 0, 2, 11, 21, 25, 21, 8, 1, 0, 0, 0, 1, 10, 28, 40, 41, 28, 9, 1, 0, 0, 0, 1, 9, 31, 60, 71, 63, 36, 10, 1

Offset: 0

Peter Bala, Jul 17 2019

We define two types of plane triangles of unit area - up-triangles with vertices at the lattice points (x, y), (x+1, y+1) and (x+2, y) and down-triangles with vertices at the lattice points (x, y), (x-1, y+1) and (x+1, y+1).
To construct a triangle stack of large Schröder type we start with a horizontal row of k contiguous down-triangles forming the base row of the stack. Subsequent rows of the stack are formed by placing up-triangles on some, all or none of the down-triangles of the previous row. In the spaces between pairs of adjacent up-triangles further down-triangles may be placed. For an example, see the illustration in the Links section. There is an obvious bijective correspondence between triangle stacks of large Schröder type with a base of k down-triangles and large Schröder paths of semilength k. For another version of this array see A129179.
For triangle stacks of small Schröder type, where the base row consists of contiguous up-triangles, see A224704.

			Triangle begins
   n\k  0  1   2   3   4   5   6   7   8   9  10
   - - - - - - - - - - - - - - - - - - - - - - -
   0 |  1
   1 |  0  1
   2 |  0  1   1
   3 |  0  0   2   1
   4 |  0  0   1   3   1
   5 |  0  0   1   3   4   1
   6 |  0  0   1   3   6   5   1
   7 |  0  0   0   4   7  10   6   1
   8 |  0  0   0   3  10  14  15   7   1
   9 |  0  0   0   2  11  21  25  21   8  1
  10 |  0  0   0   1  10  28  40  41  28  9  1
...

P. Bala, Example of triangle stack of large Schröder type
P. Bala, Triangle Stacks of large Schröder type

Row sums A088352. Column sums A006318. Cf. A047998, A129179, A224704.

O.g.f. as a continued fraction: (q marks the area of the stack and b marks down-triangles in the base of the stack)
A(q,b) = 1/(1 - q*b - q^2*b/(1 - q^3*b - q^4*b/(1 - q^5*b - q^6*b/( (...) )))) = 1 + b*q + (b + b^2)*q^2 + (2*b^2 + b^3)*q^3 + (b^2 + 3*b^3 + b^4)*q^4 + ....
A(q,b) = 1/(1 - (q + q^2)*b/(1 - q^4*b/(1 - (q^3 + q^6)*b/(1 - q^8*b/(1 - (q^5 + q^10)*b/(1 - q^12*b/( (...) ))))))).
O.g.f. as a ratio of q-series: N(q,b)/D(q,b), where N(q,b) = Sum_{n >= 0} (-1)^n*q^(2*n^2+2*n)*b^n/( (Product_{k = 1..n} 1 - q^(2*k)) * (Product_{k = 1..n+1} 1 - q^(2*k-1)*b) ) and D(q,b) = Sum_{n >= 0} (-1)^n*q^(2*n^2)*b^n/( (Product_{k = 1..n} 1 - q^(2*k)) * (Product_{k = 1..n} 1 - q^(2*k-1)*b) ).

A292800 Expansion of 1/(1 - x - x^3/(1 - x^5 - x^7/(1 - x^9 - x^11/(1 - x^13 - x^15/(1 - ...))))), a continued fraction.

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 6, 9, 14, 21, 32, 49, 74, 113, 172, 262, 399, 607, 925, 1409, 2146, 3269, 4979, 7584, 11552, 17596, 26803, 40826, 62187, 94725, 144287, 219782, 334776, 509939, 776752, 1183167, 1802230, 2745201, 4181558, 6369454, 9702111, 14778499, 22510979, 34289286, 52230301, 79558503

Offset: 0

Ilya Gutkovskiy, Sep 23 2017

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A088352, A292802.

nmax = 45; CoefficientList[Series[1/(1 - x + ContinuedFractionK[-x^(4 k - 1), 1 - x^(4 k + 1), {k, 1, nmax}]), {x, 0, nmax}], x]

a(n) ~ c * d^n, where d = 1.523225094265657459818421502249017511338863636291677936060889201502867407829... and c = 0.47457266943464547141454496057039844482970984208404015222172896259335... - Vaclav Kotesovec, Sep 24 2017

A292802 Expansion of 1/(1 - x^2 - x^3/(1 - x^5 - x^7/(1 - x^11 - x^13/(1 - ... - x^prime(2k)/(1 - x^prime(2k+1) - ...))))), a continued fraction.

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 2, 3, 5, 5, 10, 11, 17, 25, 31, 50, 64, 93, 134, 178, 266, 360, 512, 731, 1001, 1447, 2003, 2829, 4011, 5575, 7939, 11097, 15634, 22085, 30909, 43724, 61369, 86389, 121810, 171007, 241216, 339125, 477292, 672364, 945252, 1331677, 1873473, 2636704, 3712653

Offset: 0

Ilya Gutkovskiy, Sep 23 2017

nonn

Cf. A000040, A088352, A285407, A292800, A292801.

nmax = 48; CoefficientList[Series[1/(1 - x^2 + ContinuedFractionK[-x^Prime[2 k], 1 - x^Prime[2 k + 1], {k, 1, nmax}]), {x, 0, nmax}], x]

A292855 Expansion of 1/(1 - x - 2x^2/(1 - 3x^3 - 4x^4/(1 - 5x^5 - 6x^6/(1 - 7x^7 - 8*x^8/(1 - ...))))), a continued fraction.

Original entry on oeis.org

1, 1, 3, 5, 11, 27, 63, 143, 341, 799, 1865, 4417, 10401, 24433, 57619, 135749, 319683, 753427, 1775207, 4182359, 9855389, 23222687, 54718921, 128937361, 303821873, 715906625, 1686933723, 3975020013, 9366551195, 22070960907, 52007117407, 122547413479, 288765804957, 680436157615

Offset: 0

Ilya Gutkovskiy, Sep 25 2017

nonn

Cf. A088352, A088357.

nmax = 33; CoefficientList[Series[1/(1 - x + ContinuedFractionK[-2 k x^(2 k), 1 - (2 k + 1) x^(2 k + 1), {k, 1, nmax}]), {x, 0, nmax}], x]

a(n) ~ c * d^n, where d = 2.35636016857596143712421472862749989350673596686819... and c = 0.353844135039289092297842723019941866883167102736... - Vaclav Kotesovec, Sep 25 2017

A293855 G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies A(x) = 1/(1 - x^a(1) - x^a(2)/(1 - x^a(3) - x^a(4)/(1 - x^a(5) - x^a(6)/(1 - ... )))), a continued fraction.

Original entry on oeis.org

1, 1, 2, 3, 5, 9, 15, 27, 47, 82, 145, 253, 445, 781, 1369, 2405, 4219, 7405, 12998, 22809, 40035, 70263, 123316, 216434, 379854, 666680, 1170079, 2053582, 3604217, 6325695, 11102130, 19485175, 34198108, 60020567, 105341129, 184882533, 324484395

Offset: 0

Ilya Gutkovskiy, Oct 17 2017

nonn

			G.f. =  1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 9*x^5 + 15*x^6 + 27*x^7 + 47*x^8 + 82*x^9 + 145*x^10 + ... = 1/(1 - x - x^2/(1 - x^3 - x^5/(1 - x^9 - x^15/(1 - x^27 - x^47/(1 - x^82 - x^145/(1 - ...)))))).

Cf. A088352, A213411.

A301411 G.f. A(x) satisfies: A(x) = 1/(1 - xA(x) - xA(x)^2/(1 - xA(x)^3 - xA(x)^4/(1 - xA(x)^5 - xA(x)^6/(1 - ...)))), a continued fraction.

Original entry on oeis.org

1, 2, 12, 108, 1192, 14848, 200432, 2866752, 42853392, 663565616, 10579117744, 172911177584, 2888445810864, 49203276384624, 853289008064304, 15047071017842928, 269585532569464752, 4904425594952671344, 90570287337341726256, 1697589267552262891760, 32295562088556275945136

Offset: 0

Ilya Gutkovskiy, Mar 20 2018

nonn

			G.f. A(x) = 1 + 2*x + 12*x^2 + 108*x^3 + 1192*x^4 + 14848*x^5 + 200432*x^6 + 2866752*x^7 + 42853392*x^8 + ...

Cf. A088352, A092869, A192737, A301410.

A306575 Expansion of 1/(1 - x - x^2/(1 - x^2 - x^3/(1 - x^3 - x^4/(1 - x^4 - x^5/(1 - ...))))), a continued fraction.

Original entry on oeis.org

1, 1, 2, 3, 6, 11, 21, 40, 77, 148, 285, 550, 1061, 2049, 3957, 7644, 14768, 28535, 55138, 106549, 205902, 397906, 768967, 1486070, 2871932, 5550233, 10726300, 20729542, 40061784, 77423250, 149628008, 289170949, 558851751, 1080037175, 2087280839, 4033881485

Offset: 0

Ilya Gutkovskiy, Jun 25 2019

nonn

Cf. A029838, A083365, A088352.

nmax = 35; CoefficientList[Series[1/(1 - x + ContinuedFractionK[-x^k, 1 - x^k, {k, 2, nmax}]), {x, 0, nmax}], x]

a(n) ~ c * d^n, where d = 1.9326019136649450138850556203... and c = 0.389707331111778150048054243... - Vaclav Kotesovec, Jul 01 2019

cp's OEIS Frontend

A088354 G.f. = continued fraction: A(x)=1/(1-x-x/(1-x^2-x^2/(1-x^3-x^3/(1-x^4-x^4/(...))))).

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A326676 Triangular array: T(n,k) equals the number of n triangle stacks of large Schröder type with k down-triangles in the bottom row of the stack.

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A292800 Expansion of 1/(1 - x - x^3/(1 - x^5 - x^7/(1 - x^9 - x^11/(1 - x^13 - x^15/(1 - ...))))), a continued fraction.

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A292802 Expansion of 1/(1 - x^2 - x^3/(1 - x^5 - x^7/(1 - x^11 - x^13/(1 - ... - x^prime(2*k)/(1 - x^prime(2*k+1) - ...))))), a continued fraction.

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A292855 Expansion of 1/(1 - x - 2*x^2/(1 - 3*x^3 - 4*x^4/(1 - 5*x^5 - 6*x^6/(1 - 7*x^7 - 8*x^8/(1 - ...))))), a continued fraction.

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A293855 G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies A(x) = 1/(1 - x^a(1) - x^a(2)/(1 - x^a(3) - x^a(4)/(1 - x^a(5) - x^a(6)/(1 - ... )))), a continued fraction.

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A301411 G.f. A(x) satisfies: A(x) = 1/(1 - x*A(x) - x*A(x)^2/(1 - x*A(x)^3 - x*A(x)^4/(1 - x*A(x)^5 - x*A(x)^6/(1 - ...)))), a continued fraction.

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A306575 Expansion of 1/(1 - x - x^2/(1 - x^2 - x^3/(1 - x^3 - x^4/(1 - x^4 - x^5/(1 - ...))))), a continued fraction.

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A292802 Expansion of 1/(1 - x^2 - x^3/(1 - x^5 - x^7/(1 - x^11 - x^13/(1 - ... - x^prime(2k)/(1 - x^prime(2k+1) - ...))))), a continued fraction.

A292855 Expansion of 1/(1 - x - 2x^2/(1 - 3x^3 - 4x^4/(1 - 5x^5 - 6x^6/(1 - 7x^7 - 8*x^8/(1 - ...))))), a continued fraction.

A301411 G.f. A(x) satisfies: A(x) = 1/(1 - xA(x) - xA(x)^2/(1 - xA(x)^3 - xA(x)^4/(1 - xA(x)^5 - xA(x)^6/(1 - ...)))), a continued fraction.