A055879 -id:A055879 - cp's OEIS frontend

A024311 Catacondensed simply-connected monopentapolyhexes.

0, 0, 1, 7, 30, 132, 559, 2416, 10483, 46072, 204155, 912779, 4110644, 18636572, 84985825, 389586145, 1794268460, 8298524480, 38527095859, 179487051589, 838820171722, 3931498431052, 18475618863389, 87036535974062, 410947146076475

Offset: 1

N. J. A. Sloane, May 03 2000

nonn

S. J. Cyvin, B. N. Cyvin, J. Brunvoll, Zhang Fuji, Guo Xiaofeng, R. Tosic, Graph-theoretical studies on fluoranthenoids and fluorenoids: enumeration of some catacondensed systems, J. Molec. Struct. (Theochem), 285 (1993), 179-185.

Cf. A007317, A046697.

a(n) = (1/2) * (A002212(n) - 2*A002212(n - 1) - A055879(n)). - Sean A. Irvine, Jun 26 2019

G.f.: (6*x^3 - 12*x^2 + 6*x + (2*x^2-3*x+1) * sqrt(1-x) * sqrt(1-5*x) - sqrt(1-x^2) * sqrt(1-5*x^2)) / (4*x*(x-1)). - Sean A. Irvine, Jun 26 2019

More terms from Sean A. Irvine, Jun 26 2019

A376277 The least increasing sequence starting with 1, such that the determinants of the Hankel matrices H1 = [a(0), a(1), ..., a(n); ...; a(n), a(n+1), ..., a(2n)] and H2 = [a(1), a(2), ..., a(n+1); ...; a(n+1), a(n+2), ..., a(2n+1)] are > 0.

Original entry on oeis.org

1, 2, 5, 13, 35, 98, 287, 883, 2858, 9708, 34411, 126337, 476767, 1836851, 7185420, 28420613, 113317776, 454468077, 1830556209, 7397188271, 29965426959, 121620119888, 494365414071, 2011965781648, 8196475452837, 33419092543257, 136353532725534, 556669705441210

Offset: 0

Thomas Scheuerle, Sep 23 2024

A Stieltjes moment sequence by its definition.
The Hankel sequence transform gives {1, 1, 1, 1, 1, ...}.
The definition causes that the Hankel sequence transform starting with the second term of this sequence becomes {2, 1, 1, 1, ...}. This single exceptional 2 causes high complexity in the generating function and makes a nice combinatorial interpretation less likely, therefore the keyword "less" was considered.

Cf. A000108 (We obtain the Catalan numbers if we use "least positive sequence" in the definition instead of "least increasing").
Cf. A375181 (Binomial transform).
Cf. A034807, A011973.
Cf. also A055877, A055878, A055879, A350349.

hankelok(s) = {my(m1=floor((#s+1)/2)); my(m2=floor(#s/2)); my(h1=matrix(m1,m1,x,y,s[x+y-1]));  my(h2=matrix(m2,m2,x,y,s[x+y])); return((matdet(h1) > 0) && (matdet(h2) > 0))}
a(max_n) = {my(s=[1,2],k=3); while(#s < max_n, while(hankelok(concat(s,[k]))==0,k=k+1); s=concat(s,[k])); return(s)}

my(N=30, x='x+O('x^N)); Vec(1/(1-2*x/(1-(1/2)*x/(1-(1/2)*x/(1-2*x/(1-((1-sqrt(1-4*x))/(2*x))*x))))))

a(n) = if(n<3, [1, 2, 5][n+1], sum(k=1, floor((n+1)/2), (binomial(n-k+1, k)+binomial(n-k, k-1)-binomial(n-k-3, k-4))*(-1)^(k+1)*a(n-k)))

G.f.: 1/(1-2*x/(1-(1/2)*x/(1-(1/2)*x/(1-2*x/(1-C(x)*x))))), C(x) is the generating function of the Catalan numbers.
G.f.: (1 - sqrt(1 - 4*x)*(-1 + x) - 5*x + 2*x^2)/(1 - 7*x + 11*x^2 + sqrt(1 - 4*x)*(1 - 3*x + x^2)).
(sqrt((x - 4)/x) + 2*x*(13 + (x - 7)*x) - 9)/(2*((x - 4)*(x - 3)*(x - 2)*x - 1)) = Sum_{k>=0} a(k)/x^(k+1).
a(n) = Sum_{k=1..floor((n+1)/2)} (binomial(n-k+1, k) + binomial(n-k, k-1) - binomial(n-k-3, k-4))*(-1)^(k+1)*a(n-k), for n >= 3.
a(n) = Sum_{k=1..floor((n+1)/2)} (A034807(n+1, k) - A011973(n+1, k-4))*(-1)^(k+1)*a(n-k), for n >= 3.

A044044 Catafusenes (see reference for precise definition).

Original entry on oeis.org

0, 1, 2, 2, 6, 10, 19, 36, 72, 135, 274, 543, 1084, 2219, 4438, 9280, 18570, 39587, 79169, 171369, 342738, 751221, 1502472, 3328218, 6656421, 14878455, 29756910, 67030734, 134061570, 304036170, 608072289, 1387247580, 2774495160

Offset: 0

N. J. A. Sloane

nonn

S. J. Cyvin et al., Enumeration and classification of certain polygonal systems...: annelated catafusenes, J. Chem. Inform. Comput. Sci., 34 (1994), 1174-1180.

Cf. A002212 (U), A055879 (V).

G.f.: U(z^2)*V(z) + V(z)/z - V(z^3) - 1, where U(z) = (1 - 3*z - sqrt(1-6*z+ 5*z^2)) / (2*z) and V(z) = (1 - z^2 - sqrt(1-6*z^2+5*z^4)) / (2*z*(1-z)) (from eq. (29) of the Cyvin et al. paper). - Emeric Deutsch, May 02 2004

More terms from Emeric Deutsch, May 02 2004

A044049 Catafusenes (see reference for precise definition).

Original entry on oeis.org

0, 0, 1, 7, 31, 139, 609, 2677, 11827, 52648, 236071, 1065774, 4841428, 22115678, 101531221, 468224323, 2168076055, 10076177264, 46986612235, 219777649588, 1030892581741, 4848039893338, 22853638285655, 107970202258065

Offset: 0

N. J. A. Sloane

nonn

S. J. Cyvin et al., Enumeration and Classification of Certain Polygonal Systems Representing Polycyclic Conjugated Hydrocarbons: Annelated Catafusenes, J. Chem. Inform. Comput. Sci., 34 (1994), 1174-1180.

Cf. A002212 (U), A055879 (V).

G.f.: (U(z) + V(z) + U(z)^2 + U(z^2) - U(z^2)*V(z) + V(z^3))/2 + (U(z)^3 - U(z^3))/6 - V(z)/z + 1, where U(z) = (1 - 3*z - sqrt(1-6*z+5*z^2)) / (2*z) and V(z) = (1 - z^2 - sqrt(1-6*z^2+5*z^4)) / (2*z*(1-z)) (from eq. (26)-(29) of the Cyvin et al. paper). - Emeric Deutsch, May 02 2004

More terms from Emeric Deutsch, May 02 2004

cp's OEIS Frontend

A024311 Catacondensed simply-connected monopentapolyhexes.

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A376277 The least increasing sequence starting with 1, such that the determinants of the Hankel matrices H1 = [a(0), a(1), ..., a(n); ...; a(n), a(n+1), ..., a(2n)] and H2 = [a(1), a(2), ..., a(n+1); ...; a(n+1), a(n+2), ..., a(2n+1)] are > 0.

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PARI

PARI

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A044044 Catafusenes (see reference for precise definition).

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Author

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Extensions

A044049 Catafusenes (see reference for precise definition).

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Author

Keywords

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Extensions

cp's OEIS Frontend

A024311 Catacondensed simply-connected monopentapolyhexes.

Views

Author

Keywords

Links

Crossrefs

Formula

Extensions

A376277 The least increasing sequence starting with 1, such that the determinants of the Hankel matrices H1 = [a(0), a(1), ..., a(n); ...; a(n), a(n+1), ..., a(2*n)] and H2 = [a(1), a(2), ..., a(n+1); ...; a(n+1), a(n+2), ..., a(2*n+1)] are > 0.

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Author

Keywords

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Programs

PARI

PARI

PARI

Formula

A044044 Catafusenes (see reference for precise definition).

Views

Author

Keywords

References

Crossrefs

Formula

Extensions

A044049 Catafusenes (see reference for precise definition).

Views

Author

Keywords

Links

Crossrefs

Formula

Extensions

A376277 The least increasing sequence starting with 1, such that the determinants of the Hankel matrices H1 = [a(0), a(1), ..., a(n); ...; a(n), a(n+1), ..., a(2n)] and H2 = [a(1), a(2), ..., a(n+1); ...; a(n+1), a(n+2), ..., a(2n+1)] are > 0.