A002215 -id:A002215 - cp's OEIS frontend

A002216 Harary-Read numbers: restricted hexagonal polyominoes (cata-polyhexes) with n cells.

0, 1, 1, 2, 5, 12, 37, 123, 446, 1689, 6693, 27034, 111630, 467262, 1981353, 8487400, 36695369, 159918120, 701957539, 3101072051, 13779935438, 61557789660, 276327463180, 1245935891922, 5640868033058, 25635351908072, 116911035023017

Offset: 0

N. J. A. Sloane

Named after the American mathematician Frank Harary (1921-2005) and the British mathematician Ronald Cedric Read (1924-2019). - Amiram Eldar, Jun 22 2021

S. J. Cyvin, J. Brunvoll, X. F. Guo and F. J. Zhang, Number of perifusenes with one internal vertex, Rev. Roumaine Chem., Vol. 38, No. 1 (1993), pp. 65-77.
S. J. Cyvin, B. N. Cyvin, and J. Brunvoll, Enumeration of tree-like octagonal systems: catapolyoctagons, ACH Models in Chem., Vol. 134, No. 1 (1997), pp. 55-70.
J. L. Faulon, D. Visco and D. Roe, Enumerating Molecules, In: Reviews in Computational Chemistry Vol. 21, Ed. K. Lipkowitz, Wiley-VCH, 2005.
Wenchen He and Wenjie He, Generation and enumeration of planar polycyclic aromatic hydrocarbons, Tetrahedron, Vol. 42, No. 19 (1986), pp. 5291-5299. See Table 3.
J. V. Knop, K. Szymansky, Željko Jeričević and Nenad Trinajstić, On the total number of polyhexes, Match, Vol. 16 (1984), pp. 119-134.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
N. Trinajstich, Z. Jerievi, J. V. Knop, W. R. Muller and K. Szymanski, Computer generation of isomeric structures, Pure & Appl. Chem., Vol. 55, No. 2 (1983), pp. 379-390.

T. D. Noe, Table of n, a(n) for n = 0..200
L. W. Beineke and R. E. Pippert, On the enumeration of planar trees of hexagons, Glasgow Math. J., Vol. 15, No. 2 (1974), pp. 131-147.
L. W. Beineke and R. E. Pippert, On the enumeration of planar trees of hexagons, Glasgow Math. J., Vol. 15, No. 2 (1974), pp. 131-147. [Annotated scanned copy]
S. J. Cyvin, J. Brunvoll and B. N. Cyvin, Harary-Read numbers for catafusenes: Complete classification according to symmetry, Journal of mathematical chemistry, Vol. 9, No. 1 (1992), pp. 19-31 and 33-38. See Table 2.
F. Harary and R. C. Read, The enumeration of tree-like polyhexes, Proc. Edinb. Math. Soc., Vol. 17, No. 1 (1970), pp. 1-13; alternative link.
J. V. Knop, K. Szymanski, Ž. Jeričević, and N. Trinajstić, On the total number of polyhexes, Match, No. 16 (1984), 119-134.
R. C. Read, Letter to N. J. A. Sloane, Feb 12 1971. (includes 40 terms of A002212 and A002216)
Eric Weisstein's World of Mathematics, Polyhex.
Eric Weisstein's World of Mathematics, Fusene.

Cf. A036359, A005963, A000228, A001998.

Cf. A002212, A002213, A002214, A002215.

CoefficientList[Series[(12+(1-5*x)^(3/2)*(1-x)^(3/2)+24*x-48*x^2- 24*x^3- 3*(3+5 x)*Sqrt[1-5*x^2]*Sqrt[1-x^2]-4*Sqrt[1-5*x^3]*Sqrt[1-x^3])/ (24*x^2),{x,0,40}],x] (* Harvey P. Dale, Dec 23 2013 *)

G.f.: (1/(24*x^2))*(12+24*x-48*x^2-24*x^3 +(1-x)^(3/2)*(1-5*x)^(3/2)-3*(3+5*x)*(1-x^2)^(1/2)*(1-5*x^2)^(1/2) -4*(1-x^3)^(1/2)*(1-5*x^3)^(1/2)).
a(n) = (1/2)[A002214(n)+A002215(n)], n>=1. - Emeric Deutsch, Dec 23 2003
a(n) ~ 5^(n+1/2)/(4*sqrt(Pi)*n^(5/2)). - Vaclav Kotesovec, Aug 09 2013

A323850 Irregular triangle read by rows: T(n,k) (n>=2, 0<=k<=n) = total number of unbranched k-catapolyheptagons with k pentagons and n-k heptagons.

Original entry on oeis.org

1, 1, 1, 2, 3, 3, 1, 6, 12, 16, 6, 2, 20, 58, 82, 53, 18, 3, 72, 256, 432, 352, 174, 40, 6, 272, 1160, 2208, 2256, 1380, 498, 100, 10, 1056, 5120, 11088, 13312, 9992, 4672, 1388, 224, 20, 4160, 22560, 54432, 75344, 66448, 38600, 14840, 3644, 520, 36, 16512, 98304, 262528, 409600, 416192, 286720, 136448, 44032, 9352, 1152, 72

Offset: 2

N. J. A. Sloane, Feb 09 2019

			Triangle begins:
1, 1, 1,
2, 3, 3, 1,
6, 12, 16, 6, 2,
20, 58, 82, 53, 18, 3,
72, 256, 432, 352, 174, 40, 6,
272, 1160, 2208, 2256, 1380, 498, 100, 10,
1056, 5120, 11088, 13312, 9992, 4672, 1388, 224, 20,
4160, 22560, 54432, 75344, 66448, 38600, 14840, 3644, 520, 36,
16512, 98304, 262528, 409600, 416192, 286720, 136448, 44032, 9352, 1152, 72,
...

B. N. Cyvin et al., Isomer enumeration of unbranched catacondensed polygonal systems with pentagons and heptagons, Match, No. 34 (Oct 1996), pp. 109-121. See Table 4.

B. N. Cyvin et al., Isomer enumeration of unbranched catacondensed polygonal systems with pentagons and heptagons, 1996 [Annotated scanned copy of pages 118, 119 only].

The first two columns are A063376, A038177. The right-hand edge is probably either A002215 or A005418.

Equation (32) on page 118 of the scan gives an explicit formula.

A002213 Number of tree-like polyhexes rooted at a hexagon and containing n hexagons.

Original entry on oeis.org

1, 1, 5, 20, 84, 354, 1540, 6704, 29610, 131745, 591049, 2669346, 12131148, 55431285, 254539897, 1174027598, 5436826110, 25269402555, 117838870833, 551192276450, 2585418254532, 12158383558066, 57313008207960

Offset: 1

N. J. A. Sloane

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..200
F. Harary and R. C. Read, The enumeration of tree-like polyhexes, Proc. Edinburgh Math. Soc. (2) 17 (1970), 1-13.

Cf. A002212, A002214, A002215, A002216.

G.f.: x + x*U(x) + (3/2)*x*U(x)^2 + (1/2)*x*U(x^2) + (1/3)*x*U(x)^3 + (2/3)*x*U(x^3), where U(x) = (1 - 3*x - sqrt((1-x)*(1-5*x)))/(2*x).

a(n) ~ 5^(n+1/2)/(2*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Aug 13 2013

Edited by Emeric Deutsch, Feb 18 2004

A120652 Triangle read by rows: numbers of isomers of unbranched a-4-catapolypentagons.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 2, 3, 5, 2, 1, 3, 8, 11, 8, 3, 1, 6, 16, 30, 22, 13, 3, 1, 10, 38, 72, 74, 45, 18, 4, 1, 20, 80, 182, 208, 164, 73, 25, 4, 1, 36, 180, 434, 598, 527, 310, 119, 32, 5, 1, 72, 384, 1048, 1600, 1652, 1120, 548, 172, 41, 5, 1

Offset: 0

N. J. A. Sloane, Aug 18 2006

			Triangle begins
1
1 1
1 1 1
1 2 2 1
2 3 5 2 1
3 8 11 8 3 1
6 16 30 22 13 3 1

J. Brunvoll, S. J. Cyvin and B. N. Cyvin, Isomer enumeration of polygonal systems..., J. Molec. Struct. (Theochem), 364 (1996), 1-13.

Columns give A002215, A121133-A121136.

cp's OEIS Frontend

A002216 Harary-Read numbers: restricted hexagonal polyominoes (cata-polyhexes) with n cells.

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A323850 Irregular triangle read by rows: T(n,k) (n>=2, 0<=k<=n) = total number of unbranched k-catapolyheptagons with k pentagons and n-k heptagons.

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A002213 Number of tree-like polyhexes rooted at a hexagon and containing n hexagons.

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A120652 Triangle read by rows: numbers of isomers of unbranched a-4-catapolypentagons.

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