A026106 -id:A026106 - cp's OEIS frontend

A026118 Number of polyhexes of class PF2 (with two catafusenes annealated to pyrene).

5, 20, 100, 431, 1937, 8548, 38199, 171001, 770934, 3492251, 15905897, 72785480, 334571647, 1544203452, 7154247842, 33260560977, 155126129968, 725639264293, 3403612632885, 16004969728270, 75437244856898, 356337397010035, 1686618801843050

Offset: 6

N. J. A. Sloane

nonn

See reference for precise definition.
From Petros Hadjicostas, Jan 13 2019: (Start)
This sequence is defined by eq. (34), p. 536, in Cyvin et al. (1992). It is denoted by 2^Q_{4+n} (for n >= 2). Thus, a(n+4) = 2^Q_{4+n} for n >= 2 (and that is why the offset here is 6).
For n >= 2, we have a(n+4) = (3/4)*(1 + (-1)^n)*N(floor(n/2)) + (1/4)*(L(n) + 13*Sum_{1 <= i <= n-1} N(i)*N(n-i)), where N(n) = A002212(n) and L(n) = A039658(n).
The sequence (N(n): n >= 1) = (A002212(n): n >= 1) is given by eq. (1), p. 533, in Cyvin et al. (1992), while its g.f. is given by eqs. (2)-(4), p. 1174, in Cyvin et al. (1994). (The g.f. of N(n) = A002212(n) appears also in Harary and Read (1970) as eq. (9) on p. 4.)
The sequence (L(n): n >= 1) = (A039658(n): n >= 1) is given by eq. (22), p. 535, in Cyvin et al (1992), while its g.f. is given by eq. (9), p. 1175, in Cyvin et al. (1994).
The g.f. of the current sequence (a(m): m >= 6) (see below) is given in eq. (A2), p. 1180, in Cyvin et al. (1994), but it can be derived by the above formulae using standard techniques for the calculation of g.f.'s.
For the number of polyhexes of class PF2, we have 1^Q_h = A026106(h) (h >= 5, one catafusene annealated to pyrene), 3^Q_h = A026298(h) (h >= 7, three catafusenes annealated to pyrene), and 4^Q_h = A030519(h) (h >= 8, four catafusenes annealated to pyrene).
(Apparently, the word "annealated" in Cyvin et al. (1992) is spelled "annelated" in Cyvin et al. (1994).)
(End)

S. J. Cyvin, Zhang Fuji, B. N. Cyvin, Guo Xiaofeng, and J. Brunvoll, Enumeration and classification of benzenoid systems. 32. Normal perifusenes with two internal vertices, J. Chem. Inform. Comput. Sci., 32 (1992), 532-540.
S. J. Cyvin, B. N. Cyvin, J. Brunvoll, and E. Brendsdal, Enumeration and classification of certain polygonal systems representing polycyclic conjugated hydrocarbons: annelated catafusenes, J. Chem. Inform. Comput. Sci., 34 (1994), 1174-1180.
F. Harary and R. C. Read, The enumeration of tree-like polyhexes, Proc. Edinburgh Math. Soc. (2) 17 (1970), 1-13.
Eric Weisstein's World of Mathematics, Fusenes.
Eric Weisstein's World of Mathematics, Polyhex.

Cf. A002212, A026106, A026118, A026298, A030519, A030520, A030525, A030529, A030532, A030534, A039658.

From Petros Hadjicostas, Jan 14 2019: (Start)
a(n+4) = (3/4)*(1 + (-1)^n)*N(floor(n/2)) + (1/4)*(L(n) + 13*Sum_{1 <= i <= n-1} N(i)*N(n-i)) for n >= 2, where N(n) = A002212(n) and L(n) = A039658(n).
G.f.: (x^2/4)*(1-x)^(-1)*(10 - 48*x + 74*x^2 - 38*x^3) - (x^2/8)*[13*(1 - 3*x)*(1 - x)^(1/2)*(1 - 5*x)^(1/2) + (1 - x)^(-1)*(7 - 5*x)*(1 - x^2)^(1/2)*(1 - 5*x^2)^(1/2)] (see eq. (A2), p. 1180, in Cyvin et al. (1994)).
(End)

Name edited by Petros Hadjicostas, Jan 13 2019

Terms a(17)-a(28) computed by Petros Hadjicostas, Jan 13 2019 using a g.f. in Cyvin et al. (1994)

A030525 Number of polyhexes of class PF2 with a particular symmetry.

Original entry on oeis.org

0, 1, 2, 5, 10, 21, 41, 85, 167, 345, 680, 1421, 2823, 5953, 11902, 25341, 50993, 109409, 221372, 478189, 972346, 2112181, 4313451, 9415165, 19301199, 42303241, 87015978, 191398741, 394888368, 871297141, 1802488991, 3987998101, 8270169571, 18342194721

Offset: 4

N. J. A. Sloane

nonn

See reference for precise definition.

Cyvin has incorrect a(13)=273 and a(14)=608 in Table III due to using incorrect values for A026298(13) and A026298(14) in Table II. - Sean A. Irvine, Apr 03 2020

S. J. Cyvin, Zhang Fuji, B. N. Cyvin, Guo Xiaofeng, and J. Brunvoll, Enumeration and classification of benzenoid systems. 32. Normal perifusenes with two internal vertices, J. Chem. Inform. Comput. Sci., 32 (1992), 532-540.

Cf. A026106, A026118, A026298, A030519, A030520, A030525, A030529, A030532, A030534.

a(13) and a(14) corrected and more terms from Sean A. Irvine, Apr 03 2020

A030529 Number of polyhexes of class PF2 with a particular symmetry.

Original entry on oeis.org

0, 0, 1, 4, 17, 66, 269, 1102, 4635, 19768, 85659, 375524, 1664015, 7438862, 33515027, 152016610, 693622315, 3181516040, 14661568795, 67850245684, 315187594779, 1469195413102, 6869889480447, 32215398047474, 151467333043437, 713881813137776, 3372142135461789

Offset: 2

N. J. A. Sloane

nonn

See references for precise definition.

Column D_{2h}(b) and Eq. 50 in Cyvin et al. (1994). - Sean A. Irvine, Mar 27 2021

S. J. Cyvin, J. Brunvoll, and B. N. Cyvin, Harary-Read numbers for catafusenes: Complete classification according to symmetry, Journal of mathematical chemistry 9.1 (1992): 19-31 and 33-38. See pages 30 and 38.
S. J. Cyvin, B. N. Cyvin, J. Brunvoll and E. Brendsdal, Enumeration and Classification of Certain Polygonal Systems Representing Polycyclic Conjugated Hydrocarbons: Annelated Catafusenes, Journal of Chemical Information and Modeling [formerly, J. Chem. Inform. Comput. Sci.], 34 (1994), pp. 1174-1180.
S. J. Cyvin, Zhang Fuji, B. N. Cyvin, Guo Xiaofeng, and J. Brunvoll, Enumeration and classification of benzenoid systems. 32. Normal perifusenes with two internal vertices, J. Chem. Inform. Comput. Sci., 32 (1992), 532-540.
Sean A. Irvine, Java program (github)

Cf. A026106, A026118, A026298, A030519, A030520, A030525, A030529, A030532, A030534.

A055879(n)= my(A); if( n<1, 0, n--; A = O(x); for( k = 0, n\2, A = 1 / (1 - x - x^2 / (1 + x - x^2 * A))); polcoeff( A, n));
b(n) = (A055879(2*n+1) - A055879(2*n) - A055879(n)) / 2;
a(n) = if( n<=2, 0, b(n - 2)); \\ Michel Marcus, Apr 03 2020

a(2)=0, a(n+2) = (M(2*n+1) - M(2*n) - M(n)) / 2 where M(n) = A055879(n) [Cyvin Eq. (54)]. - Sean A. Irvine, Apr 03 2020

More terms from Sean A. Irvine, Apr 03 2020

A030532 Number of polyhexes of class PF2 with symmetry point group C_s.

Original entry on oeis.org

0, 1, 6, 35, 168, 807, 3738, 17326, 79909, 369330, 1709087, 7929590, 36880231, 171981241, 804008476, 3767969067, 17699758030, 83328230588, 393123455667, 1858351021018, 8801159427825, 41756067216508, 198437454009869, 944521139813575, 4502419756667924

Offset: 4

N. J. A. Sloane

nonn

See reference for precise definition.

Cyvin has incorrect a(13)=369366 and a(14)=1709123 in Table III due to using incorrect values for A026298(13) and A026298(14) in Table II.

S. J. Cyvin, Zhang Fuji, B. N. Cyvin, Guo Xiaofeng, and J. Brunvoll, Enumeration and classification of benzenoid systems. 32. Normal perifusenes with two internal vertices, J. Chem. Inform. Comput. Sci., 32 (1992), 532-540.
Sean A. Irvine, Java program (github)

Cf. A026106, A026118, A026298, A030519, A030520, A030525, A030529, A030532, A030534.

L(n) = my(x = 'x + O('x^(n+4))); polcoeff((1+x)*(1-3*x^2-sqrt(1-6*x^2+5*x^4))/(2*x^2*(1-x)), n); \\ A039658
Lp(n) = my(x = 'x + O('x^(n+4))); polcoeff((1+x)*(1-6*x^2+7*x^4-(1-3*x^2)*sqrt(1-6*x^2+5*x^4))/(2*x^4*(1-x)), n); \\ A039660
M(n)= my(A); if( n<1, 0, n--; A = O(x); for( k = 0, n\2, A = 1 / (1 - x - x^2 / (1 + x - x^2 * A))); polcoeff( A, n)); \\ A055879
N(n) = polcoeff( (1 - x - sqrt(1 - 6*x + 5*x^2 + x^2 * O(x^n))) / 2, n+1); \\ A002212
Mp(n) = N(n) - sum(j=0, n-1, N(j)); \\ A039919
b(n) = N(n+3) - 6*N(n+2) - Mp(floor((n+1)/2)) + (41*N(n+1)-21*N(n)-L(n))/4 - (M(n+3)-M(n+2)+M(n)-if (!(n%2),M(n/2))+Lp(n))/2;
a(n) = if (n<=4, 0, b(n-4)); \\ Michel Marcus, Apr 05 2020

a(n+4) = N(n+3) - 6*N(n+2) - M'(floor((n+1)/2)) + (41*N(n+1)-21*N(n)-L(n))/4 - (M(n+3)-M(n+2)+M(n)-e(n)*M(n/2)+L'(n))/2 where N(n)=A002212(n), M(n)=A055879(n), M'(n)=A039919(n), L(n)=A039658(n), L'(n)=A039660(n), e(n)=1 if n is even and 0 if n is odd. - Sean A. Irvine, Apr 03 2020

a(13) and a(14) corrected, title improved, and more terms from Sean A. Irvine, Apr 03 2020

A030534 Number of polyhexes of class PF2.

Original entry on oeis.org

1, 2, 10, 40, 185, 828, 3805, 17411, 80177, 369675, 1710173, 7931011, 36884730, 171987194, 804027444, 3767994408, 17699839325, 83328339997, 393123808821, 1858351499207, 8801160980038, 41756069328689, 198437460900302, 944521149228740, 4502419787519360

Offset: 4

N. J. A. Sloane

nonn

See reference for precise definition.

Cyvin has incorrect a(13)=369639 and a(14)=1710137 in Table III due to using incorrect values for A026298(13) and A026298(14) in Table II. - Sean A. Irvine, Apr 02 2020

S. J. Cyvin, Zhang Fuji, B. N. Cyvin, Guo Xiaofeng, and J. Brunvoll, Enumeration and classification of benzenoid systems. 32. Normal perifusenes with two internal vertices, J. Chem. Inform. Comput. Sci., 32 (1992), 532-540.

Cf. A026106, A026118, A026298, A030519, A030520, A030525, A030529, A030532, A030534.

a(4)=1, a(n) = A026106(n) + A026118(n) + A026298(n) + A030519(n) for n > 4. - Sean A. Irvine, Apr 02 2020

a(13) and a(14) corrected and more terms from Sean A. Irvine, Apr 02 2020

A026298 Number of polyhexes of class PF2.

Original entry on oeis.org

4, 28, 176, 950, 4908, 24402, 119240, 575348, 2757460, 13157752, 62638788, 297832008, 1415550920, 6728600060, 31998023632, 152271569872, 725231959452, 3457304575812, 16497751608120, 78804354881238, 376806016649964, 1803539487096138, 8641075826669256, 41441524062045660

Offset: 7

N. J. A. Sloane

nonn

See reference for precise definition.

Cyvin et al. has incorrect a(13) = 119204 and a(14) = 575312 due to using incorrect value for A039919(5); cf. A039659. - Sean A. Irvine, Sep 24 2019

S. J. Cyvin, Zhang Fuji, B. N. Cyvin, Guo Xiaofeng, and J. Brunvoll, Enumeration and classification of benzenoid systems. 32. Normal perifusenes with two internal vertices, J. Chem. Inform. Comput. Sci., 32 (1992), 532-540.

Cf. A026106, A026118, A030519, A030520, A030525, A030529, A030532, A030534, A002212, A039919.

a(n + 4) = 3 * (N(n+2) - 6*N(n+1) + 8*N(n)) + A039919(floor((n+1)/2)) where N(n) = A002212(n) [from Cyvin]. - Sean A. Irvine, Sep 24 2019

Corrected and extended by Sean A. Irvine, Sep 24 2019

A030519 Number of polyhexes of class PF2 with four catafusenes annealated to pyrene.

Original entry on oeis.org

2, 13, 101, 619, 3641, 20028, 106812, 554352, 2828660, 14244878, 71077246, 352184306, 1736118578, 8525182798, 41741378126, 203929434766, 994680883360, 4845761306611, 23586192274443, 114731539477465, 557859497501007, 2711772157178038, 13180227306740726

Offset: 8

N. J. A. Sloane

nonn

See reference for precise definition.

S. J. Cyvin, Zhang Fuji, B. N. Cyvin, Guo Xiaofeng, and J. Brunvoll, Enumeration and classification of benzenoid systems. 32. Normal perifusenes with two internal vertices, J. Chem. Inform. Comput. Sci., 32 (1992), 532-540.
Sean A. Irvine, Java program (github)

Cf. A026106, A026118, A026298, A030519, A030520, A030525, A030529, A030532, A030534.

Lp(n)=my(x = 'x + O('x^(n+4))); polcoeff((1+x)*(1-6*x^2+7*x^4-(1-3*x^2)*sqrt(1-6*x^2+5*x^4))/(2*x^4*(1-x)), n); \\ A039660
M(n)= my(A); if( n<1, 0, n--; A = O(x); for( k = 0, n\2, A = 1 / (1 - x - x^2 / (1 + x - x^2 * A))); polcoeff( A, n)); \\ A055879
N(n) = polcoeff( (1 - x - sqrt(1 - 6*x + 5*x^2 + x^2 * O(x^n))) / 2, n+1); \\ A002212
b(n) = N(n+3) - 9*N(n+2) + 25*N(n+1) - 21*N(n) + (M(n+3) - M(n+2) - 3*M(n+1) + 3*M(n) + Lp(n))/2;
a(n) = b(n-4); \\ Michel Marcus, Apr 03 2020

a(n+4) = N(n+3) - 9*N(n+2) + 25*N(n+1) - 21*N(n) + (M(n+3) - M(n+2) - 3*M(n+1) + 3*M(n) + L'(n))/2 where N(n)=A002212(n), M(n)=A055879(n), and L'(n)=A039660(n) for n >= 4. - Sean A. Irvine, Apr 02 2020

More terms and title improved by Sean A. Irvine, Apr 02 2020

A030520 Number of polyhexes of class PF2 with C_{2n} symmetry.

Original entry on oeis.org

0, 1, 5, 20, 82, 335, 1402, 5949, 25652, 111963, 494157, 2201270, 9886034, 44712737, 203489627, 931191850, 4282171470, 19778577235, 91715812335, 426824400684, 1992828161414, 9332192498397, 43821128181652, 206288470970025, 973361629499032, 4602638827207605

Offset: 2

N. J. A. Sloane

nonn

See reference for precise definition.

S. J. Cyvin, Zhang Fuji, B. N. Cyvin, Guo Xiaofeng, and J. Brunvoll, Enumeration and classification of benzenoid systems. 32. Normal perifusenes with two internal vertices, J. Chem. Inform. Comput. Sci., 32 (1992), 532-540.

Cf. A026106, A026118, A026298, A030519, A030520, A030525, A030529, A030532, A030534.

Title improved, a(2)=0 inserted, and more terms from Sean A. Irvine, Apr 02 2020

cp's OEIS Frontend

A026118 Number of polyhexes of class PF2 (with two catafusenes annealated to pyrene).

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A030525 Number of polyhexes of class PF2 with a particular symmetry.

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A030529 Number of polyhexes of class PF2 with a particular symmetry.

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A030532 Number of polyhexes of class PF2 with symmetry point group C_s.

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A030534 Number of polyhexes of class PF2.

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A026298 Number of polyhexes of class PF2.

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A030519 Number of polyhexes of class PF2 with four catafusenes annealated to pyrene.

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PARI

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A030520 Number of polyhexes of class PF2 with C_{2n} symmetry.

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