A039660 -id:A039660 - cp's OEIS frontend

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

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

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

A045905 Catafusenes (see reference for precise definition).

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 16, 80, 473, 2517, 13431, 69357, 353943, 1780894, 8886254, 44024308, 217021967, 1065653519, 5217704239, 25491204743, 124335254416, 605720277647, 2948274685881, 14341442951945, 69732440151924

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, ^4O = ^4F_7 page 1178.

Last column of A121178.

8*a(n) = A045829(n) + 3*A045445(n/2) + 2*A002212(n/4) + 2*A039660(n) for n > 0, where sequence values with non-integer index are implicitly defined as 0. - R. J. Mathar, Jul 30 2019

cp's OEIS Frontend

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

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

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A045905 Catafusenes (see reference for precise definition).

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