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

4,3

Comments

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.

Links

Table of n, a(n) for n=4..28.

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)

Formula

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

Prog

(PARI) 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

Crossrefs

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

Sequence in context: A026957 A026987 A375248 * A333800 A026997 A014337

Adjacent sequences: A030529 A030530 A030531 * A030533 A030534 A030535

Keyword

nonn

Author

N. J. A. Sloane.

Extensions

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

Status

approved