A018212 Alkane (or paraffin) numbers l(11,n).

1, 5, 25, 85, 255, 651, 1519, 3235, 6470, 12190, 21942, 37854, 63090, 101850, 160050, 245322, 367983, 541035, 781495, 1110395, 1554553, 2146573, 2927145, 3945045, 5260060, 6942988, 9079292, 11769100, 15131700, 19305540

Offset

0,2

References

S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926.

Winston C. Yang (paper in preparation).

Links

Table of n, a(n) for n=0..29.

S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926. (Annotated scanned copy)

N. J. A. Sloane, Classic Sequences

Index entries for linear recurrences with constant coefficients, signature (5,-6,-10,29,-9,-36,36,9,-29,10,6,-5,1).

Formula

G.f.: (1+6*x^2+x^4)/((1-x)^5*(1-x^2)^4). - N. J. A. Sloane

l(c, r) = 1/2 binomial(c+r-3, r) + 1/2 d(c, r), where d(c, r) is binomial((c + r - 3)/2, r/2) if c is odd and r is even, 0 if c is even and r is odd, binomial((c + r - 4)/2, r/2) if c is even and r is even, binomial((c + r - 4)/2, (r - 1)/2) if c is odd and r is odd.

a(n) = (1/(2*8!))*(n+2)*(n+4)*(n+6)*(n+8)*((n+1)*(n+3)*(n+5)*(n+7) + 1*3*5*7) - (1/3)*(1/2^6)*(n^3+(27/2)*n^2+56*n+(279/4))*(1/2)*(1-(-1)^n). - Yosu Yurramendi Jun 23 2013

a(n) ~ n^8/80640. - Charles R Greathouse IV, May 26 2026

Mathematica

LinearRecurrence[{5, -6, -10, 29, -9, -36, 36, 9, -29, 10, 6, -5, 1}, {1, 5, 25, 85, 255, 651, 1519, 3235, 6470, 12190, 21942, 37854, 63090}, 30] (* Ray Chandler, Sep 23 2015 *)

Prog

(PARI) a(n)=my(q=n\2); q^8/315+[144*q^7+1092*q^6+4536*q^5+11277*q^4+17346*q^3+16603*q^2+9474*q+2520, 176*q^7+1652*q^6+8624*q^5+27377*q^4+54194*q^3+65463*q^2+44106*q+12600][n%2+1]/2520 \\ Charles R Greathouse IV, May 26 2026

Crossrefs

Cf. A282011.

Sequence in context: A147122 A051229 A058919 * A181477 A147274 A147034

Adjacent sequences: A018209 A018210 A018211 * A018213 A018214 A018215

Keyword

nonn,easy

Author

N. J. A. Sloane, Winston C. Yang (yang(AT)math.wisc.edu)

Status

approved