A000620 Number of monosubstituted alkanes C(n-1)H(2n-1)-X with n-1 carbon atoms that are stereoisomers.

0, 0, 0, 0, 2, 6, 20, 60, 176, 512, 1488, 4326, 12648, 37186, 109980, 327216, 979020, 2944414, 8897732, 27005290, 82288516, 251650788, 772127678, 2376238138, 7333188770, 22688297950, 70360977228, 218678818026, 681017928476, 2124840874610, 6641336507270, 20791999731518

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

Comments

Also number of monosubstituted alkanes C(n)H(2n+1)-X of the form R-CH2-X (primary) that are stereoisomers.

Let the entries in the nine columns of Blair and Henze's Table I (JACS 54 (1932), p. 1098) be denoted by Ps(n), Pn(n), Ss(n), Sn(n), Ts(n), Tn(n), As(n), An(n), T(n) respectively (here P = Primary, S = Secondary, T = Tertiary, s = stereoisomers, n = non-stereoisomers and the last column T(n) gives total).

Then Ps (and As) = this sequence, Pn (and An, Sn) = A000621, Ss = A000622, Ts = A000623, Tn = A000624, T = A000625. Recurrences generating these sequences are given in the Maple program in A000620.

References

• 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).

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 1..1930
• Jean-François Alcover, Mathematica program
• C. M. Blair and H. R. Henze, The number of stereoisomeric and non-stereoisomeric mono-substitution products of the paraffins, J. Amer. Chem. Soc., 54 (3) (1932), 1098-1106.
• C. M. Blair and H. R. Henze, The number of stereoisomeric and non-stereoisomeric mono-substitution products of the paraffins, J. Amer. Chem. Soc., 54 (3) (1932), 1098-1105. (Annotated scanned copy)

Crossrefs

Cf. A000621, A000622, A000623, A000624, A000625, A239803, A239805.

Programs

Maple

# Blair and Henze's recurrences for A000620-A000625 (see comments lines for relationship between the sequences and their symbols).
Ps := [0,0,0]; Pn := [1,1,1]; Ss := [0,0,0]; Sn := [0,0,1]; Ts := [0,0,0]; Tn := [0,0,0]; As := [0,0,0]; An := [1,1,2]; T := [1,1,2];
for n from 4 to 60 do Ps := [op(Ps),As[n-1]]; Pn := [op(Pn),An[n-1]]; t1 := add( 2*T[n-1-j]*T[j],j=1..floor((n-2)/2) ); if n mod 2 = 1 then i := (n-1)/2; t1 := t1+T[i]^2-An[i]; fi; Ss := [op(Ss),t1];
t2 := 0; if n mod 2 = 1 then i := (n-1)/2; t2 := An[i]; fi; Sn := [op(Sn),t2]; t3 := 0; for i from 1 to (n-1)/3 do for j from i+1 to (n-2)/2 do k := n-1-i-j; if j 0 and i <> j then t4 := t4+(T[i]^2-An[i])*T[j]+An[i]*As[j]; t5 := t5+An[i]*An[j]; fi; od; t6 := 0; t7 := 0; if n mod 3 = 1 then i := (n-1)/3; t6 := (2*T[i]+T[i]^3)/3-An[i]^2; t7 := An[i]^2; fi;
Ts := [op(Ts), t3+t4+t6]; Tn := [op(Tn), t5+t7]; As := [op(As), Ps[n]+Ss[n]+Ts[n]]; An := [op(An), Pn[n]+Sn[n]+Tn[n]]; T := [op(T),As[n]+An[n]]; od: Ps; Pn; Ss; Ts; Tn; T;

Mathematica

(* See links *)

Formula

See Maple program for recurrences for this sequence and A000621-A000625.

a(n) ~ c * b^n / n^(3/2), where b = 3.287112055584474991259... (see A239803), c = 0.105352133282419523497... (see A239805). - Vaclav Kotesovec, Mar 27 2014

Extensions

Additional comments from Bruce Corrigan, Nov 04 2002