A248837 -id:A248837 - cp's OEIS frontend

A248842 T(n,k)=Number of length n arrays x(i), i=1..n with x(i) in 0..i and no value appearing more than k times.

2, 2, 4, 2, 6, 8, 2, 6, 22, 16, 2, 6, 24, 96, 32, 2, 6, 24, 118, 482, 64, 2, 6, 24, 120, 686, 2736, 128, 2, 6, 24, 120, 718, 4598, 17302, 256, 2, 6, 24, 120, 720, 4994, 34872, 120576, 512, 2, 6, 24, 120, 720, 5038, 39556, 295044, 917762, 1024, 2, 6, 24, 120, 720, 5040

Offset: 1

R. H. Hardin, Oct 15 2014

Table starts
....2.......2........2........2........2........2........2........2........2
....4.......6........6........6........6........6........6........6........6
....8......22.......24.......24.......24.......24.......24.......24.......24
...16......96......118......120......120......120......120......120......120
...32.....482......686......718......720......720......720......720......720
...64....2736.....4598.....4994.....5038.....5040.....5040.....5040.....5040
..128...17302....34872....39556....40260....40318....40320....40320....40320
..256..120576...295044...351320...361636...362804...362878...362880...362880
..512..917762..2753958..3456742..3606044..3626872..3628706..3628798..3628800
.1024.7574016.28103804.37314126.39517182.39874764.39913932.39916686.39916798

			Some solutions for n=6 k=4
..0....0....0....0....0....0....1....0....0....1....0....0....0....1....1....1
..1....0....1....1....1....1....0....2....1....0....0....2....2....2....0....0
..3....1....0....1....1....3....1....3....1....0....0....2....2....2....3....2
..0....1....4....3....2....2....2....0....3....4....1....0....4....4....1....0
..1....2....5....2....3....4....3....4....0....5....5....2....3....5....0....5
..4....6....4....0....3....5....1....3....0....6....3....2....6....2....6....3

R. H. Hardin, Table of n, a(n) for n = 1..3388

Cf. A000079 (column 1), A248836 (column 2), A248837 (column 3), A248838 (column 4).

A135106 Number of physical trees of alkane structures with n carbon vertices.

Original entry on oeis.org

1, 1, 2, 6, 24, 118, 686, 4598, 34872, 295044, 2753958, 28103804, 311216626, 3716341042, 47597786154, 650812077852, 9461423560788, 145724617925326, 2370293673319292, 40600119927220706, 730458115445479734

Offset: 1

R. J. Mathar, Feb 12 2008

nonn

Similar to A000602 (alkane trees with n carbon atoms) but keeping track of the history of attaching carbon atoms (methyls) to the backbone, as if these had been labeled.

			Starting with a(1)=1, one C1 methane, we get a(2)=1, the C1-C2 backbone.
The third can be attached to either C1 ending up with C3-C1-C2, or to C2 ending up with C1-C2-C3, yielding a(3)=2 different propanes.
C4 may be attached to any of C1 to C3 in these two propanes, yielding a(4)=6 different butanes, four of which are linear and two of which are stars.

J. V. Knop, W. R. Muller, K. Szymanski et al., Computer enumeration and generation of physical trees, J. Comput. Chem. vol. 8 no. 4 (1987) pp 549-554.

A135106 := proc(n) local numb, stack,istack,N,i ; numb := array(1..n) ; for i from 1 to n do numb[i] := 0 ; od: stack := array(1..7,1..100) ; stack[1,1]:=2 ; stack[2,1]:=0 ; stack[3,1]:=0 ; stack[4,1]:=0 ; stack[5,1]:=1 ; stack[6,1]:=0 ; stack[7,1]:=1 ; istack := 1 ; while istack <> 0 do for i from 1 to 7 do stack[i,istack+1] := stack[i,istack] ; od: if stack[6,istack] = 3 then istack := istack-1 ; else stack[6,istack] := stack[6,istack]+1 ; stack[1,istack+1] := stack[1,istack]+1 ; N := stack[6,istack] ; if stack[N,istack] <> 0 then stack[N,istack+1] := stack[N,istack+1]-1 ; stack[N+1,istack+1] := stack[N+1,istack+1]+1 ; stack[5,istack+1] := stack[N,istack]*stack[5,istack] ; stack[6,istack+1] := 0 ; stack[7,istack+1] := stack[7,istack]+1 ; numb[stack[7,istack+1]]:=numb[stack[7,istack+1]]+stack[5,istack+1] ; if stack[7,istack+1] <> n then istack := istack+1 ; fi ; fi ; fi ; od: numb[n] ; end: for n from 2 do print( A135106(n)) ; end: # R. J. Mathar, Feb 18 2008

a(n) = A248837(n-2). - Georg Fischer, Oct 23 2018

More terms from R. J. Mathar, Feb 18 2008

a(20)-a(21) from Alois P. Heinz, May 27 2013

cp's OEIS Frontend

A248842 T(n,k)=Number of length n arrays x(i), i=1..n with x(i) in 0..i and no value appearing more than k times.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A135106 Number of physical trees of alkane structures with n carbon vertices.

Views

Author

Keywords

Comments

Examples

Links

Programs

Maple

Formula

Extensions