|
|
A068293
|
|
a(1) = 1; thereafter a(n) = 6*(2^(n-1) - 1).
|
|
10
|
|
|
1, 6, 18, 42, 90, 186, 378, 762, 1530, 3066, 6138, 12282, 24570, 49146, 98298, 196602, 393210, 786426, 1572858, 3145722, 6291450, 12582906, 25165818, 50331642, 100663290, 201326586, 402653178, 805306362, 1610612730, 3221225466, 6442450938, 12884901882
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,2
|
|
COMMENTS
|
1/4 the number of colorings of an n X n octagonal array with 4 colors.
Consider the planar net 3^6 (as in the top left figure in the uniform planar nets link). Then a(n) is the total number of ways that a spider starting at a point P can reach any point n steps away by using a path of length n. - N. J. A. Sloane, Feb 20 2016
Equals inverse binomial transform of A091344: (1, 7, 31, 115, 391, ...).
Equals binomial transform of (1, 5, 7, 5, 7, 5, ...). (End)
For n > 1, number of ternary strings of length n with exactly 2 different digits. - Enrique Navarrete, Nov 20 2020
|
|
LINKS
|
|
|
FORMULA
|
a(n) = 3*a(n-1) - 2*a(n-2); a(1)=1, a(2)=6, a(3)=18. - Harvey P. Dale, Nov 27 2011
|
|
MATHEMATICA
|
Transpose[NestList[{First[#]+1, 6(2^First[#]-1)}&, {1, 1}, 30]][[2]] (* or *) Join[{1}, LinearRecurrence[{3, -2}, {6, 18}, 30]] (* Harvey P. Dale, Nov 27 2011 *)
|
|
PROG
|
(PARI) a(n)=polcoeff(prod(i=1, 2, (1+i*x))/(prod(i=1, 2, (1-i*x))+x*O(x^n)), n)
for(n=0, 50, print1(a(n), ", "))
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
EXTENSIONS
|
Old definition (which is now a comment) replaced with explicit formula by N. J. A. Sloane, May 12 2010
|
|
STATUS
|
editing
|
|
|
|