login
A198633
Total number of round trips, each of length 2*n on the graph P_3 (o-o-o).
8
3, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33554432, 67108864, 134217728, 268435456, 536870912, 1073741824, 2147483648
OFFSET
0,1
COMMENTS
See the array and triangle A198632 for the general case for the graph P_N (there N is n and the length is l=2*k).
FORMULA
a(n) = w(3,2*n), n>=0, with w(3,l) the total number of closed walks on the graph P_3 (the simple path with 3 points (vertices) and 2 lines (or edges)).
O.g.f. for w(3,l) (with zeros for odd l): y*(d/dy)S(3,y)/S(3,y) with y=1/x and Chebyshev S-polynomials (coefficients A049310). See A198632, also for a rewritten form.
Empirical g.f.: (3-2*x)/(1-2*x). - Colin Barker, Jan 02 2012
This g.f. follows from the Chebyshev o.g.f. given above with x -> sqrt(x). Therefore a(0) = 3 and a(n) = 2^(n+1), n >= 1. - Wolfdieter Lang, Feb 18 2013.
EXAMPLE
With the graph P_3 as 1-2-3:
n=0: 3, from the length 0 walks starting at 1, 2 and 3.
n=2: 8, from the walks of length 4, namely 12121, 12321, 21212, 23232, 21232, 23212, 32323 and 32123.
MATHEMATICA
Join[{3}, NestList[2#&, 4, 30]] (* Harvey P. Dale, Nov 07 2020 *)
PROG
(PARI) a(n)=if(n, 2<<n, 3) \\ Charles R Greathouse IV, Jan 02 2012
CROSSREFS
Sequence in context: A202025 A227615 A049894 * A153057 A215095 A192474
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Nov 02 2011
STATUS
approved