OFFSET
0,2
COMMENTS
Counts total area under elevated Schroeder paths of length 2n+2, where horizontal steps can choose from three colors.
Case r=3 for family (1+(r-1)x)/(1-2(1+r)x+(1-r)^2*x^2). Case r=2 gives NSW numbers A002315 and case r=4 gives NSW numbers A096053.
Fifth binomial transform of (1+8x)/(1-16x^2), A107906.
If p is an odd prime, a((p-1)/2) == 1 mod p. - Altug Alkan, Mar 17 2016
LINKS
FORMULA
G.f.: (1+2*x)/(1-8*x+4*x^2). [corrected by Ralf Stephan, Nov 30 2010]
a(n) = Sum_{k=0..n} binomial(2*n+1, 2*k)*3^k.
a(n) = 8*a(n-1) - 4*a(n-2); a(0) = 1, a(1) = 10. - Lekraj Beedassy, Apr 19 2020
a(n) = 2^n*A001834(n). - Philippe Deléham, Mar 18 2023
MATHEMATICA
Table[Sum[Binomial[2 n + 1, 2 k] 3^k, {k, 0, n}], {n, 0, 20}] (* or *) CoefficientList[Series[(1 + 2 x)/(1 - 8 x + 4 x^2), {x, 0, 20}], x] (* Michael De Vlieger, Mar 17 2016 *)
PROG
(PARI) Vec((1+2*x)/(1-8*x+4*x^2) + O(x^40)) \\ Michel Marcus, Mar 17 2016
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Paul Barry, May 27 2005
EXTENSIONS
Typo corrected and link added by Johannes W. Meijer, Aug 07 2010
STATUS
approved