A107903 Generalized NSW numbers.

1, 10, 76, 568, 4240, 31648, 236224, 1763200, 13160704, 98232832, 733219840, 5472827392, 40849739776, 304906608640, 2275853910016, 16987204845568, 126794223124480, 946404965613568, 7064062832410624, 52726882796830720

Offset: 0

Author

Paul Barry, May 27 2005

Keywords

• easy
• nonn

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

• Index entries for linear recurrences with constant coefficients, signature (8,-4).

Crossrefs

Cf. A001834, A099156, A102591.

Programs

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 *)

PARI

Vec((1+2*x)/(1-8*x+4*x^2) + O(x^40)) \\ Michel Marcus, Mar 17 2016

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) = ((1+sqrt(3))*(4+2*sqrt(3))^n+(1-sqrt(3))*(4-2*sqrt(3))^n)/2 = A099156(n+1)+2*A099156(n).

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

Extensions

Typo corrected and link added by Johannes W. Meijer, Aug 07 2010