A107903 Generalized NSW numbers.

1, 10, 76, 568, 4240, 31648, 236224, 1763200, 13160704, 98232832, 733219840, 5472827392, 40849739776, 304906608640, 2275853910016, 16987204845568, 126794223124480, 946404965613568, 7064062832410624, 52726882796830720, 393558811045003264, 2937562957172703232

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+8*x)/(1-16*x^2), A107906.

If p is an odd prime, a((p-1)/2) == 1 mod p. - Altug Alkan, Mar 17 2016

Links

Table of n, a(n) for n=0..21.

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

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.

a(n) = 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

E.g.f.: exp(4*x) * (cosh(2*sqrt(3)*x) + sqrt(3)*sinh(2*sqrt(3)*x)). - Amiram Eldar, Feb 03 2026

Mathematica

Table[Sum[Binomial[2 n + 1, 2 k] 3^k, {k, 0, n}], {n, 0, 20}] (* Michael De Vlieger, Mar 17 2016 *)
(* Alternative: *)
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

Cf. A001834, A002315, A096053, A099156, A102591, A107906.

Sequence in context: A198692 A215465 A169584 * A075489 A184273 A185986

Adjacent sequences: A107900 A107901 A107902 * A107904 A107905 A107906

Keyword

easy,nonn,changed

Author

Paul Barry, May 27 2005

Extensions

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

Status

approved