A126393 Number of base 6 n-digit numbers with adjacent digits differing by two or less.

1, 6, 24, 100, 418, 1748, 7310, 30570, 127842, 534628, 2235784, 9349922, 39100844, 163517514, 683820978, 2859700582, 11959105792, 50012302772, 209148616298, 874647662172, 3657726962214, 15296406894730, 63968706878962

Offset: 0

Author

R. H. Hardin, Dec 28 2006

Keywords

• nonn
• base

Comments

a(base,n) = a(base-1,n) + 5^(n-1) for base >= 2*n - 1.

a(base,n) = a(base-1,n) + 5^(n-1) - 2 when base = 2*(n-1).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Sergey Kitaev and Jeffrey Remmel, (a,b)-rectangle patterns in permutations and words, arXiv:1304.4286 [math.CO], 2013.
• Index entries for linear recurrences with constant coefficients, signature (4,1,-1).

Crossrefs

Cf. Base 6 differing by one or less A126360.

Cf. A364705.

Programs

Magma

I:=[1,6,24,100]; [n le 4 select I[n] else 4*Self(n-1) +Self(n-2) -Self(n-3): n in [1..41]]; // G. C. Greubel, Aug 08 2023

Mathematica

LinearRecurrence[{4,1,-1}, {1,6,24,100}, 41] (* G. C. Greubel, Aug 08 2023 *)

SageMath

@CachedFunction
def a(n): # A126393
    if (n<4): return (1,6,24,100)[n]
    else: return 4*a(n-1) +a(n-2) -a(n-3)
[a(n) for n in range(41)] # G. C. Greubel, Aug 08 2023

Formula

G.f.: 1 + 2*x*(3-x^2)/(1-4*x-x^2+x^3). - R. J. Mathar, Jun 06 2013

a(n) = [n=0] + 6*A364705(n) - 2*A364705(n-2). - G. C. Greubel, Aug 08 2023