A124807 Number of base-6 circular n-digit numbers with adjacent digits differing by 2 or less.

1, 6, 24, 84, 332, 1336, 5478, 22658, 94196, 392664, 1639274, 6849002, 28627874, 119688094, 500456806, 2092720174, 8751273556, 36596513060, 153042707976, 640011807436, 2676483843602, 11192882945426, 46807955443900

Offset: 0

Author

R. H. Hardin, Dec 28 2006

Keywords

• nonn
• easy

Comments

[Empirical] a(base,n) = a(base-1,n) + A005191(n+1) for base >= 2*floor(n/2) + 1.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• OEIS Wiki, Number of base k circular n-digit numbers with adjacent digits differing by d or less
• Index entries for linear recurrences with constant coefficients, signature (6,-6,-8,5,2,-1).

Crossrefs

Cf. A005191, A006054, A364705.

Programs

Magma

I:=[1,6,24,84,332,1336,5478]; [n le 7 select I[n] else 6*Self(n-1) -6*Self(n-2) -8*Self(n-3) +5*Self(n-4) +2*Self(n-5) -Self(n-6): n in [1..41]]; // G. C. Greubel, Aug 04 2023

Mathematica

LinearRecurrence[{6,-6,-8,5,2,-1}, {1,6,24,84,332,1336,5478}, 35] (* G. C. Greubel, Aug 04 2023 *)

SageMath

@CachedFunction
def a(n): # a = A124807
    if (n<7): return (1,6,24,84,332,1336,5478)[n]
    else: return 6*a(n-1) -6*a(n-2) -8*a(n-3) +5*a(n-4) +2*a(n-5) -a(n-6)
[a(n) for n in range(41)] # G. C. Greubel, Aug 04 2023

Formula

From Colin Barker, Jun 04 2017: (Start)

G.f.: (1 - 6*x^2 - 16*x^3 + 15*x^4 + 8*x^5 - 5*x^6) / ((1 - 4*x - x^2 + x^3)*(1 - 2*x - x^2 + x^3)).

a(n) = 6*a(n-1) - 6*a(n-2) - 8*a(n-3) + 5*a(n-4) + 2*a(n-5) - a(n-6) for n > 6.

(End)

a(n) = -5*[n=0] + 3*A006054(n+2) - 4*A006054(n+1) - A006054(n) + 3*A364705(n) - 8*A364705(n-1) - A364705(n-2). - G. C. Greubel, Aug 04 2023