A364705 -id:A364705 - cp's OEIS frontend

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

R. H. Hardin, Dec 28 2006

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

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).

Cf. A005191, A006054, A364705.

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

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

@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

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

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

Original entry on oeis.org

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

R. H. Hardin, Dec 28 2006

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).

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).

Cf. Base 6 differing by one or less A126360.

Cf. A364705.

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

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

@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

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

cp's OEIS Frontend

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

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