%I #20 Nov 05 2025 15:22:05

%S 1,6,24,100,418,1748,7310,30570,127842,534628,2235784,9349922,

%T 39100844,163517514,683820978,2859700582,11959105792,50012302772,

%U 209148616298,874647662172,3657726962214,15296406894730,63968706878962

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

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

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

%H G. C. Greubel, <a href="/A126393/b126393.txt">Table of n, a(n) for n = 0..1000</a>

%H Sergey Kitaev and Jeffrey Remmel, <a href="https://arxiv.org/abs/1304.4286">(a,b)-rectangle patterns in permutations and words</a>, arXiv:1304.4286 [math.CO], 2013.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (4,1,-1).

%F G.f.: 1 + 2*x*(3-x^2)/(1-4*x-x^2+x^3). - _R. J. Mathar_, Jun 06 2013

%F a(n) = [n=0] + 6*A364705(n) - 2*A364705(n-2). - _G. C. Greubel_, Aug 08 2023

%t LinearRecurrence[{4,1,-1}, {1,6,24,100}, 41] (* _G. C. Greubel_, Aug 08 2023 *)

%o (S/R) stvar $[N]:(0..M-1) init $[]:=0 asgn $[]->{*} kill +[i in 0..N-2](($[i]`-$[i+1]`>2)+($[i+1]`-$[i]`>2))

%o (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

%o (SageMath)

%o @CachedFunction

%o def a(n): # A126393

%o if (n<4): return (1,6,24,100)[n]

%o else: return 4*a(n-1) +a(n-2) -a(n-3)

%o [a(n) for n in range(41)] # _G. C. Greubel_, Aug 08 2023

%Y Cf. Base 6 differing by one or less A126360.

%Y Cf. A364705.

%K nonn,base

%O 0,2

%A _R. H. Hardin_, Dec 28 2006