This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A124805 #33 Apr 30 2025 14:08:06
%S A124805 1,4,14,46,162,574,2042,7270,25890,92206,328394,1169590,4165554,
%T A124805 14835838,52838618,188187526,670239810,2387094478,8501763050,
%U A124805 30279478102,107841960402,384084837406,1367938433018,4871984973862
%N A124805 Number of circular n-letter words over the alphabet {0,1,2,3} with adjacent letters differing by at most 2.
%C A124805 Empirical: a(base, n) = a(base-1, n) + A005191(n+1) for base >= 2*floor(n/2) + 1 where base is the number of letters in the alphabet.
%C A124805 Sequence appears to have generating function (1-x^2-4*x^3)/((1-x)*(1-3*x-2*x^2)). The degree of the numerator would drop by one if the initial term were changed from 1 to 3: (3-8*x+x^2)/((1-x)*(1-3*x-2*x^2)). - _Creighton Dement_, Aug 20 2007
%H A124805 G. C. Greubel, <a href="/A124805/b124805.txt">Table of n, a(n) for n = 0..1000</a>
%H A124805 OEIS Wiki, <a href="/wiki/Number_of_base_k_circular_n-digit_numbers_with_adjacent_digits_differing_by_d_or_less">Number of base k circular n-digit numbers with adjacent digits differing by d or less</a>
%H A124805 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (4,-1,-2).
%F A124805 a(n) = 1 + A206776(n) for n > 0. - _Bruno Berselli_, Jan 11 2013
%F A124805 From _Colin Barker_, Jun 02 2017: (Start)
%F A124805 G.f.: (1 - x^2 - 4*x^3) / ((1 - x)*(1 - 3*x - 2*x^2)).
%F A124805 a(n) = 1 + ((3-sqrt(17))/2)^n + ((3+sqrt(17))/2)^n for n>0.
%F A124805 a(n) = 4*a(n-1) - a(n-2) - 2*a(n-3) for n > 3. (End)
%t A124805 LinearRecurrence[{4,-1,-2}, {1,4,14,46}, 40] (* _G. C. Greubel_, Aug 03 2023 *)
%o A124805 (S/R) stvar $[N]:(0..M-1) init $[]:=0 asgn $[]->{*} kill +[i in 0..N-1](($[i]`-$[(i+1)mod N]`>2)+($[(i+1)mod N]`-$[i]`>2))
%o A124805 (Magma) I:=[1,4,14,46]; [n le 4 select I[n] else 4*Self(n-1) -Self(n-2) -2*Self(n-3): n in [1..40]]; // _G. C. Greubel_, Aug 03 2023
%o A124805 (SageMath)
%o A124805 A206776=BinaryRecurrenceSequence(3,2,2,3)
%o A124805 [1+A206776(n) -2*int(n==0) for n in range(41)] # _G. C. Greubel_, Aug 03 2023
%Y A124805 Cf. A005191, A124806, A206776.
%K A124805 nonn,easy
%O A124805 0,2
%A A124805 _R. H. Hardin_, Dec 28 2006