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 A277236 #27 Jul 12 2025 19:05:50
%S A277236 1,4,10,26,66,170,434,1114,2850,7306,18706,47930,122754,314474,805490,
%T A277236 2063386,5285346,13538890,34680274,88835834,227556930,582900266,
%U A277236 1493127986,3824729050,9797240994,25096157194,64285121170,164669749946,421810234626,1080489234410,2767730172914
%N A277236 Number of strings of length n composed of symbols from the circular list [1,2,3,4] such that adjacent symbols in the string must be adjacent in the list. No runs of length 2 or more are allowed for symbols 1 and 3.
%C A277236 To generalize to strings composed of symbols from the circular list [1,2,3,...2m], m>=2, with no runs of 2 or more allowed for symbols 1,3,5,...2m-1, use the same recurrence given below with initial values a(1)=2m, a(2)=5m, see A277237 for the m=3 case.
%H A277236 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (1,4).
%F A277236 G.f.: (1+3*x+2*x^2)/(1-x-4*x^2).
%F A277236 For n>=3, the recurrence is a(n) = a(n-1) + 4*a(n-2), a(1)=4, a(2)=10.
%F A277236 a(n) = ((13+3*sqrt(17))*z1^n-(13-3*sqrt(17))*z2^n)/(4*sqrt(17)) where z1=(1+sqrt(17))/2 and z2=(1-sqrt(17))/2.
%e A277236 For n=3 the 26 strings are 121, 122, 123, 141, 143, 144, 212, 214, 221, 222, 223, 232, 234, 321, 322, 323, 341, 343, 344, 412, 414, 432, 434, 441, 443, 444.
%e A277236 For n=4 the 66 strings are 1212, 1214, 1221, 1222, 1223, 1232, 1234, 1412, 1414, 1432, 1434, 1441, 1443, 1444, 2121, 2122, 2123, 2141, 2143, 2144, 2212, 2214, 2221, 2222, 2223, 2232, 2234, 2321, 2322, 2323, 2341, 2343, 2344, 3212, 3214, 3221, 3222, 3223, 3232, 3234, 3412, 3414, 3432, 3434, 3441, 3443, 3444, 4121, 4122, 4123, 4141, 4143, 4144, 4321, 4322, 4323, 4341, 4343, 4344, 4412, 4414, 4432, 4434, 4441, 4443, 4444.
%t A277236 CoefficientList[Series[(1 + 3 x + 2 x^2)/(1 - x - 4 x^2), {x, 0, 30}], x] (* _Michael De Vlieger_, Oct 07 2016 *)
%t A277236 LinearRecurrence[{1,4},{1,4,10},40] (* _Harvey P. Dale_, Jul 12 2025 *)
%o A277236 (PARI) Vec((1+3*z+2*z^2)/(1-z-4*z^2) + O(z^40)) \\ _Michel Marcus_, Oct 06 2016
%Y A277236 Cf. A222132 (z1), A277237.
%K A277236 nonn,easy
%O A277236 0,2
%A A277236 _Stefan Hollos_, Oct 06 2016