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 A349700 #16 Jan 05 2025 19:51:42
%S A349700 -6,4,6,0,18,24,30,92,138,236,518,856,1570,3072,5374,9972,18714,33684,
%T A349700 62406,115440,210482,388776,715934,1312460,2419114,4449532,8174406,
%U A349700 15049672,27675714,50884368,93629694,172187364,316668474,582540836,1071371910,1970517728
%N A349700 Difference between 4*A001590(n+2) and A075092(n).
%C A349700 Tribonacci numbers 4*A001590(n+2) (i.e., tribonacci numbers t(n) = t(n-1) + t(n-2) + t(n-3) with t(0) = 0, t(1) = 4, t(2) = 8) for n > 2 are numbers of ternary sequences (q(1),q(2),...,q(n)), q(i) = 0,1,2, of length n such that all triples (q(i),q(i+1),q(i+2)) contain digits 0 and 1 at least once. In the other words {0,1} is a subset of each {q(i),q(i+1),q(i+2)}. Similarly, A075092(n), for n > 2, presents a number of ternary cyclic sequences with this property. Hence, a(n), for n > 2, gives a number of (ordinary) sequences which do not lead to cyclic sequences with triples (q(n-1),q(n),q(1)) and (q(n),q(1),q(2)) satisfying the above formulated condition. The special cases, n = 1,2, can be included in a way proposed in A001644.
%C A349700 The recurrence formula is the same as this for A075092, with different initial conditions.
%H A349700 Wojciech Florek, <a href="http://doi.org/10.1016/j.amc.2018.06.014">A class of generalized Tribonacci sequences applied to counting problems</a>, Appl. Math. Comput., 338 (2018), 809-821.
%H A349700 W. O. J. Moser, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/31-1/moser.pdf">Cyclic binary strings without long runs of like (alternating) bits</a>, Fibonacci Quart. 31(1) (1993), 2-6.
%H A349700 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,1,4,1,0,-1).
%F A349700 a(n) = a(n-2) + 4*a(n-3) + a(n-4) - a(n-6), a(0) = -6, a(1) = 4, a(2) = 6, a(3) = 0, a(4) = 18, a(5) = 24.
%F A349700 G.f.: (4*x^5-2*x^4-20*x^3-12*x^2-4*x+6)/(-x^6+x^4+4*x^3+x^2-1).
%e A349700 If n = 4, then there are 24 finite sequences: 0010, 0011, 0012, 0100, 0101, 0102, 0110, 0120, 0210, nine analogous starting with 1, 2010, 2011, 2012, 2100, 2101, and 2102. Only six of them, namely 0011, 0101, 0110, 1001, 1010, 1100, yield cyclic sequences satisfying the restriction imposed. Therefore, a(4) = 24 - 6 = 18.
%t A349700 nterms=50;LinearRecurrence[{0,1,4,1,0,-1},{-6,4,6,0,18,24},nterms] (* _Paolo Xausa_, Nov 26 2021 *)
%Y A349700 Cf. A001590, A001644, A075092, A294627.
%K A349700 sign,easy
%O A349700 0,1
%A A349700 _Wojciech Florek_, Nov 25 2021