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 A214997 #57 May 28 2025 00:11:57
%S A214997 4,13,45,153,523,1785,6095,20809,71047,242569,828183,2827593,9654007,
%T A214997 32960841,112535351,384219721,1311808183,4478793289,15291556791,
%U A214997 52208640585,178251448759,608588513865,2077851157943,7094227604041,24221208100279,82696377193033
%N A214997 Power ceiling-floor sequence of 2+sqrt(2).
%C A214997 See A214992 for a discussion of power ceiling-floor sequence and power ceiling-floor function, p3(x) = limit of a(n,x)/x^n. The present sequence is a(n,r), where r = 2+sqrt(2), and the limit p3(r) = 3.8478612632206289...
%C A214997 a(n) is the number of words over {0,1,2,3} of length n+1 that avoid 23, 32, and 33. As an example, a(2)=45 corresponds to the 45 such words of length 3; these are all 64 words except for the 19 prohibited cases which are 320, 321, 322, 323, 230, 231, 232, 233, 330, 331, 332, 333, 023, 123, 223, 032, 132, 033, 133. - _Greg Dresden_ and _Mina BH Arsanious_, Aug 09 2023
%C A214997 Let M denote the 4 X 4 matrix = [[1,1,1,1], [1,1,1,1], [1,1,1,0], [1,1,0,0]] and A(n) = the column vector (p(n),q(n),r(n),s(n)) = M^n * A(0), where A(0) = (1,1,1,1), then a(n) = p(n)+q(n)+r(n)+s(n) = p(n+1). - _Mina BH Arsanious_, Jan 18 2025
%C A214997 Sum_{k=0..n} a(k) = (r(n-2)-3)/2 where r(n) is defined in previous comment. - _Mina BH Arsanious_, May 21 2025
%H A214997 Clark Kimberling, <a href="/A214997/b214997.txt">Table of n, a(n) for n = 0..250</a>
%H A214997 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,2,-2).
%F A214997 a(n) = floor(x*a(n-1)) if n is odd, a(n) = ceiling(x*a(n-1)) if n is even, where x = 2+sqrt(2) and a(0) = ceiling(x).
%F A214997 a(n) = 3*a(n-1) + 2*a(n-2) - 2*a(n-3).
%F A214997 G.f.: (4 + x - 2*x^2)/(1 - 3*x - 2*x^2 + 2*x^3).
%F A214997 a(n) = (1/14)*(2*(-1)^n + (27-19*sqrt(2))*(2-sqrt(2))^n + (2+sqrt(2))^n*(27+19*sqrt(2))). - _Colin Barker_, Nov 13 2017
%e A214997 a(0) = ceiling(r) = 4, where r = 2+sqrt(2);
%e A214997 a(1) = floor(4*r) = 13; a(2) = ceiling(13*r) = 45.
%t A214997 (See A214996.)
%t A214997 CoefficientList[Series[(4+x-2*x^2)/(1-3*x-2*x^2+2*x^3), {x,0,50}], x] (* _G. C. Greubel_, Feb 01 2018 *)
%o A214997 (PARI) Vec((4 + x - 2*x^2) / ((1 + x)*(1 - 4*x + 2*x^2)) + O(x^40)) \\ _Colin Barker_, Nov 13 2017
%o A214997 (Magma) Q:=Rationals(); R<x>:=PowerSeriesRing(Q, 40); Coefficients(R!((4 +x-2*x^2)/(1-3*x-2*x^2+2*x^3))); // _G. C. Greubel_, Feb 01 2018
%Y A214997 Cf. A214992, A007052, A214996, A007070.
%K A214997 nonn,easy
%O A214997 0,1
%A A214997 _Clark Kimberling_, Nov 10 2012