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 A033193 #27 Aug 22 2025 14:49:25
%S A033193 1,2,6,19,62,207,704,2430,8486,29903,106098,378391,1354700,4863834,
%T A033193 17499302,63055947,227465414,821215295,2966571096,10721076118,
%U A033193 38757594758,140143505031,506827217210,1833150646599,6630915738212,23986989146162,86775559512774,313930265564035
%N A033193 Binomial transform of A033192.
%C A033193 Number of (s(0), s(1), ..., s(2n)) such that 0 < s(i) < 10 and |s(i) - s(i-1)| = 1 for i = 1,2,...,2n, s(0) = 3, s(2n) = 3. - _Herbert Kociemba_, Jun 16 2004
%H A033193 Michael De Vlieger, <a href="/A033193/b033193.txt">Table of n, a(n) for n = 0..1791</a>
%H A033193 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>.
%H A033193 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (8,-21,20,-5).
%F A033193 G.f.: (x^4-7*x^3+11*x^2-6*x+1)/((1-3*x+x^2)*(1-5*x+5*x^2)).
%F A033193 a(n) = (1/5)*Sum_{r=1..9} sin(3*r*Pi/10)^2*(2*cos(r*Pi/10))^(2*n), n >= 1. - _Herbert Kociemba_, Jun 16 2004
%F A033193 For n > 0, a(n) = (phi^(2*n+1) + 1/phi^(2*n+1))/(2*sqrt(5)) + 5^(n/2-1)*(phi^(n+2) + 1/phi^(n+2))/2, where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - _Vaclav Kotesovec_, Aug 22 2025
%t A033193 CoefficientList[Series[(x^4 - 7 x^3 + 11 x^2 - 6 x + 1)/((1 - 3 x + x^2) (1 - 5 x + 5 x^2)), {x, 0, 23}], x] (* _Michael De Vlieger_, Feb 12 2022 *)
%o A033193 (PARI) Vec((x^4-7*x^3+11*x^2-6*x+1)/((1-3*x+x^2)*(1-5*x+5*x^2)) + O(x^24)) \\ _Stefano Spezia_, Aug 22 2025
%Y A033193 Cf. A033192.
%K A033193 nonn,easy
%O A033193 0,2
%A A033193 _N. J. A. Sloane_