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 A254729 #32 Sep 08 2022 08:46:11
%S A254729 1,1,2,3,4,7,11,18,29,47,76,123,199,322,521,843,1364,2207,3571,5778,
%T A254729 9349,15127,24476,39603,64079,103682,167761,271443,439204,710647,
%U A254729 1149851,1860498,3010349,4870847,7881196,12752043,20633239,33385282,54018521,87403803
%N A254729 Number of numbers j + k*sqrt(2) of length n, where the length is the least number of steps to reach 0, the allowable steps being x -> x + 1 and x -> x*sqrt(2).
%C A254729 See the MathOverflow link for a proof that the sequence coincides with the Lucas sequence, A000032, beginning at 4.
%C A254729 Therefore also the same as A080023 (beginning at 2). - _Georg Fischer_, Oct 09 2018
%H A254729 Clark Kimberling, <a href="/A254729/b254729.txt">Table of n, a(n) for n = 0..1000</a>
%H A254729 MathOverflow, <a href="http://mathoverflow.net/questions/195207/a-possibly-surprising-appearance-of-lucas-numbers">A possibly surprising appearance of Lucas numbers</a>
%H A254729 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (1,1).
%F A254729 a(n) = a(n-1) + a(n-2) for n >= 6.
%F A254729 G.f.: (-1 + x^4)/(-1 + x + x^2).
%e A254729 One can view the minimal paths in a tree having generation g(0) = {0} followed by generations g(1) = {1}, g(2) = {2, sqrt(2)}, g(3) = {3, 2*sqrt(2), 1+sqrt(2)}, and so on. Duplicates are removed as they occur. Also, a(n) = |g(n)| for n >= 0.
%t A254729 t = NestList[DeleteDuplicates[Flatten[Map[{# + {0, 1}, {Last[#], 2*First[#]}} &, #], 1]] &, {{0, 0}}, 25] ; s[0] = t[[1]]; s[n_] := s[n] = Union[t[[n + 1]], s[n - 1]]; g[n_] := Complement[s[n], s[n - 1]]; g[0] = {{0, 0}}; Table[Length[g[z]], {z, 0, 25}]
%t A254729 CoefficientList[Series[(-1 + x^4)/(-1 + x + x^2), {x, 0, 39}], x] (* _Robert G. Wilson v_, Feb 28 2015 *)
%o A254729 (PARI) x='x+O('x^40); Vec((1-x^4)/(1-x-x^2)) \\ _G. C. Greubel_, Sep 30 2018
%o A254729 (Magma) m:=40; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-x^4)/(1-x-x^2))); // _G. C. Greubel_, Sep 30 2018
%Y A254729 Cf. A000032, A080023, A169985, A252864.
%K A254729 nonn,easy
%O A254729 0,3
%A A254729 _Clark Kimberling_, Feb 06 2015