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 A084075 #36 Nov 16 2024 18:15:54
%S A084075 1,2,5,12,33,86,249,680,2033,5722,17485,50260,156033,455534,1431281,
%T A084075 4228752,13412193,40003058,127840085,384232156,1235575201,3737280582,
%U A084075 12080678505,36736735672,119276490193,364372758986,1187542872989
%N A084075 Length of list created by n substitutions k -> Range( -abs(k+1), abs(k-1), 2) starting with {1}.
%H A084075 G. C. Greubel, <a href="/A084075/b084075.txt">Table of n, a(n) for n = 0..1000</a>
%F A084075 G.f. is the series reversion of (-1 -6*x -8*x^2 + (1+2*x)^2 * sqrt(1+4*x))/(2*(x +4*x^2 +4*x^3)).
%F A084075 a(2*n) = A027307(n)/2, n >= 1.
%F A084075 a(n) = ( 6*(35*n^2 +15*n -96)*a(n-1) + (275*n^4 +330*n^3 -863*n^2 +120*n +126)*a(n-2) - 6*(5*n^2 +15*n -18)*a(n-3) + (n-3)*(n-1)*(25*n^2 +55*n -18)*a(n-4) )/((n+1)*(n+3)*(25*n^2 +5*n -48)), n >= 4. - _G. C. Greubel_, Nov 24 2022
%e A084075 {1}, {-2,0}, {-1,1,3,-1,1}, {0,2,-2,0,-4,-2,0,2,0,2,-2,0}
%t A084075 Rest@CoefficientList[InverseSeries[Series[ (-1-6n-8n^2+(1+2n)^2 Sqrt[1+4n])/( 2(n+4n^2+4n^3)), {n, 0, 40}]], n]
%t A084075 Length/@Flatten/@NestList[ #/.k_Integer:>Range[-Abs[k+1], Abs[k-1], 2] &, {1}, 8]
%o A084075 (Magma) I:=[1,2,5,12]; [n le 4 select I[n] else (6*(35*n^2-55*n-76)*Self(n-1) + (275*n^4-770*n^3-203*n^2+1736*n-912)*Self(n-2) -6*(5*n^2+5*n-28)*Self(n-3) + (n-4)*(n-2)*(25*n^2+5*n-48)*Self(n-4))/(n*(n+2)*(25*n^2-45*n-28)): n in [1..41]]; // _G. C. Greubel_, Nov 24 2022
%o A084075 (SageMath)
%o A084075 @CachedFunction
%o A084075 def a(n): # a = A084075
%o A084075 if n < 4: return (1, 2, 5, 12)[n]
%o A084075 else: return (6*(35*n^2 +15*n -96)*a(n-1) +(275*n^4+330*n^3-863*n^2+120*n+126)*a(n-2) -6*(5*n^2+15*n-18)*a(n-3) +(n-3)*(n-1)*(25*n^2+55*n-18)*a(n-4))/((n+1)*(n+3)*(25*n^2+5*n-48))
%o A084075 [a(n) for n in range(41)] # _G. C. Greubel_, Nov 24 2022
%o A084075 (Python) # replace iterates lists as described in Example.
%o A084075 def replace(L):
%o A084075 return [i for k in L for i in range(-abs(k + 1), 1 + abs(k - 1), 2)]
%o A084075 def a(n):
%o A084075 L = [1]
%o A084075 for k in range(n): L=replace(L)
%o A084075 return len(L)
%o A084075 print([a(n) for n in range(12)]) # _F. Chapoton_, Nov 15 2024
%Y A084075 Cf. A084076, A084077, A084078.
%Y A084075 Cf. A027307, A215067, A034015 (even bisection).
%K A084075 nonn
%O A084075 0,2
%A A084075 _Wouter Meeussen_, May 11 2003