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%I #32 Aug 25 2017 03:23:00
%S 1,1,1,2,3,5,8,13,22,36,60,99,164,272,450,746,1235,2046,3389,5613,
%T 9299,15402,25514,42262,70005,115962,192084,318182,527053,873043,
%U 1446161,2395504,3968060,6572925,10887788,18035177,29874537,49485965,81971484,135782448
%N G.f.: 1/(1 - x/(1 - x^2/(1 - x^5/(1 - x^12/(1 - x^29/(1 - x^70/(1 -...- x^Pell(n)/(1 -...)))))))), a continued fraction.
%C From _Clark Kimberling_, Jun 12 2016: (Start)
%C Number of real integers in n-th generation of tree T(2i) defined as follows.
%C Let T* be the infinite tree with root 0 generated by these rules: if p is in T*, then p+1 is in T* and x*p is in T*. Let g(n) be the set of nodes in the n-th generation, so that g(0) = {0}, g(1) = {1}, g(2) = {2,x}, g(3) = {3,2x,x+1,x^2}, etc. Let T(r) be the tree obtained by substituting r for x.
%C For r = 2i, then g(3) = {3,2r,r+1, r^2}, in which the number of real integers is a(3) = 2.
%C See A274142 for a guide to related sequences. (End)
%H Kenny Lau, <a href="/A206743/b206743.txt">Table of n, a(n) for n = 0..1000</a>
%F a(n) ~ c * d^n, where d = 1.6564594309887754808836889708489581749625897572527517021957723319642053908... and c = 0.3844078703275069072126260832303344589497793302955451672191630264983... - _Vaclav Kotesovec_, Aug 25 2017
%e G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 3*x^4 + 5*x^5 + 8*x^6 + 13*x^7 +...
%p A206743 := proc(r)
%p local gs,n,gs2,el,a ;
%p gs := [2,r] ;
%p for n from 3 do
%p gs2 := [] ;
%p for el in gs do
%p gs2 := [op(gs2),el+1,r*el] ;
%p end do:
%p gs := gs2 ;
%p a := 0 ;
%p for el in gs do
%p if type(el,'realcons') then
%p a := a+1 :
%p end if;
%p end do:
%p print(n,a) ;
%p end do:
%p end proc: # _R. J. Mathar_, Jun 16 2016
%t z = 18; t = Join[{{0}}, Expand[NestList[DeleteDuplicates[Flatten[Map[{# + 1, x*#} &, #], 1]] &, {1}, z]]]; u = Table[t[[k]] /. x -> 2 I, {k, 1, z}]; Table[Count[Map[IntegerQ, u[[k]]], True], {k, 1, z}] (* _Clark Kimberling_, Jun 12 2016 *)
%o (PARI) {Pell(n)=polcoeff(x/(1-2*x-x^2+x*O(x^n)),n)}
%o {a(n)=local(CF=1+x*O(x^n),M=ceil(log(2*n+1)/log(2.4))); for(k=0, M, CF=1/(1-x^Pell(M-k+1)*CF)); polcoeff(CF, n, x)}
%o for(n=0,55,print1(a(n),", "))
%o (Python)
%o N = 1000
%o pell = [0,1]
%o c = 2
%o while c < N:
%o ....pell.append(c)
%o ....c = pell[-1]*2 + pell[-2]
%o pell.reverse()
%o gf = [0]*(N+1)
%o for p in pell:
%o ....gf = [-x for x in gf]
%o ....gf[0] += 1
%o ....quotient = [0]*(N+1)
%o ....remainder = [0]*(N+1)
%o ....remainder[p] = 1
%o ....for n in range(N+1):
%o ........q = remainder[n]//gf[0]
%o ........for i in range(n,N+1):
%o ............remainder[i] -= q*gf[i-n]
%o ........quotient[n] = q
%o ....gf = quotient
%o for i in range(N+1):
%o ....print(i,gf[i])
%o # _Kenny Lau_, Aug 01 2017
%Y Cf. A000621, A206741, A274142.
%K nonn
%O 0,4
%A _Paul D. Hanna_, Feb 12 2012
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