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A206743 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.

%I A206743 #34 May 22 2025 10:21:35
%S A206743 1,1,1,2,3,5,8,13,22,36,60,99,164,272,450,746,1235,2046,3389,5613,
%T A206743 9299,15402,25514,42262,70005,115962,192084,318182,527053,873043,
%U A206743 1446161,2395504,3968060,6572925,10887788,18035177,29874537,49485965,81971484,135782448
%N A206743 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 A206743 From _Clark Kimberling_, Jun 12 2016: (Start)
%C A206743 Number of real integers in n-th generation of tree T(2i) defined as follows.
%C A206743 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 A206743 For r = 2i, then g(3) = {3,2r,r+1, r^2}, in which the number of real integers is a(3) = 2.
%C A206743 See A274142 for a guide to related sequences. (End)
%H A206743 Kenny Lau, <a href="/A206743/b206743.txt">Table of n, a(n) for n = 0..1000</a>
%F A206743 a(n) ~ c * d^n, where d = 1.6564594309887754808836889708489581749625897572527517021957723319642053908... and c = 0.3844078703275069072126260832303344589497793302955451672191630264983... - _Vaclav Kotesovec_, Aug 25 2017
%e A206743 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 A206743 := proc(r)
%p A206743 local gs,n,gs2,el,a ;
%p A206743 gs := [2,r] ;
%p A206743 for n from 3 do
%p A206743 gs2 := [] ;
%p A206743 for el in gs do
%p A206743 gs2 := [op(gs2),el+1,r*el] ;
%p A206743 end do:
%p A206743 gs := gs2 ;
%p A206743 a := 0 ;
%p A206743 for el in gs do
%p A206743 if type(el,'realcons') then
%p A206743 a := a+1 :
%p A206743 end if;
%p A206743 end do:
%p A206743 print(n,a) ;
%p A206743 end do:
%p A206743 end proc: # _R. J. Mathar_, Jun 16 2016
%t A206743 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 A206743 (PARI) {Pell(n)=polcoeff(x/(1-2*x-x^2+x*O(x^n)),n)}
%o A206743 {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 A206743 for(n=0,55,print1(a(n),", "))
%o A206743 (Python)
%o A206743 N = 1000
%o A206743 pell = [0,1]
%o A206743 c = 2
%o A206743 while c < N:
%o A206743     pell.append(c)
%o A206743     c = pell[-1]*2 + pell[-2]
%o A206743 pell.reverse()
%o A206743 gf = [0]*(N+1)
%o A206743 for p in pell:
%o A206743     gf = [-x for x in gf]
%o A206743     gf[0] += 1
%o A206743     quotient = [0]*(N+1)
%o A206743     remainder = [0]*(N+1)
%o A206743     remainder[p] = 1
%o A206743     for n in range(N+1):
%o A206743         q = remainder[n]//gf[0]
%o A206743         for i in range(n,N+1):
%o A206743             remainder[i] -= q*gf[i-n]
%o A206743         quotient[n] = q
%o A206743     gf = quotient
%o A206743 for i in range(N+1):
%o A206743     print(i,gf[i])
%o A206743 # _Kenny Lau_, Aug 01 2017
%Y A206743 Cf. A000621, A206741, A274142.
%K A206743 nonn
%O A206743 0,4
%A A206743 _Paul D. Hanna_, Feb 12 2012