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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. 5
 1, 1, 1, 2, 3, 5, 8, 13, 22, 36, 60, 99, 164, 272, 450, 746, 1235, 2046, 3389, 5613, 9299, 15402, 25514, 42262, 70005, 115962, 192084, 318182, 527053, 873043, 1446161, 2395504, 3968060, 6572925, 10887788, 18035177, 29874537, 49485965, 81971484, 135782448 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS From Clark Kimberling, Jun 12 2016: (Start) Number of real integers in n-th generation of tree T(2i) defined as follows. 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. For r = 2i, then g(3) = {3,2r,r+1, r^2}, in which the number of real integers is a(3) = 2. See A274142 for a guide to related sequences. (End) LINKS Kenny Lau, Table of n, a(n) for n = 0..1000 FORMULA a(n) ~ c * d^n, where d = 1.6564594309887754808836889708489581749625897572527517021957723319642053908... and c = 0.3844078703275069072126260832303344589497793302955451672191630264983... - Vaclav Kotesovec, Aug 25 2017 EXAMPLE G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 3*x^4 + 5*x^5 + 8*x^6 + 13*x^7 +... MAPLE A206743 := proc(r) local gs, n, gs2, el, a ; gs := [2, r] ; for n from 3 do gs2 := [] ; for el in gs do gs2 := [op(gs2), el+1, r*el] ; end do: gs := gs2 ; a := 0 ; for el in gs do if type(el, 'realcons') then a := a+1 : end if; end do: print(n, a) ; end do: end proc: # R. J. Mathar, Jun 16 2016 MATHEMATICA 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 *) PROG (PARI) {Pell(n)=polcoeff(x/(1-2*x-x^2+x*O(x^n)), n)} {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)} for(n=0, 55, print1(a(n), ", ")) (Python) N = 1000 pell = [0, 1] c = 2 while c < N: ....pell.append(c) ....c = pell[-1]*2 + pell[-2] pell.reverse() gf = [0]*(N+1) for p in pell: ....gf = [-x for x in gf] ....gf[0] += 1 ....quotient = [0]*(N+1) ....remainder = [0]*(N+1) ....remainder[p] = 1 ....for n in range(N+1): ........q = remainder[n]//gf[0] ........for i in range(n, N+1): ............remainder[i] -= q*gf[i-n] ........quotient[n] = q ....gf = quotient for i in range(N+1): ....print(i, gf[i]) # Kenny Lau, Aug 01 2017 CROSSREFS Cf. A000621, A206741, A274142. Sequence in context: A320356 A004697 A245271 * A186085 A018151 A227374 Adjacent sequences: A206740 A206741 A206742 * A206744 A206745 A206746 KEYWORD nonn AUTHOR Paul D. Hanna, Feb 12 2012 STATUS approved

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Last modified February 5 11:26 EST 2023. Contains 360084 sequences. (Running on oeis4.)