

A086893


a(n) is the index of F(n+1) at the unique occurrence of the ordered pair of reversed consecutive terms (F(n+1),F(n)) in Stern's diatomic sequence A002487, where F(k) denotes the kth term of the Fibonacci sequence A000045.


33



1, 3, 5, 13, 21, 53, 85, 213, 341, 853, 1365, 3413, 5461, 13653, 21845, 54613, 87381, 218453, 349525, 873813, 1398101, 3495253, 5592405, 13981013, 22369621, 55924053, 89478485, 223696213, 357913941, 894784853, 1431655765, 3579139413
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OFFSET

1,2


COMMENTS

If the Fibonacci pairs are kept in the natural order (F(n),F(n+1)), it appears that the first term of the pair occurs in A002487 at the index given by A061547(n).
Starting at n=3, begin subtracting from (2^(n1)1)/2^(n1): 3/4  1/2 = 1/4 with 1+4=5=a(3); 7/8  1/4 = 5/8 with 5+8=13=a(4); 15/16  5/8 = 5/16 with 5+16=21= a(5); 31/32  5/16 = 21/32 with 21+32=53=a(6); 63/64  21/32 = 21/64 with 21+64=85=a(7) and so on. For n odd in the first fraction (2^(n1)1)/2^(n1), the result approaches 1/3, and for n even in the first fraction, the result approaches 2/3.  J. M. Bergot, May 08 2015
Also, the decimal representation of the xaxis, from the left edge to the origin, of the nth stage of growth of the twodimensional cellular automaton defined by "Rule 678", based on the 5celled von Neumann neighborhood, initialized with a single black (ON) cell at stage zero. See A283641.  Robert Price, Mar 12 2017


LINKS



FORMULA

It appears that a(n)=(4^((n+1)/2)1)/3 if n is odd and a(n)=(a(n1)+a(n+1))/2 if n is even.
G.f.: (1+2*x2*x^2)/((1x)*(14*x^2)); a(n) = 2^(n1)(3(1)^n/3)1/3 (offset 0); a(n) = Sum{k=0..n+1, 4^floor(k/2)/2} (offset 0); a(2n) = A002450(n+1) (offset 0); a(2n+1) = A072197(n) (offset 0).  Paul Barry, May 21 2004
a(n+2) = 4*a(n) + 1, a(1) = 1, a(2) = 3, n > 0.  Yosu Yurramendi, Mar 07 2017


EXAMPLE

A002487 begins 0,1,1,2,1,3,2,... with offset 0. Thus a(1)=1 since (F(2),F(1)) = (1,1) occurs at term 1 of A002487. Similarly, a(2)=3 and a(3)=5, since (F(3),F(2))=(2,1) occurs at term 3 and (F(4),F(3))=(3,2) at term 5 of A002487.


MATHEMATICA

f[n_] := Module[{a = 1, b = 0, m = n}, While[m > 0, If[OddQ@ m, b = a + b, a = a + b]; m = Floor[m/2]]; b]; a = Table[f[n], {n, 0, 10^6}]; b = Reverse /@ Partition[Map[Fibonacci, Range[Ceiling@ Log[GoldenRatio, Max@ a] + 1]], 2, 1]; Map[If[Length@ # > 0, #[[1, 1]]  1, 0] &@ SequencePosition[a, #] &, b] (* Michael De Vlieger, Mar 15 2017, Version 10.1, after JeanFrançois Alcover at A002487 *)


PROG

(Python)


CROSSREFS

Positive terms of A096773 in ascending order.


KEYWORD

nonn,easy


AUTHOR



EXTENSIONS



STATUS

approved



