

A022290


Replace 2^k in binary expansion of n with Fibonacci(k+2).


11



0, 1, 2, 3, 3, 4, 5, 6, 5, 6, 7, 8, 8, 9, 10, 11, 8, 9, 10, 11, 11, 12, 13, 14, 13, 14, 15, 16, 16, 17, 18, 19, 13, 14, 15, 16, 16, 17, 18, 19, 18, 19, 20, 21, 21, 22, 23, 24, 21, 22, 23, 24, 24, 25, 26, 27, 26, 27, 28, 29, 29, 30, 31, 32, 21, 22, 23, 24, 24, 25, 26
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OFFSET

0,3


LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000


FORMULA

G.f.: 1/(1x) * Sum_{k>=0} F(k+2)*x^2^k/(1+x^2^k), where F = A000045.
a(n) = Sum_{k>=0} A030308(n,k)*A000045(k+2).  Philippe Deléham, Oct 15 2011
a(A003714(n)) = n.  R. J. Mathar, Jan 31 2015
a(A000225(n)) = A001911(n).  Philippe Deléham, Jun 05 2015
From Jeffrey Shallit, Jul 17 2018: (Start)
Can be computed from the recurrence:
a(4*k) = a(k)+a(2*k),
a(4*k+1) = a(k)+a(2*k+1),
a(4*k+2) = a(k)a(2*k)+2*a(2*k+1),
a(4*k+3) = a(k)2*a(2*k)+3*a(2*k+1),
and the initial terms a(0) = 0, a(1) = 1. (End)


EXAMPLE

n=4 = 2^2 is replaced by A000045(2+2) =3. n=5 =2^2+2^0 is replaced by A000045(2+2)+A000045(0+2) = 3+1=4.  R. J. Mathar, Jan 31 2015
This sequence regarded as a triangle with rows of lengths 1, 1, 2, 4, 8, 16, ...:
0
1
2, 3
3, 4, 5, 6
5, 6, 7, 8, 8, 9, 10, 11
8, 9, 10, 11, 11, 12, 13, 14, 13, 14, 15, 16, 16, 17, 18, 19
...  Philippe Deléham, Jun 05 2015


MAPLE

A022290 := proc(n)
dgs := convert(n, base, 2) ;
add( op(i, dgs)*A000045(i+1), i=1..nops(dgs)) ;
end proc: # R. J. Mathar, Jan 31 2015


MATHEMATICA

Table[Reverse[#].Fibonacci[1 + Range[Length[#]]] &@ IntegerDigits[n, 2], {n, 0, 54}] (* IWABUCHI Yu(u)ki, Aug 01 2012 *)


PROG

(Haskell)
a022290 0 = 0
a022290 n = h n 0 $ drop 2 a000045_list where
h 0 y _ = y
h x y (f:fs) = h x' (y + f * r) fs where (x', r) = divMod x 2
 Reinhard Zumkeller, Oct 03 2012


CROSSREFS

Other sequences that are built by replacing 2^k in the binary representation with other numbers: A029931 (naturals), A059590 (factorials), A089625 (primes).
Sequence in context: A290801 A322815 A244041 * A185363 A103827 A094182
Adjacent sequences: A022287 A022288 A022289 * A022291 A022292 A022293


KEYWORD

nonn,tabf


AUTHOR

Marc LeBrun


STATUS

approved



