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A022290 Replace 2^k in binary expansion of n with Fibonacci(k+2). 36
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
FORMULA
G.f.: 1/(1-x) * 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)
a(A003754(n)) = n-1. - Rémy Sigrist, Jan 28 2020
From Rémy Sigrist, Aug 04 2022: (Start)
Empirically:
- a(2*A003714(n)) = A022342(n+1),
- a(3*A003714(n)) = a(4*A003714(n)) = A026274(n) for n > 0.
(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
From Philippe Deléham, Jun 05 2015: (Start)
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
...
(End)
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
# second Maple program:
b:= (n, i, j)-> `if`(n=0, 0, j*irem(n, 2, 'q')+b(q, j, i+j)):
a:= n-> b(n, 1$2):
seq(a(n), n=0..127); # Alois P. Heinz, Jan 26 2022
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
(PARI) my(m=Mod('x, 'x^2-'x-1)); a(n) = subst(lift(subst(Pol(binary(n)), 'x, m)), 'x, 2); \\ Kevin Ryde, Sep 22 2020
(Python)
def A022290(n):
a, b, s = 1, 2, 0
for i in bin(n)[-1:1:-1]:
s += int(i)*a
a, b = b, a+b
return s # Chai Wah Wu, Sep 10 2022
CROSSREFS
Other sequences that are built by replacing 2^k in the binary representation with other numbers: A029931 (naturals), A054204 (even index Fibonaccis), A062877 (odd index Fibonaccis), A059590 (factorials), A089625 (primes).
Sequence in context: A322815 A244041 A331835 * A344350 A185363 A103827
KEYWORD
nonn,tabf,base
AUTHOR
STATUS
approved

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Last modified March 19 01:57 EDT 2024. Contains 370952 sequences. (Running on oeis4.)