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A022290 Replace 2^k in binary expansion of n with Fibonacci(k+2). 36

%I #67 Sep 11 2022 00:39:29

%S 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,

%T 16,17,18,19,13,14,15,16,16,17,18,19,18,19,20,21,21,22,23,24,21,22,23,

%U 24,24,25,26,27,26,27,28,29,29,30,31,32,21,22,23,24,24,25,26

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

%H Reinhard Zumkeller, <a href="/A022290/b022290.txt">Table of n, a(n) for n = 0..10000</a>

%F G.f.: 1/(1-x) * Sum_{k>=0} F(k+2)*x^2^k/(1+x^2^k), where F = A000045.

%F a(n) = Sum_{k>=0} A030308(n,k)*A000045(k+2). - _Philippe Deléham_, Oct 15 2011

%F a(A003714(n)) = n. - _R. J. Mathar_, Jan 31 2015

%F a(A000225(n)) = A001911(n). - _Philippe Deléham_, Jun 05 2015

%F From _Jeffrey Shallit_, Jul 17 2018: (Start)

%F Can be computed from the recurrence:

%F a(4*k) = a(k)+a(2*k),

%F a(4*k+1) = a(k)+a(2*k+1),

%F a(4*k+2) = a(k)-a(2*k)+2*a(2*k+1),

%F a(4*k+3) = a(k)-2*a(2*k)+3*a(2*k+1),

%F and the initial terms a(0) = 0, a(1) = 1. (End)

%F a(A003754(n)) = n-1. - _Rémy Sigrist_, Jan 28 2020

%F From _Rémy Sigrist_, Aug 04 2022: (Start)

%F Empirically:

%F - a(2*A003714(n)) = A022342(n+1),

%F - a(3*A003714(n)) = a(4*A003714(n)) = A026274(n) for n > 0.

%F (End)

%e 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

%e From _Philippe Deléham_, Jun 05 2015: (Start)

%e This sequence regarded as a triangle with rows of lengths 1, 1, 2, 4, 8, 16, ...:

%e 0

%e 1

%e 2, 3

%e 3, 4, 5, 6

%e 5, 6, 7, 8, 8, 9, 10, 11

%e 8, 9, 10, 11, 11, 12, 13, 14, 13, 14, 15, 16, 16, 17, 18, 19

%e ...

%e (End)

%p A022290 := proc(n)

%p dgs := convert(n,base,2) ;

%p add( op(i,dgs)*A000045(i+1),i=1..nops(dgs)) ;

%p end proc: # _R. J. Mathar_, Jan 31 2015

%p # second Maple program:

%p b:= (n, i, j)-> `if`(n=0, 0, j*irem(n, 2, 'q')+b(q, j, i+j)):

%p a:= n-> b(n, 1$2):

%p seq(a(n), n=0..127); # _Alois P. Heinz_, Jan 26 2022

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

%o (Haskell)

%o a022290 0 = 0

%o a022290 n = h n 0 $ drop 2 a000045_list where

%o h 0 y _ = y

%o h x y (f:fs) = h x' (y + f * r) fs where (x',r) = divMod x 2

%o -- _Reinhard Zumkeller_, Oct 03 2012

%o (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

%o (Python)

%o def A022290(n):

%o a, b, s = 1,2,0

%o for i in bin(n)[-1:1:-1]:

%o s += int(i)*a

%o a, b = b, a+b

%o return s # _Chai Wah Wu_, Sep 10 2022

%Y 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).

%Y Cf. A003754, A022342, A026274, .

%K nonn,tabf,base

%O 0,3

%A _Marc LeBrun_

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Last modified March 28 13:21 EDT 2024. Contains 371254 sequences. (Running on oeis4.)