A062877 -id:A062877 - cp's OEIS frontend

A062878 a(n) is the position of A050614(n) in A062877.

1, 3, 6, 15, 24, 60, 102, 255, 384, 960, 1632, 4080, 6168, 15420, 26214, 65535, 98304, 245760, 417792, 1044480, 1579008, 3947520, 6710784, 16776960, 25166208, 62915520, 106956384, 267390960, 404232216, 1010580540, 1717986918, 4294967295, 6442450944

Offset: 0

Antti Karttunen, Jun 26 2001

nonn

In binary this sequence looks like 1, 11, 110, 1111, 11000, 111100, 1100110, 11111111, 110000000, 1111000000, 11001100000, 111111110000, 1100000011000, 11110000111100, 110011001100110, ...
Sequence A282387 may be the same, but I cannot prove nor disprove this beyond a(22). - Robert Price, Feb 13 2017
Agrees with A282387 for at least 1000 terms. - Sean A. Irvine, Apr 14 2023

A050614 = Table[k = Floor[Log[2, n + 1]]; Product[j = 2^(i + 1); l = Fibonacci[j + 1] + Fibonacci[j - 1]; If[BitAnd[2^i, n] == 0, b = 0, b = 1]; l^b, {i, 0, k}], {n, 0, 200}]; A062877 = Union[Total /@ Subsets[Fibonacci[Range[1, 46, 2]]]]; Flatten[Table[Position[ A062877, A050614[[i]] ] - 1, {i, 1, 25}]] (* Robert Price, Feb 13 2017 *)

a(2^n-1) = 2^(2^n) - 1. - Philippe Deléham, Apr 05 2007

a(n) = Sum_{k=0..n} A127872(n,k)*2^k. - Philippe Deléham, Oct 09 2007

a(15)-a(22) from Robert Price, Feb 13 2017

More terms from Sean A. Irvine, Apr 14 2023

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

Original entry on oeis.org

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

Offset: 0

Marc LeBrun

			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)

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

Other sequences that are built by replacing 2^k in the binary representation with other numbers: A029931 (naturals), A054204 (even-indexed Fibonacci numbers), A062877 (odd-indexed Fibonacci numbers), A059590 (factorials), A089625 (primes).

Cf. A003754, A022342, A026274.

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

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

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

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

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

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)

A050614 Products of distinct terms of A001566: a(n) = Product_{i=0..floor(log_2(n+1))} L(2^(i+1))^bit(n,i).

Original entry on oeis.org

1, 3, 7, 21, 47, 141, 329, 987, 2207, 6621, 15449, 46347, 103729, 311187, 726103, 2178309, 4870847, 14612541, 34095929, 102287787, 228929809, 686789427, 1602508663, 4807525989, 10749959329, 32249877987, 75249715303

Offset: 0

Antti Karttunen, Dec 02 1999

nonn

Each subset a(0..(2^k)-1) gives all the divisors of F(2^(k+1)) up to k=4 (F_32) and after that a subset of such divisors. E.g., the terms a(0)-a(7) are the divisors of F_16 = 987 (A018760).

Robert Price, Table of n, a(n) for n = 0..200
A. Karttunen, On Pascal's Triangle Modulo 2 in Fibonacci Representation, Fibonacci Quarterly, 42 (2004), 38-46.

Bisection of A075149 and A050613 (see there for the other Maple procedures), subset of A062877. Cf. also A050615.

[seq(A050614(n),n=0..30)]; A050614 := n -> product('luc(2^(i+1))^bit_i(n,i)','i'=0..floor_log_2(n+1));

Table[k = Floor[Log[2, n + 1]]; Product[j = 2^(i + 1); l = Fibonacci[j + 1] + Fibonacci[j - 1]; If[BitAnd[2^i, n] == 0, b = 0, b = 1]; l^b, {i, 0, k}], {n, 0, 200}] (* Robert Price, Feb 13 2017 *)

a(n) = Sum_{k=0..n} A127872(n,k)*Fibonacci(2*k+1), see A000045 and A001519. - Philippe Deléham, Aug 30 2007

A054204 Integers expressible as sums of distinct even-subscripted Fibonacci numbers.

Original entry on oeis.org

1, 3, 4, 8, 9, 11, 12, 21, 22, 24, 25, 29, 30, 32, 33, 55, 56, 58, 59, 63, 64, 66, 67, 76, 77, 79, 80, 84, 85, 87, 88, 144, 145, 147, 148, 152, 153, 155, 156, 165, 166, 168, 169, 173, 174, 176, 177, 199, 200, 202, 203, 207, 208, 210, 211, 220, 221, 223, 224, 228, 229

Offset: 1

Marjorie Bicknell-Johnson (marjohnson(AT)earthlink.net), Apr 30 2000

nonn

Number of partitions of a(n) into sums of distinct Fibonacci numbers is (n+1)st term of Stern's Diatomic series A002487. This sequence has A046815 as a subsequence.

			a(9)=22 since 9=2^3+2^0 and 22=F(2(3+1)) + F(2(0+1)) = F(8) + F(2).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Marjorie Bicknell-Johnson, The least integer having p Fibonacci representations (p prime), Fibonacci Quarterly 40 (2002), pp. 260-265.
Marjorie Bicknell-Johnson, Stern's Diatomic Array Applied to Fibonacci Representations, Fibonacci Quarterly 41 (2003), pp. 169-180.
Sam Northshield, Some generalizations of a formula of Reznick, SUNY Plattsburgh (2022).
Index entries for sequences related to Stern's sequences

Cf. A002487, A014417, A046815.
Cf. A022290, A062877 (odd-indexed Fibonaccis).
Distinct additive closure of A001906.

fibEvenCount[n_] := Plus @@ (Reverse @ IntegerDigits[n, 2])[[2 ;; -1 ;; 2]]; evenIndexed = fibEvenCount /@ Select[Range[10000], BitAnd[#, 2 #] == 0 &]; Position[evenIndexed, ?(# == 0 &)] // Flatten (* _Amiram Eldar, Jan 20 2020*)

my(m=Mod('x,'x^2-3*'x+1)); a(n) = subst(lift(subst(Pol(binary(n)), 'x,m)), 'x,3); \\ Kevin Ryde, Nov 25 2020

Subscripts in Zeckendorf representation of a(n) are 2(e+1) where e is exponent used to write n as sum of powers of 2.

A062879 Integers whose Zeckendorf expansion does not contain ones at even positions.

Original entry on oeis.org

0, 2, 5, 7, 13, 15, 18, 20, 34, 36, 39, 41, 47, 49, 52, 54, 89, 91, 94, 96, 102, 104, 107, 109, 123, 125, 128, 130, 136, 138, 141, 143, 233, 235, 238, 240, 246, 248, 251, 253, 267, 269, 272, 274, 280, 282, 285, 287, 322, 324, 327, 329, 335, 337, 340, 342, 356

Offset: 1

Antti Karttunen, Jun 26 2001

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Bisection of A062877.
Subset of A022342.
Cf. A003714, A014417, A062880, A165276.

fibOddCount[n_] := Plus @@ (Reverse@IntegerDigits[n, 2])[[1 ;; -1 ;; 2]]; oddIndexed = fibOddCount /@ Select[Range[0, 10000], BitAnd[#, 2 #] == 0 &]; -1 + Position[oddIndexed, ?(# == 0 &)] // Flatten (* _Amiram Eldar, Jan 20 2020  *)

A062880(n) = A003714(a(n)).

A165276(a(n)) = 0. - Amiram Eldar, Jan 20 2020

Offset corrected by Amiram Eldar, Jan 20 2020

A356326 The terms in the negaFibonacci representation of a(n) are the terms in common in the negaFibonacci representations of n and -n.

Original entry on oeis.org

0, 0, 0, 0, -1, 0, 0, 0, 0, -1, -3, -3, -1, 0, 0, 0, 0, 4, 0, 0, 0, 0, -1, -3, -3, -1, -8, -8, -8, -8, -1, -3, -3, -1, 0, 0, 0, 0, 4, 0, 0, 0, 0, 12, 10, 10, 12, 0, 0, 0, 0, 4, 0, 0, 0, 0, -1, -3, -3, -1, -8, -8, -8, -8, -1, -3, -3, -1, -21, -21, -21, -21, -17

Offset: 0

Rémy Sigrist, Aug 03 2022

			For n = 11:
- using F(-k) = A039834(k):
- 11 = F(-1) + F(-4) + F(-7),
- -11 = F(-4) + F(-6),
- so a(11) = F(-4) = -3.

Rémy Sigrist, Table of n, a(n) for n = 0..10946
Rémy Sigrist, PARI program
Wikipedia, NegaFibonacci coding
Index entries for sequences related to Zeckendorf expansion of n

Cf. A000045, A039834, A062877, A215024, A215025, A356325, A356327.

PARI
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See Links section.
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a(n) = 0 iff n belongs to A062877.
a(n) = A356327(A215024(n) AND A215025(n)) (where AND denotes the bitwise AND operator).
Empirically:
- a(A000045(k)+m) = a(A000045(k+1)-m) for k >= 0, m = 0..A000045(k+1)-A000045(k).

cp's OEIS Frontend

A062878 a(n) is the position of A050614(n) in A062877.

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

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A050614 Products of distinct terms of A001566: a(n) = Product_{i=0..floor(log_2(n+1))} L(2^(i+1))^bit(n,i).

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A054204 Integers expressible as sums of distinct even-subscripted Fibonacci numbers.

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A062879 Integers whose Zeckendorf expansion does not contain ones at even positions.

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A356326 The terms in the negaFibonacci representation of a(n) are the terms in common in the negaFibonacci representations of n and -n.

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