A004685 -id:A004685 - cp's OEIS frontend

A350700 a(n) is the number of 1's minus the number of 0's in A004685(n).

-1, 1, 1, 0, 2, 1, -2, 2, 1, -2, 4, 1, -4, 2, 3, -2, 6, 3, -4, -3, 3, -2, 1, 7, -4, -5, 1, 4, 3, 5, -4, 1, -4, 4, 1, -2, 0, 3, -6, -2, 5, 6, 0, 3, 6, -1, 11, -6, -9, 3, 2, -1, -1, -2, -5, 6, 4, -7, 8, 0, -9, -4, 10, 3, -4, 6, -7, 6, -17, -1, -2, -5, 1, 4, -3

Offset: 0

Karl-Heinz Hofmann, Jan 18 2022

			A004685(0) = 0; this term has 0 ones and 1 zero. So a(0) = 0 - 1 = -1.
A004685(7) = 1101; this term has 3 ones and 1 zero. So a(7) = 3 - 1 = 2.

Karl-Heinz Hofmann, Table of n, a(n) for n = 0..10000

Cf. A000045, A004685, A011373, A145037, A177718, A214852.

a[n_] := Subtract @@ DigitCount[Fibonacci[n], 2, {1, 0}]; Array[a, 75, 0] (* Amiram Eldar, Jan 22 2022 *)

from sympy import fibonacci
print([(bin(fibonacci(n))[2:].count("1") - bin(fibonacci(n))[2:].count("0")) for n in range (0,100)])

a(n) = A145037(A000045(n)) for n >= 1.

a(n) = 0 if and only if n is in A214852. - Amiram Eldar, Jan 22 2022

A036284 Periodic vertical binary vectors of Fibonacci numbers.

Original entry on oeis.org

6, 24, 1440, 5728448, 92568198012160, 26494530374406845814111659520, 2095920895719545919920115988669687683503034097906010941440, 13128614603426246034591796912897206548807135027496968025827278400248602613784037111736380004928525614173642247188480

Offset: 0

Antti Karttunen, Nov 01 1998

The sequence can be also computed with a recurrence that does not explicitly refer to Fibonacci numbers. See the given Maple and C programs.

Conjecture: For n>=1, each term a(n), when considered as a GF(2)[X]-polynomial, is divisible by GF(2)[X] -polynomial (x^3 + 1) ^ A000225(n-1). If this holds, then for n>=1, a(n) = A048720bi(A136380(n),A048723bi(9,A000225(n-1))). Conjecture 2: there is also one extra (x^1 + 1) factor present, see A136384.

			When Fibonacci numbers are written in binary (see A004685), under each other as:
0000000 (0)
0000001 (1)
0000001 (1)
0000010 (2)
0000011 (3)
0000101 (5)
0001000 (8)
0001101 (13)
0010101 (21)
0100010 (34)
0110111 (55)
1011001 (89)
it can be seen that the bits in the n-th column from right repeat after a period of A007283(n): 3, 6, 12, 24, ... (See also A001175). This sequence is formed from those bits: 011, reversed is 110, is binary for 6, thus a(0) = 6. 000110, reversed is 11000, is binary for 24, thus a(1) = 24, 000001011010, reversed is 10110100000, is binary for 1440, thus a(2) = 1440.

A. Karttunen, Table of n, a(n) for n = 0..10
A. Karttunen, C program for computing this sequence

Same sequence in octal base: A036285. Bits reversed: A036286. See also A136378, A136379, A136380, A136382, A136384, A037096, A037093, A000045.

A036284:=proc(n) option remember; local a, b, c, i, j, k, l, s, x, y, z; if (0 = n) then (6) else a := 0; b := 0; s := 0; x := 0; y := 0; k := 3*(2^(n-1)); l := 3*(2^n); j := 0; for i from 0 to l do z := bit_i(A036284(n-1),(j)); c := (a + b + (`if`((x = y),x,(z+1))) mod 2); if(c <> 0) then s := s + (2^i); fi; a := b; b := c; x := y; y := z; j := j + 1; if(j = k) then j := 0; fi; od; RETURN(s); fi; end:
bit_i := (x,i) -> `mod`(floor(x/(2^i)),2);

a[n_] := Sum[Mod[Fibonacci[k]/2^n // Floor, 2]* 2^k, {k, 0, 3*2^n - 1}]; Table[a[n], {n, 0, 7}] (* Jean-François Alcover, Mar 04 2016 *)

a(n) = Sum_{k=0..A007283(n)-1} ([A000045(k)/(2^n)] mod 2) * 2^k, where [] stands for floor function, i.e. Sum (bit n of Fibonacci(k))*(2^k), k = 0 ... (3*(2^n))-1.

Entry revised Dec 29 2007

A037093 "Sloping binary representation" of Fibonacci numbers, slope = +1.

Original entry on oeis.org

0, 1, 3, 14, 57, 229, 916, 7761, 29567, 117474, 469113, 3973641, 15138352, 60146777, 240187355, 2070207870, 7733090689, 30791909229, 260408711716, 991495872825, 3942106110215, 15739612088946, 133333733918417

Offset: 0

Antti Karttunen, Jan 28 1999

			When Fibonacci numbers are written in binary (see A004685), under each other as:
0000000 (0)
0000001 (1)
0000001 (1)
0000010 (2)
0000011 (3)
0000101 (5)
0001000 (8)
0001101 (13)
0010101 (21)
0100010 (34)
0110111 (55)
1011001 (89)
and one starts collecting their bits from column-0 to SW-direction (from the least to the most significant end), one gets 000... (0), ...00001 (1), ...00011 (3), ...001110 (14), etc. (See A102370 for similar transformation done on nonnegative integers).

Same sequence in octal: A037098. Cf. also: A102370, A000045, A037094-A037095, A036284.

a(n) := Sum(bit_n(A000045(n+i), i)*(2^i), i=0..inf) [ bit_n := (x, n) -> `mod`(floor(x/(2^n)), 2); ]

In practice, n can be used as an upper limit instead of infinity.

Entry revised Dec 29 2007

A272170 Second most significant bit of Fibonacci numbers > 1 written in base 2.

Original entry on oeis.org

0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0

Offset: 3

Andres Cicuttin, Apr 21 2016

It is conjectured that there are no more than two consecutive "0's" or “1’s” (tested up to n=10^5). The sequence looks quasiperiodic and its Fourier spectrum seems to have a fractal structure.

			(second MSB in parenthesis)
  n   A000045(n)      A004685(n)
  3      2       ->   1(0)
  4      3       ->   1(1)
  5      5       ->   1(0)1
  6      8       ->   1(0)00
  7      13      ->   1(1)01
  8      21      ->   1(0)101
  9      34      ->   1(0)0010
  10     55      ->   1(1)0111
...

Chai Wah Wu, Table of n, a(n) for n = 3..10000

Cf. A000045, A004685, A079944, A271591.

nmax = 120; Table[IntegerDigits[Fibonacci[j], 2][[2]], {j, 3, nmax}]

a(n) = binary(fibonacci(n))[2]; \\ Michel Marcus, Apr 25 2016

A272170_list, a, b = [], 1 ,1
for n in range(3,10001):
    a, b = b, a+b
    A272170_list.append(int(bin(b)[3])) # Chai Wah Wu, Feb 07 2018

a(n) = floor(A000045(n)/(2^(ceiling(log_2(A000045(n) + 1)) - 2))) - 2.

a(n) = A079944(A000045(n)-2). - Michel Marcus, Apr 22 2016

A030324 Triangle read by rows, where row k consists of the binary digits of Fibonacci(k+1).

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 0, 0

Offset: 1

Clark Kimberling

			Triangle starts
1
1, 0
1, 1
1, 0, 1
1, 0, 0, 0
1, 1, 0, 1
1, 0, 1, 0, 1
1, 0, 0, 0, 1, 0

Robert Israel, Table of n, a(n) for n = 1..10098 (rows 1 to 170, flattened)

Cf. A000045, A004685, A272170 (second column).

for n from 2 to 30 do
  ListTools:-Reverse(convert(combinat:-fibonacci(n),base,2))
od; # Robert Israel, Sep 12 2018

Flatten[Map[IntegerDigits[#, 2] &, Table[Fibonacci[n], {n, 50}], {1}]] (* Ben Branman, Feb 14 2011 *)
IntegerDigits[#,2]&/@Fibonacci[Range[2,20]]//Flatten (* Harvey P. Dale, May 29 2021 *)

Edited by Robert Israel, Sep 12 2018

A222601 Conjectured number of Fibonacci numbers with exactly n 0-bits in their binary representation.

Original entry on oeis.org

3, 5, 2, 4, 3, 1, 2, 3, 5, 1, 3, 5, 3, 3, 2, 1, 6, 1, 2, 3, 3, 2, 0, 3, 5, 4, 4, 3, 4, 2, 1, 2, 6, 1, 2, 2, 5, 4, 3, 5, 2, 2, 2, 1, 2, 2, 2, 5, 6, 3, 2, 2, 3, 1, 5, 1, 1, 0, 8, 4, 3, 3, 3, 3, 5, 4, 4, 2, 2, 2, 2, 3, 2, 6, 3, 0, 0, 2, 5, 5, 1, 6, 5, 0, 3, 5, 1

Offset: 0

T. D. Noe, Mar 08 2013

nonn

T. D. Noe, Table of n, a(n) for n = 0..1000

Cf. A004685 (Fibonacci numbers in binary), A214853 (one 0-bit), A222602.

Cf. A222295, A222296.

f = Fibonacci[Range[0,100]]; Table[Length[Select[f, Count[IntegerDigits[#, 2], 0] == n &]], {n, 0, 20}]

A222602 Irregular triangle of conjectured Fibonacci numbers with exactly n 0-bits in their binary representation.

Original entry on oeis.org

1, 1, 3, 0, 2, 5, 13, 55, 21, 987, 8, 89, 233, 377, 34, 1597, 28657, 6765, 144, 610, 17711, 196418, 514229, 2584, 4181, 10946, 121393, 317811, 3524578, 46368, 1346269, 1836311903, 75025, 5702887, 24157817, 102334155, 165580141, 832040, 14930352, 701408733

Offset: 0

T. D. Noe, Mar 08 2013

			The irregular triangle begins
{1, 1, 3},
{0, 2, 5, 13, 55},
{21, 987},
{8, 89, 233, 377},
{34, 1597, 28657},
{6765},
{144, 610},
{17711, 196418, 514229},
{2584, 4181, 10946, 121393, 317811},
{3524578}, {46368, 1346269, 1836311903},
{75025, 5702887, 24157817, 102334155, 165580141},
{832040, 14930352, 701408733}

T. D. Noe, Rows n = 0..500 of irregular triangle, flattened

Cf. A004685 (Fibonacci numbers in binary), A214853 (one 0-bit), A222601.

Cf. A222295, A222296, A222757, A222758.

f = Fibonacci[Range[0,1000]]; Table[Select[f, Count[IntegerDigits[#, 2], 0] == n &], {n, 0, 20}]

A222296 Irregular triangle read by rows: row n lists the Fibonacci numbers with exactly n 1's in their binary representation.

Original entry on oeis.org

0, 1, 1, 2, 8, 3, 5, 34, 144, 13, 21

Offset: 0

T. D. Noe, Feb 22 2013

Besides those listed in Example section, there are no additional terms with small number of 1's in the first 10^12 Fibonacci numbers. In particular, if A000120(Fibonacci(n)) < 100, then n <= 319 or n > 10^12. - Charles R Greathouse IV, Mar 06 2014

For the theorem about S-units that Noam Elkies quotes (in the MathOverflow link), see Chapter 1 of Storey-Tijdemann, 1986. - N. J. A. Sloane, Jan 28 2017

			The irregular table begins
{0},
{1, 1, 2, 8},
{3, 5, 34, 144},
{13, 21, ...}.
It is conjectured that the previous (n=3) row is complete, and that the subsequent rows are:
{89, 610, 2584},
{55, 233, 4181},
{377, 10946, 46368, 75025},
{1597},
{987, 6765, 17711, 832040},
{121393, 2178309},
{39088169},
{28657, 196418, 317811, 1346269, 9227465},
{514229, 5702887, 14930352, 63245986, 4807526976},
{3524578, 2971215073}
...

T. N. Shorey and R. Tijdeman, Exponential Diophantine Equations, Cambridge Tracts in Mathematics, 1986.

Charles Greathouse and Noam D. Elkies, Hamming weight of Fibonacci numbers, MathOverflow, 2014
Charles Greathouse and Noam D. Elkies, Hamming weight of Fibonacci numbers, MathOverflow, 2014 [Cached copy of three screen shots]
T. D. Noe, Conjectured values for rows n = 0..500 of irregular triangle, flattened
David Terr, On the sums of digits of Fibonacci numbers, Fibonacci Quarterly 34, Aug. 1996, pp. 349-355.

Cf. A004685 (Fibonacci numbers in binary), A221158 (weight 2), A222295, A222601, A222602, A222757, A222758.

f = Fibonacci[Range[0,100]]; Table[Select[f, Total[IntegerDigits[#, 2]] == n &], {n, 0, 20}]

row(n)=my(k=-1,t); while(1,t=fibonacci(k++); if(hammingweight(t)==n, print1(t", "))) \\ Charles R Greathouse IV, Mar 04 2014

a(9)-a(10) from Noam D. Elkies, via Charles R Greathouse IV, Mar 04 2014
Truncated to established terms by Max Alekseyev, May 13 2014
Edited by Max Alekseyev, Sep 08 2016

A222757 Irregular table of conjectured indices of Fibonacci numbers with exactly n 0-bits in their binary representation.

Original entry on oeis.org

1, 2, 4, 0, 3, 5, 7, 10, 8, 16, 6, 11, 13, 14, 9, 17, 23, 20, 12, 15, 22, 27, 29, 18, 19, 21, 26, 28, 33, 24, 31, 46, 25, 34, 37, 40, 41, 30, 36, 44, 32, 35, 43, 39, 42, 49, 38, 45, 50, 55, 58, 62, 56, 51, 52, 47, 53, 65, 59, 63, 67, 48, 54, 57, 61, 73

Offset: 0

T. D. Noe, Mar 11 2013

Every nonnegative integer appears.

			The irregular table begins
{1, 2, 4},
{0, 3, 5, 7, 10},
{8, 16},
{6, 11, 13, 14},
{9, 17, 23},
{20},
{12, 15},
{22, 27, 29},
{18, 19, 21, 26, 28},
{33},
{24, 31, 46},
{25, 34, 37, 40, 41}

T. D. Noe, Rows n = 0..1000 of irregular triangle, flattened

Cf. A004685 (Fibonacci numbers in binary), A222601, A222602, A222758.

nn = 100; f = Fibonacci[Range[0, nn]]; t2 = Transpose[{Range[0, nn], f}]; Table[Select[Range[nn + 1], Count[IntegerDigits[t2[[#, 2]], 2], 0] == n &] - 1, {n, 0, nn/5}]
Insert[Flatten[Module[{nn=100,dc},dc=DigitCount[Fibonacci[Range[nn]],2,0];Table[Position[dc,n],{n,0,30}]]],0,4] (* Harvey P. Dale, Mar 17 2024 *)

A222758 Irregular table of conjectured indices of Fibonacci numbers with exactly n 1-bits in their binary representation.

Original entry on oeis.org

0, 1, 2, 3, 6, 4, 5, 9, 12, 7, 8, 11, 15, 18, 10, 13, 19, 14, 21, 24, 25, 17, 16, 20, 22, 30, 26, 32, 38, 23, 27, 28, 31, 35, 29, 34, 36, 39, 48, 33, 47, 37, 42, 45, 68, 40, 43, 54, 57, 60, 41, 51, 52, 53, 44, 49, 50, 61, 66, 59, 64, 46, 56, 55, 71, 77

Offset: 0

T. D. Noe, Mar 11 2013

Every nonnegative integer appears.

			The irregular table begins
{0},
{1, 2, 3, 6},
{4, 5, 9, 12},
{7, 8},
{11, 15, 18},
{10, 13, 19},
{14, 21, 24, 25},
{17},
{16, 20, 22, 30},
{26, 32},
{38},
{23, 27, 28, 31, 35}

T. D. Noe, Rows n = 0..1000 of irregular triangle, flattened

Cf. A004685 (Fibonacci numbers in binary), A222601, A222602, A222757.

nn = 100; f = Fibonacci[Range[0, nn]]; t2 = Transpose[{Range[0, nn], f}]; Table[Select[Range[nn + 1], Count[IntegerDigits[t2[[#, 2]], 2], 1] == n &] - 1, {n, 0, nn/5}]

cp's OEIS Frontend

A350700 a(n) is the number of 1's minus the number of 0's in A004685(n).

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A036284 Periodic vertical binary vectors of Fibonacci numbers.

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A037093 "Sloping binary representation" of Fibonacci numbers, slope = +1.

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A272170 Second most significant bit of Fibonacci numbers > 1 written in base 2.

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A030324 Triangle read by rows, where row k consists of the binary digits of Fibonacci(k+1).

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A222601 Conjectured number of Fibonacci numbers with exactly n 0-bits in their binary representation.

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A222602 Irregular triangle of conjectured Fibonacci numbers with exactly n 0-bits in their binary representation.

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A222296 Irregular triangle read by rows: row n lists the Fibonacci numbers with exactly n 1's in their binary representation.

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