A219608 -id:A219608 - cp's OEIS frontend

A026351 a(n) = floor(n*phi) + 1, where phi = (1+sqrt(5))/2.

1, 2, 4, 5, 7, 9, 10, 12, 13, 15, 17, 18, 20, 22, 23, 25, 26, 28, 30, 31, 33, 34, 36, 38, 39, 41, 43, 44, 46, 47, 49, 51, 52, 54, 56, 57, 59, 60, 62, 64, 65, 67, 68, 70, 72, 73, 75, 77, 78, 80, 81, 83, 85, 86, 88, 89, 91, 93, 94, 96, 98, 99

Offset: 0

N. J. A. Sloane, Clark Kimberling

a(n)=least k such that s(k)=n, where s=A026350.
a(n)=position of n-th 1 in A096270.
From Wolfdieter Lang, Jun 27 2011: (Start)
a(n) = A(n)+1, with Wythoff sequence A(n)=A000201(n), n>=1, and A(0)=0.
a(n) = -floor(-n*phi). Recall that floor(-x) = -(floor(x)+1) if x is not integer and -floor(x) otherwise.
An exhaustive and disjoint decomposition of the integers is given by the following two Wythoff sequences A' and B: A'(0):=-1 (not 0), A'(-n):=-a(n)=-(A(n)+1), n>=1, A'(n) = A(n), n>=1, and B(-n):=-(B(n)+1)= -A026352(n), n>=1, with B(n)=A001950(n), n>=1, and B(0)=0.
(End)
Where odd terms in A060142 occur: A060142(a(n)) = A219608(n). - Reinhard Zumkeller, Nov 26 2012

Carmine Suriano, Table of n, a(n) for n = 0..10000
Jon Asier Bárcena-Petisco, Luis Martínez, María Merino, Juan Manuel Montoya, and Antonio Vera-López, Fibonacci-like partitions and their associated piecewise-defined permutations, arXiv:2503.19696 [math.CO], 2025. See p. 3.
Benoit Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence, J. Integer Seqs., Vol. 6 (2003), #03.2.2.
Benoit Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence, arXiv:math/0305308 [math.NT], 2003.
J. H. Conway and N. J. A. Sloane, Notes on the Para-Fibonacci and related sequences
Eric Friedman, Scott M. Garrabrant, Ilona K. Phipps-Morgan, A. S. Landsberg and Urban Larsson, Geometric analysis of a generalized Wythoff game, in Games of no Chance 5, MSRI publ. Cambridge University Press, 2019.
N. J. A. Sloane, Classic Sequences

Essentially same as A004956. Cf. A000201.
Complement of A026352.
Cf. A283733 (partial sums).

import Data.List (findIndices)
a026351 n = a026351_list !! n
a026351_list = findIndices odd a060142_list
-- Reinhard Zumkeller, Nov 26 2012

Table[Floor[n*GoldenRatio] + 1, {n, 0, 100}] (* T. D. Noe, Apr 15 2011 *)

from math import isqrt
def A026351(n): return (n+isqrt(5*n**2)>>1)+1 # Chai Wah Wu, Aug 17 2022

A060142 Ordered set S defined by these rules: 0 is in S and if x is in S then 2x+1 and 4x are in S.

Original entry on oeis.org

0, 1, 3, 4, 7, 9, 12, 15, 16, 19, 25, 28, 31, 33, 36, 39, 48, 51, 57, 60, 63, 64, 67, 73, 76, 79, 97, 100, 103, 112, 115, 121, 124, 127, 129, 132, 135, 144, 147, 153, 156, 159, 192, 195, 201, 204, 207, 225, 228, 231, 240, 243, 249, 252, 255, 256, 259, 265, 268, 271

Offset: 0

Clark Kimberling, Mar 05 2001

nonn

After expelling 0 and 1, the numbers 4x occupy same positions in S that 1 occupies in the infinite Fibonacci word (A003849).
a(A026351(n)) = A219608(n); a(A004957(n)) = 4 * a(n). - Reinhard Zumkeller, Nov 26 2012
Apart from the initial term, this lists the indices of the 1's in A086747. - N. J. A. Sloane, Dec 05 2019
From Gus Wiseman, Jun 10 2020: (Start)
Numbers k such that the k-th composition in standard order has all odd parts, or numbers k such that A124758(k) is odd. The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. For example, the sequence of all compositions into odd parts begins:
()              57: (1,1,3,1)          135: (5,1,1,1)
(1)             60: (1,1,1,3)          144: (3,5)
(1,1)           63: (1,1,1,1,1,1)      147: (3,3,1,1)
(3)             64: (7)                153: (3,1,3,1)
(1,1,1)         67: (5,1,1)            156: (3,1,1,3)
(3,1)           73: (3,3,1)            159: (3,1,1,1,1,1)
(1,3)           76: (3,1,3)            192: (1,7)
(1,1,1,1)       79: (3,1,1,1,1)        195: (1,5,1,1)
(5)             97: (1,5,1)            201: (1,3,3,1)
(3,1,1)        100: (1,3,3)            204: (1,3,1,3)
(1,3,1)        103: (1,3,1,1,1)        207: (1,3,1,1,1,1)
(1,1,3)        112: (1,1,5)            225: (1,1,5,1)
(1,1,1,1,1)    115: (1,1,3,1,1)        228: (1,1,3,3)
(5,1)          121: (1,1,1,3,1)        231: (1,1,3,1,1,1)
(3,3)          124: (1,1,1,1,3)        240: (1,1,1,5)
(3,1,1,1)      127: (1,1,1,1,1,1,1)    243: (1,1,1,3,1,1)
(1,5)          129: (7,1)              249: (1,1,1,1,3,1)
(1,3,1,1)      132: (5,3)              252: (1,1,1,1,1,3)
(End)
Numbers whose binary representation has the property that every run of consecutive 0's has even length. - Harry Richman, Jan 31 2024

			From _Harry Richman_, Jan 31 2024: (Start)
In the following, dots are used for zeros in the binary representation:
   n  binary(a(n))  a(n)
   0:    .......     0
   1:    ......1     1
   2:    .....11     3
   3:    ....1..     4
   4:    ....111     7
   5:    ...1..1     9
   6:    ...11..    12
   7:    ...1111    15
   8:    ..1....    16
   9:    ..1..11    19
  10:    ..11..1    25
  11:    ..111..    28
  12:    ..11111    31
  13:    .1....1    33
  14:    .1..1..    36
  15:    .1..111    39
  16:    .11....    48
  17:    .11..11    51
  18:    .111..1    57
  19:    .1111..    60
  20:    .111111    63
  21:    1......    64
  22:    1....11    67
(End)

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Clark Kimberling, A Self-Generating Set and the Golden Mean, J. Integer Sequences, 3 (2000), #00.2.8.
Clark Kimberling, Fusion, Fission, and Factors, Fib. Q., 52 (2014), 195-202.
Lukasz Merta, Composition inverses of the variations of the Baum-Sweet sequence, arXiv:1803.00292 [math.NT], 2018. See l(n) p. 5.
Gus Wiseman, Statistics, classes, and transformations of standard compositions

Cf. A219608 (odd terms), A060138, A060139, A060140, A060141, A086747.
Cf. A003714 (no consecutive 1's in binary expansion).
Odd partitions are counted by A000009.
Numbers with an odd number of 1's in binary expansion are A000069.
Numbers whose binary expansion has odd length are A053738.
All of the following pertain to compositions in standard order (A066099):
- Length is A000120.
- Compositions without odd parts are A062880.
- Sum is A070939.
- Product is A124758.
- Strict compositions are A233564.
- Heinz number is A333219.
- Number of distinct parts is A334028.

import Data.Set (singleton, deleteFindMin, insert)
a060142 n = a060142_list !! n
a060142_list = 0 : f (singleton 1) where
   f s = x : f (insert (4 * x) $ insert (2 * x + 1) s') where
       (x, s') = deleteFindMin s
-- Reinhard Zumkeller, Nov 26 2012

Take[Nest[Union[Flatten[# /. {{i_Integer -> i}, {i_Integer -> 2 i + 1}, {i_Integer -> 4 i}}]] &, {1}, 5], 32]  (* Or *)
Select[Range[124], FreeQ[Length /@ Select[Split[IntegerDigits[#, 2]], First[#] == 0 &], ?OddQ] &] (* _Birkas Gyorgy, May 29 2012 *)

is(n)=if(n<3, n<2, if(n%2,is(n\2),n%4==0 && is(n/4))) \\ Charles R Greathouse IV, Oct 21 2013

Corrected by T. D. Noe, Nov 01 2006

Definition simplified by Charles R Greathouse IV, Oct 21 2013

A375456 Expansion of g.f. A(x) satisfying x = A( A(x) - 2A(x)^2A'(x) ).

Original entry on oeis.org

1, 1, 5, 40, 414, 5096, 71465, 1113432, 18964415, 349252420, 6899717360, 145360352592, 3250782038728, 76887080836140, 1917401350590001, 50284361717695424, 1383636099826635216, 39865319955874291412, 1200467734347938040895, 37718141663144558046536, 1234556743772762830508484

Offset: 1

Paul D. Hanna, Sep 06 2024

nonn

It appears that a(A219608(n)) is odd for n >= 1, and that the only other odd term is a(2) = 1.

			G.f.: A(x) = x + x^2 + 5*x^3 + 40*x^4 + 414*x^5 + 5096*x^6 + 71465*x^7 + 1113432*x^8 + 18964415*x^9 + 349252420*x^10 + ...
where x = A( A(x) - 2*A(x)^2 * A'(x) ).
RELATED SERIES.
Let R(x) be the series reversion of A(x) so that R(A(x)) = x, then
R(x) = x - x^2 - 3*x^3 - 20*x^4 - 190*x^5 - 2240*x^6 - 30759*x^7 - 475116*x^8 - 8081145*x^9 - 149243380*x^10 + ...
where R(x) = A(x) - 2*A(x)^2 * A'(x).
A(x)^2 = x^2 + 2*x^3 + 11*x^4 + 90*x^5 + 933*x^6 + 11420*x^7 + 158862*x^8 + 2453874*x^9 + ...

Paul D. Hanna, Table of n, a(n) for n = 1..500

Cf. A219608, A259606.

{a(n) = my(A=[0,1],Ax=x); for(i=1,n, A=concat(A,0); Ax=Ser(A);
A[#A] = -polcoeff( subst(Ax,x, Ax - 2*Ax^2*Ax')/2, #A-1); ); H=Ax; A[n+1]}
for(n=1,30,print1(a(n),", "))

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) x = A( A(x) - 2*A(x)^2 * A'(x) ).
(2) A(A(x)) = x + 2*A(A(x))^2 * A'(A(x)).
(3) R(R(x)) = x - 2*x^2 * A'(R(x)), where A(R(x)) = x.
a(n) ~ c * d^n * n! * n^alpha, where d = 1.3534821142256737694364485294..., alpha = 2.7625501039589..., c = 0.0101323266748276... - Vaclav Kotesovec, Sep 07 2024

A307096 Positive integers m such that for any positive integer k the last k bits of the binary expansion of m is not a multiple of 3.

Original entry on oeis.org

1, 5, 13, 17, 29, 37, 49, 61, 65, 77, 101, 113, 125, 133, 145, 157, 193, 205, 229, 241, 253, 257, 269, 293, 305, 317, 389, 401, 413, 449, 461, 485, 497, 509, 517, 529, 541, 577, 589, 613, 625, 637, 769, 781, 805, 817, 829, 901, 913, 925, 961, 973, 997, 1009

Offset: 1

John Rickert, Mar 24 2019

The number of terms less than 2^n is the n-th Fibonacci number F(n), A000045.
The number of terms between 2^(n-1) and 2^n in the sequence is the Fibonacci number F(n-2), A000045.
If 2^(n-1) <= x < 2^n, then x is in the sequence if and only if x is not divisible by 3 and x - 2^(n-1) is in the sequence. - Robert Israel, Apr 25 2019

			29 is 11101_2 and none of 11101_2, 1101_2, 101_2, 1_2 are divisible by 3.

Cf. A060142, A219608, A000045.

f := n-> if(n != 0, add(2^(k-1)*`if`((n mod 2^k) mod 3 = 0, 1, 0), k = 1 .. ceil(log(n)/log(2))), 0);
ker := []; for n from 1 to 1024 do if f(n) = 0 then ker := [op(ker), n] end if end do; ker;
# Alternative:
A1:= {1}: A2:= {}:
for d from 1 to 12 do
  if d::odd then A1:= A1 union map(`+`,A2,2^d)
  else A2:= A2 union map(`+`,A1,2^d)
  fi
od:
sort(convert(A1 union A2,list)); # Robert Israel, Apr 25 2019

Select[Range[10^3], Function[s, NoneTrue[Array[FromDigits[Take[s, -#], 2] &, Length@ s], Mod[#, 3] == 0 &]]@ IntegerDigits[#, 2] &] (* Michael De Vlieger, Mar 24 2019 *)

isok(n) = {if (n % 3, my(b=binary(n)); for (k=1, #b-1, b[k] = 0; if ((fromdigits(b, 2) % 3) == 0, return (0));); return (1);); return (0);} \\ Michel Marcus, Apr 24 2019

(a(n)+1)/2 = A219608(n), the n-th odd term in A060142.

cp's OEIS Frontend

A219609 Half of first differences of A219608.

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A026351 a(n) = floor(n*phi) + 1, where phi = (1+sqrt(5))/2.

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A060142 Ordered set S defined by these rules: 0 is in S and if x is in S then 2x+1 and 4x are in S.

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A375456 Expansion of g.f. A(x) satisfying x = A( A(x) - 2*A(x)^2*A'(x) ).

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A307096 Positive integers m such that for any positive integer k the last k bits of the binary expansion of m is not a multiple of 3.

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Mathematica

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Formula

A375456 Expansion of g.f. A(x) satisfying x = A( A(x) - 2A(x)^2A'(x) ).