A095775 -id:A095775 - cp's OEIS frontend

A036990 Numbers n such that, in the binary expansion of n, reading from right to left, the number of 1's never exceeds the number of 0's.

0, 2, 4, 8, 10, 12, 16, 18, 20, 24, 32, 34, 36, 40, 42, 44, 48, 50, 52, 56, 64, 66, 68, 72, 74, 76, 80, 82, 84, 88, 96, 98, 100, 104, 112, 128, 130, 132, 136, 138, 140, 144, 146, 148, 152, 160, 162, 164, 168, 170, 172, 176, 178, 180, 184, 192, 194, 196, 200, 202, 204

Offset: 1

N. J. A. Sloane

A036989(a(n)) = 1. - Reinhard Zumkeller, Jul 31 2013

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
H. Niederreiter and M. Vielhaber, Tree complexity and a doubly exponential gap between structured and random sequences, J. Complexity, 12 (1996), 187-198.

Cf. A036988, A036991, A036992, A061854, A125086.
Each term is 2^n * some term of A014486 (n >= 0).
Cf. A030308.

a036990 n = a036990_list !! (n-1)
a036990_list = filter ((== 1) . a036989) [0..]
-- Reinhard Zumkeller, Jul 31 2013

fQ[n_] := Block[{od = ev = k = 0, id = Reverse@IntegerDigits[n, 2], lmt = Floor@Log[2, n] + 1}, While[k < lmt && od < ev + 1, If[OddQ@id[[k + 1]], od++, ev++ ]; k++ ]; If[k == lmt && od < ev + 1, True, False]]; Select[ Range[0, 204, 2], fQ@# &] (* Robert G. Wilson v, Jan 11 2007 *)
(* b = A036989 *) b[0] = 1; b[n_?EvenQ] := b[n] = Max[b[n/2]-1, 1]; b[n_] := b[n] = b[(n-1)/2]+1; Select[Range[0, 300, 2], b[#] == 1 &] (* Jean-François Alcover, Nov 05 2013, after Reinhard Zumkeller *)

a(n) = 2*A095775(n). - Robert G. Wilson v

More terms from Erich Friedman.

A003160 a(1) = a(2) = 1, a(n) = n - a(a(n-1)) - a(a(n-2)).

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 7, 8, 9, 9, 9, 10, 11, 12, 12, 12, 13, 14, 15, 15, 15, 16, 16, 16, 17, 17, 17, 18, 19, 20, 20, 20, 21, 21, 21, 22, 22, 22, 23, 24, 25, 25, 25, 26, 26, 26, 27, 27, 27, 28, 29, 30, 30, 30, 31, 32, 33, 33, 33, 34, 35, 36, 36, 36, 37, 37, 37, 38

Offset: 1

N. J. A. Sloane

nonn

Sequence of indices n where a(n-1) < a(n) appears to be given by A003156. - Joerg Arndt, May 11 2010

The number n appears A080426(n+1) times. - John Keith, Dec 31 2020

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
L. Carlitz, R. Scoville, and V. E. Hoggatt, Jr., Representations for a special sequence, Fibonacci Quarterly 10.5 (1972), 499-518, 550.

Cf. A003156, A005206, A080426, A095774, A095775.

a003160 n = a003160_list !! (n-1)
a003160_list = 1 : 1 : zipWith (-) [3..] (zipWith (+) xs $ tail xs)
   where xs = map a003160 a003160_list
-- Reinhard Zumkeller, Aug 02 2013

Block[{a = {1, 1}}, Do[AppendTo[a, i - a[[ a[[-1]] ]] - a[[ a[[-2]] ]] ], {i, 3, 76}]; a] (* Michael De Vlieger, Dec 31 2020 *)

PARI
```
a(n)=if(n<3,1,n-a(a(n-1))-a(a(n-2)))
```

@CachedFunction
def a(n): return 1 if (n<3) else n - a(a(n-1)) - a(a(n-2))
[a(n) for n in range(1, 81)] # G. C. Greubel, Nov 06 2022

a(n) is asymptotic to n/2.

Conjecture: a(n) = E/2 where we start with A := n + 1, B := 0, L := A085423(A), C := A000975(L-1), D := 0, E := C and until A = B consecutively apply B := A, A := 2*C - A - (L mod 2) + 2, L := A085423(A), C := A000975(L-1), D := D + 1, E := (1 + [A = B])*E + (-1)^D*C. - Mikhail Kurkov, May 12 2025

Edited by Benoit Cloitre, Jan 01 2003

A095773 a(1)=1, a(n) = 1 + a(n - a(a(a(n-1)))).

Original entry on oeis.org

1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 6, 7, 7, 7, 8, 8, 8, 9, 9, 9, 10, 10, 10, 10, 11, 11, 11, 11, 12, 12, 12, 12, 13, 13, 13, 13, 14, 14, 14, 14, 15, 15, 15, 15, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 18, 18, 18, 18, 18, 19, 19, 19, 19, 19, 20, 20, 20, 20, 20, 21, 21, 21, 21, 21, 22

Offset: 1

Benoit Cloitre, Jun 05 2004

nonn

A generalization of Golomb's sequence.

a(10^n): 1, 6, 26, 124, 611, 2963, 14172, ...

J. Grytczuk, Another variation on Conway's recursive sequence, Discr. Math. 282 (2004), 149-161.

Cf. A001462, A095774, A095775.

a[1] = 1; a[n_] := a[n] = 1 + a[n - a[a[a[n - 1]]]]; Table[ a[n], {n, 80}] (* Robert G. Wilson v, Jun 09 2004 *)

v=vector(1000,j,1);for(n=2,1000,g=v[n-v[v[v[n-1]]]]+1;v[n]=g);a(n)=v[n]

Is a(n) asymptotic to r^(r-1)*n^r where r is the positive root of x^3+x=1 and so r=0.682327803828019327...?

cp's OEIS Frontend

A036990 Numbers n such that, in the binary expansion of n, reading from right to left, the number of 1's never exceeds the number of 0's.

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A003160 a(1) = a(2) = 1, a(n) = n - a(a(n-1)) - a(a(n-2)).

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A095773 a(1)=1, a(n) = 1 + a(n - a(a(a(n-1)))).

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Mathematica

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