A036989 -id:A036989 - cp's OEIS frontend

A130161 A054525 * A036989 as a diagonalized matrix.

1, -1, 2, -1, 0, 1, 0, -2, 0, 3, -1, 0, 0, 0, 1, 1, -2, -1, 0, 0, 2, -1, 0, 0, 0, 0, 0, 2, 0, 0, 0, -3, 0, 0, 0, 4, 0, 0, -1, 0, 0, 0, 0, 0, 1, 1, -2, 0, 0, -1, 0, 0, 0, 0, 2

Offset: 0

Gary W. Adamson, May 13 2007

Left border = mu(n), A008683.
Right border = A036989, (1, 2, 1, 3, 1, 2, 2, 4, 1, 2, ...) = the inverse Moebius transform (A051731) of the Thue-Morse sequence, offset 1: (1, 1, 0, 1, 0, 0, 1, ...).
Row sums = the Thue-Morse sequence starting with "1".

			First few rows of the triangle:
   1;
  -1,  2;
  -1,  0,  1;
   0, -2,  0,  3;
  -1,  0,  0,  0,  1;
   1, -2, -1,  0,  0,  2;
  -1,  0,  0,  0,  0,  0,  2;
  ...

Cf. A036989, A051731, A054525, A010060, A008683.

Moebius transform of an infinite lower triangular matrix with A036989 in the main diagonal and the rest zeros.

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.

Original entry on oeis.org

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.

A036988 Has simplest possible tree complexity of all transcendental sequences.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
H. Niederreiter and M. Vielhaber, Tree complexity and a doubly exponential gap between structured and random sequences, J. Complexity, 12 (1996), 187-198.

Cf. A036989. Characteristic function of A036990.

a036988 = a063524 . a036989  -- Reinhard Zumkeller, Jul 31 2013

(* 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; a[n_] := Boole[b[n] == 1]; Table[a[n], {n, 0, 104}] (* Jean-François Alcover, Nov 05 2013, after Reinhard Zumkeller *)

a(n) = 1 iff, in the binary expansion of n, reading from right to left, the number of 1's never exceeds the number of 0's.

a(n) = A063524(A036989(n)). - Reinhard Zumkeller, Jul 31 2013

More terms from Larry Reeves (larryr(AT)acm.org), Sep 25 2000

A126387 Read binary expansion of n from the left; keep track of the excess of 1's over 0's that have been seen so far; sequence gives maximum(excess of 1's over 0's).

Original entry on oeis.org

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

Offset: 0

Franklin T. Adams-Watters, Dec 26 2006

			59 in binary is 111011, excess from left to right is 1,2,3,2,3,4, maximum is 4, so a(59) = 4.

Cf. A036989.

a(0) = 0, a(2^i) = 1, if n = 2^i + 2^j + m with j < i and 0 <= m < 2^j, then a(n) = max(a(2^j+m) + j + 2 - i, 1).

cp's OEIS Frontend

A130161 A054525 * A036989 as a diagonalized matrix.

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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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A036988 Has simplest possible tree complexity of all transcendental sequences.

Views

Author

Keywords

Links

Crossrefs

Programs

Haskell

Mathematica

Formula

Extensions

A126387 Read binary expansion of n from the left; keep track of the excess of 1's over 0's that have been seen so far; sequence gives maximum(excess of 1's over 0's).

Views

Author

Keywords

Examples

Crossrefs

Formula