A052499 -id:A052499 - cp's OEIS frontend

A283165 a(0) = 0; a(1) = 1; a(2n) = 2a(n), a(2n+1) = 2a(n) + (-1)^a(n+1).

0, 1, 2, 3, 4, 3, 6, 7, 8, 7, 6, 7, 12, 11, 14, 15, 16, 15, 14, 15, 12, 11, 14, 15, 24, 23, 22, 23, 28, 27, 30, 31, 32, 31, 30, 31, 28, 27, 30, 31, 24, 23, 22, 23, 28, 27, 30, 31, 48, 47, 46, 47, 44, 43, 46, 47, 56, 55, 54, 55, 60, 59, 62, 63, 64, 63, 62, 63, 60, 59, 62, 63, 56, 55, 54, 55, 60, 59, 62, 63, 48, 47, 46, 47, 44, 43, 46, 47, 56, 55, 54

Offset: 0

Ilya Gutkovskiy, Mar 02 2017

nonn

			a(0) = 0;
a(1) = 1;
a(2) = a(2*1) = 2*a(1) = 2;
a(3) = a(2*1+1) = 2*a(1) + (-1)^a(2) = 2*1 + (-1)^2 = 3;
a(4) = a(2*2) = 2*a(2) = 2*2 = 4;
a(5) = a(2*2+1) = 2*a(2) + (-1)^a(3) = 2*2 + (-1)^3 = 3, etc.

Indranil Ghosh, Table of n, a(n) for n = 0..10000
Michael Gilleland, Some Self-Similar Integer Sequences
Ilya Gutkovskiy, Extended graphical example

Cf. A023758 (fixed points), A052499 (records), A080100, A087808.

a[0] = 0; a[1] = 1; a[n_] := If[EvenQ[n], 2 a[n/2], 2 a[(n - 1)/2] + (-1)^a[(n + 1)/2]]; Table[a[n], {n, 0, 90}]

a(n) = if (n<2, n, if (n%2==0, 2*a(n/2), 2*a((n-1)/2)+(-1)^(a(n+1)/2)));
tabl(nn)={for (n=0, nn, print1(a(n), ", "); ); };
tabl(90); \\ Indranil Ghosh, Mar 03 2017

def a(n):
    if n<2: return n
    if n%2==0: return 2*a(n//2)
    else: return 2*a((n-1)//2)+(-1)**a((n+1)//2) # Indranil Ghosh, Mar 03 2017

A380358 Numbers whose binary expansion ends with 11 and does not contain adjacent zeros.

Original entry on oeis.org

3, 7, 11, 15, 23, 27, 31, 43, 47, 55, 59, 63, 87, 91, 95, 107, 111, 119, 123, 127, 171, 175, 183, 187, 191, 215, 219, 223, 235, 239, 247, 251, 255, 343, 347, 351, 363, 367, 375, 379, 383, 427, 431, 439, 443, 447, 471, 475, 479, 491, 495, 503, 507, 511, 683

Offset: 1

R. J. Cintra, Jan 22 2025

The numbers in this sequence appear in the conversion of conventional binary numbers to the canonical signed-digit representation.

			183 is in the sequence because its binary expansion is 10110111.

J. L. Smith and A. Weinberger, "Shortcut Multiplication for Binary Digital Computers", in Methods for High-Speed Addition and Multiplication, National Bureau of Standards Circular 591, Sec. 1, February, 1958, page 21.

R. J. Cintra, A note on the conversion of nonnegative integers to the canonical signed-digit representation, arXiv:2501.10908 [eess.SP], 2025.

Cf. A003754, A052499, A247648, A365808, A365809.

Select[4*Range[0, 170] + 3, SequencePosition[IntegerDigits[#, 2], {0, 0}] == {} &] (* Amiram Eldar, Feb 05 2025 *)

from itertools import count, islice
def A380358_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:n&3==3 and not '00' in bin(n),count(max(startvalue,1)))
A380358_list = list(islice(A380358_gen(),20)) # Chai Wah Wu, Feb 12 2025

a(n) = 2 * A247648(n) + 1.
From Hugo Pfoertner, Feb 07 2025: (Start)
a(n) = 4*A052499(n) - 1.
a(n) = 4*(A365808(n+1) + 1)/3 - 1.
a(n) = 2*(A365809(n) + 1)/3 - 1. (End)

cp's OEIS Frontend

A283165 a(0) = 0; a(1) = 1; a(2n) = 2a(n), a(2n+1) = 2a(n) + (-1)^a(n+1).

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A380358 Numbers whose binary expansion ends with 11 and does not contain adjacent zeros.

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Formula

cp's OEIS Frontend

A283165 a(0) = 0; a(1) = 1; a(2*n) = 2*a(n), a(2*n+1) = 2*a(n) + (-1)^a(n+1).

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Author

Keywords

Examples

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Programs

Mathematica

PARI

Python

A380358 Numbers whose binary expansion ends with 11 and does not contain adjacent zeros.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Python

Formula

A283165 a(0) = 0; a(1) = 1; a(2n) = 2a(n), a(2n+1) = 2a(n) + (-1)^a(n+1).