A280049 Canonical representation of n as a sum of distinct Jacobsthal numbers J(n) (A001045) (see Comments for details); also binary numbers that end in an even number of zeros.

1, 11, 100, 101, 111, 1001, 1011, 1100, 1101, 1111, 10000, 10001, 10011, 10100, 10101, 10111, 11001, 11011, 11100, 11101, 11111, 100001, 100011, 100100, 100101, 100111, 101001, 101011, 101100, 101101, 101111, 110000, 110001, 110011, 110100, 110101, 110111

Offset

1,2

Comments

Every positive integer has a unique expression as a sum of distinct Jacobsthal numbers in which the index of the smallest summand is odd, with J(1) = 1 and J(2) = 1 both allowed. [Carlitz-Scoville-Hoggatt, 1972]. - Based on a comment in A001045 from Ira M. Gessel, Dec 31 2016.

The highest-order bits are on the left. Interpreting these as binary numbers we get A003159.

Links

Lars Blomberg, 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.

Formula

a(n) = A007088(A003159(n)). - Amiram Eldar, Jul 14 2023

Example

9 = 5+3+1 = J(4)+J(3)+J(1) = 1101.

Mathematica

FromDigits[IntegerDigits[#, 2]] & /@ Select[Range[100], EvenQ[IntegerExponent[#, 2]] &] (* Amiram Eldar, Jul 14 2023 *)

Prog

(PARI) lista(kmax) = for(k = 1, kmax, if(!(valuation(k, 2)%2), print1(fromdigits(binary(k), 10), ", "))); \\ Amiram Eldar, Jul 14 2023
(Python)
from itertools import count, islice
def A280049_gen(): # generator of terms
    return map(lambda n:int(bin(n)[2:]), filter(lambda n:(n&-n).bit_length()&1, count(1)))
A280049_list = list(islice(A280049_gen(), 20)) # Chai Wah Wu, Mar 19 2024
(Python)
def A280049(n):
    def f(x):
        c, s = n+x, bin(x)[2:]
        l = len(s)
        for i in range(l&1^1, l, 2):
            c -= int(s[i])+int('0'+s[:i], 2)
        return c
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return int(bin(m)[2:]) # Chai Wah Wu, Jan 30 2025

Crossrefs

Cf. A001045, A003159, A007088.

Sequence in context: A287984 A261757 A377925 * A066329 A309870 A219896

Adjacent sequences: A280046 A280047 A280048 * A280050 A280051 A280052

Keyword

nonn,base

Author

N. J. A. Sloane, Dec 31 2016

Extensions

Corrected a(5), a(16) and more terms from Lars Blomberg, Jan 02 2017

Status

approved