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

Author

N. J. A. Sloane, Dec 31 2016

Keywords

• nonn
• base

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.

Examples

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

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.

Crossrefs

Cf. A001045, A003159, A007088.

Programs

Mathematica

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

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

Formula

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

Extensions

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