

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..


3



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
(list;
graph;
refs;
listen;
history;
text;
internal format)



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. [CarlitzScovilleHoggatt, 1972].  Based on a comment in A001045 from Ira M. Gessel, Dec 31 2016.
The highestorder 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), 499518, 550


EXAMPLE

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


MATHEMATICA

Table[Min@ Map[FromDigits@ Function[w, Function[t, Reverse@ ReplacePart[t, Map[# > 1 &, w]]]@ ConstantArray[0, Max@w]]@ Join[Flatten@ Map[Position[s, #] &, Select[#, # > 1 &]], Range@ Count[#, 1]] &, Select[IntegerPartitions[n], And[SubsetQ[s, #], Reverse@ Union@ # == # &@ DeleteCases[#, 1], Count[#, 1] <= 2] &]], {n, 37}] (* Michael De Vlieger, Jan 02 2017 *)


CROSSREFS

Cf. A001045, A003159.
Sequence in context: A288402 A287984 A261757 * 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



