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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. [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

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

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Last modified June 2 11:35 EDT 2020. Contains 334771 sequences. (Running on oeis4.)