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A108730 Triangle read by rows: row n gives list of number of zeros following each 1 in binary representation of n. 4
0, 1, 0, 0, 2, 1, 0, 0, 1, 0, 0, 0, 3, 2, 0, 1, 1, 1, 0, 0, 0, 2, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 4, 3, 0, 2, 1, 2, 0, 0, 1, 2, 1, 1, 0, 1, 0, 1, 1, 0, 0, 0, 0, 3, 0, 2, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 2, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 5, 4, 0, 3, 1, 3, 0, 0, 2, 2, 2, 1, 0, 2, 0, 1, 2, 0, 0, 0, 1, 3, 1, 2, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

This is probably the simplest method for putting the nonnegative integers into one-to-one correspondence with the finite sequences of nonnegative integers and is the standard ordering for such sequences in this database.

This sequence contains every finite sequence of nonnegative integers.

This can be regarded as a table in two ways: with each composition as a row, or with the compositions of each integer as a row. The first way has A000120 as row lengths and A080791 as row sums; the second has A001792 as row lengths and A001787 as row sums. - Franklin T. Adams-Watters, Nov 06 2006

Concatenate the base-two positive integers (A030190 less the initial zero). Left to right and disallowing leading zeros, reorganize the digits into the smallest possible numbers. These will be the base-two powers-of-two of A108730. [From Hans Havermann, Nov 14 2009]

T(2^(n-1),0) = n-1 and T(m,k) < n-1 for all m < 2^n, k <= A000120(m). When the triangle is seen as a flattened list, each n occurs first at position n*2^(n-1)+1, cf. A005183. [Reinhard Zumkeller, Feb 26 2012]

LINKS

Franklin T. Adams-Watters, Table of n, a(n) for n = 1..5120 (through 10 bit numbers)

EXAMPLE

Triangle begins:

0

1

0,0

2

1,0

0,1

0,0,0

3

For example, 25 base 2 = 11001; following the 1's are 0, 2 and 0 zeros, so row 25 is 0, 2, 0.

MATHEMATICA

row[n_] := (id = IntegerDigits[n, 2]; sp = Split[id]; f[run_List] := If[First[run] == 0, run, Sequence @@ Table[{}, {Length[run] - 1}]]; len = Length /@ f /@ sp; If[Last[id] == 0, len, Append[len, 0]]); Flatten[ Table[row[n], {n, 1, 41}]] (* Jean-Fran├žois Alcover, Jul 13 2012 *)

PROG

(Haskell)

import Data.List (unfoldr, group)

a108730 n k = a108730_tabf !! (n-1) !! (k-1)

a108730_row = f . group . reverse . unfoldr

   (\x -> if x == 0 then Nothing else Just $ swap $ divMod x 2) where

   f [] = []

   f [os] = replicate (length os) 0

   f (os:zs:dss) = replicate (length os - 1) 0 ++ [length zs] ++ f dss

a108730_tabf = map a108730_row [1..]

a108730_list = concat a108730_tabf

-- Reinhard Zumkeller, Feb 26 2012

CROSSREFS

Cf: A066099, A007088, A000120, A080791, A001792, A001787, A124735.

Sequence in context: A066301 A046660 A183094 * A056973 A107782 A086017

Adjacent sequences:  A108727 A108728 A108729 * A108731 A108732 A108733

KEYWORD

easy,nice,nonn,tabf

AUTHOR

Franklin T. Adams-Watters, Jun 22 2005

EXTENSIONS

Edited by Franklin T. Adams-Watters, Nov 06 2006

STATUS

approved

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Last modified October 24 23:00 EDT 2014. Contains 248516 sequences.