[seqfan] Re: "Types/lengths of runs" in binary numbers
q1qq2qqq3qqqq at yahoo.com
Tue Sep 1 22:44:41 CEST 2009
I said: 'By "types" of runs, it is meant that..'
I probably should have said 'by "a combination of the lengths of runs", it is meant that..', or something a little more accurate as well as clear.
Sorry, English is my second language. Babbling incoherently is my first....
[ ( [ ([( [ ( ([[o0Oo0Ooo0Oo(0)oO0ooO0oO0o]]) ) ] )]) ] ) ]
--- On Tue, 9/1/09, Leroy Quet <q1qq2qqq3qqqq at yahoo.com> wrote:
> I have already posted several
> sequences involving the concept of "types of runs" in a
> binary number.
> I just posted this today.
> %I A164953
> %S A164953 1,1,1,1,2,1,1,3,2,1,1,3,4,3,1,1,4,5,6,3,1
> %N A164953 Square array read by antidiagonals: a(m,n) = the
> number of different combinations of types of runs in the
> binary representations of positive integers that contain
> exactly m 0's and n 1's in binary. (The leftmost digit must
> be 1 in each binary number.)
> %C A164953 The top row of the array is where m=0. The
> leftmost column of the array is where n=1.
> %C A164953 Clarification regarding the definition: Each
> positive integer can be thought of as a finite binary string
> with 1 as the leftmost digit. The "runs" alternate between
> those completely of 1's and those completely of 0's. Each
> run of digit b (0 or 1) is bounded by the digit 1-b or by
> the edge of the string. By "types" of runs, it is meant that
> the lengths of the runs of digit b's (b=0 or 1) form a
> permutation of the lengths of the runs of b's in all binary
> number with the same types of runs. (See example.)
> %e A164953 Consider those binary numbers with exactly four
> 1's and two 0's. There are 10 such binary numbers that each
> have a 1 as the leftmost digit. These binary numbers,
> grouped by those numbers with the same types of runs, are:
> (111100), (111010, 101110), (111001, 100111), (110110),
> (110101, 101101, 101011), (110011). There are 6 such
> groupings, so a(2,4) = 6.
> %K A164953 base,more,nonn,tabl
> %O A164953 0,5
> I know that the "clarification" is somewhat confusing.
> Perhaps there is some better terminology or wording that
> someone could suggest for these types of sequences.
> (Also, I probably should have had "lengths of runs" in the
> definition instead of "types of runs", I realize now.
> Second, also interesting is the sequence of the sums of the
> antidiagonals, the number of different combinations of
> lengths of runs in the binary representations of positive
> integers that each contain exactly n digits in binary.
> The first couple terms are: 1,2,4,7,12,20. Unfortunately,
> this matches up with a number of pre-existing sequences. So
> I haven't submitted this.
> And as always I ask, have I made any mistakes in
> calculating the sequence so far? (I figured these terms by
> Leroy Quet
> [ ( [ ([( [ ( ([[o0Oo0Ooo0Oo(0)oO0ooO0oO0o]]) ) ] )]) ] )
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