login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 59th year, we have over 358,000 sequences, and we’ve crossed 10,300 citations (which often say “discovered thanks to the OEIS”).

Other ways to Give
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A109435 Triangle read by rows: T(n,m) = number of binary numbers n digits long, which have m 0's as a substring. 2
1, 2, 1, 4, 3, 1, 8, 7, 3, 1, 16, 15, 8, 3, 1, 32, 31, 19, 8, 3, 1, 64, 63, 43, 20, 8, 3, 1, 128, 127, 94, 47, 20, 8, 3, 1, 256, 255, 201, 107, 48, 20, 8, 3, 1, 512, 511, 423, 238, 111, 48, 20, 8, 3, 1, 1024, 1023, 880, 520, 251, 112, 48, 20, 8, 3, 1, 2048, 2047, 1815, 1121, 558 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Column 0 is A000079, column 2 is A000225, column 3 is A008466, column 4 is A050231

Column 5 is A050232, column 6 is A050233, the last column is A001792.

A050227 with a leading column of powers of 2. - R. J. Mathar, Mar 25 2014

LINKS

Table of n, a(n) for n=0..70.

FORMULA

G.f. for column m: x^m/( (1 - Sum_{k=1..m} x^k)*(1-2*x) ). - Geoffrey Critzer, Jan 07 2014

EXAMPLE

Triangle begins:

n\m_0__1__2__3__4__5

0| 1 0 0 0 0 0

1| 2 1 0 0 0 0

2| 4 3 1 0 0 0

3| 8 7 3 1 0 0

4| 16 15 8 3 1 0

5| 32 31 19 8 3 1

T(5,3)=8 because there are 8 length 5 binary words that contain 000 as a contiguous substring: 00000, 00001, 00010, 00011, 01000, 10000, 10001, 11000. - Geoffrey Critzer, Jan 07 2014

MATHEMATICA

T[n_, m_] := Length[ Select[ StringPosition[ #, StringDrop[ ToString[10^m], 1]] & /@ Table[ ToString[ FromDigits[ IntegerDigits[i, 2]]], {i, 2^n, 2^(n + 1) - 1}], # != {} &]]; Flatten[ Table[ T[n, m], {n, 0, 11}, {m, 0, n}]]

nn=15; Map[Select[#, #>0&]&, Transpose[Table[CoefficientList[Series[x^m/(1-Sum[x^k, {k, 1, m}])/(1-2x), {x, 0, nn}], x], {m, 0, nn}]]]//Grid (* Geoffrey Critzer, Jan 07 2014 *)

CROSSREFS

Cf. A109433, A001792, A109436.

Sequence in context: A134626 A347633 A115450 * A134392 A048483 A276562

Adjacent sequences: A109432 A109433 A109434 * A109436 A109437 A109438

KEYWORD

base,nonn,tabl

AUTHOR

Robert G. Wilson v, Jun 28 2005

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 9 23:05 EST 2022. Contains 358710 sequences. (Running on oeis4.)