A109436 -id:A109436 - cp's OEIS frontend

A109435 Triangle read by rows: T(n,m) = number of binary numbers n digits long, which have m 0's as a substring.

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

Offset: 0

Robert G. Wilson v, Jun 28 2005

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

			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

Cf. A109433, A001792, A109436, A102712 (row sums ?).

A109435 := proc(n,k)
    option remember ;
    if n< k then
        0;
    elif n = k then
        1;
    else
        2*procname(n-1,k)+2^(n-1-k)-procname(n-1-k,k) ;
    end if;
end proc:
seq(seq( A109435(n,k),k=0..n),n=0..12) ; # R. J. Mathar, Jun 05 2025

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 *)

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

A109434 Irregular triangle read by rows: row n contains the numbers from 2^n up to (n+3)*2^(n-2), inclusive, read along their common subdiagonal of A109433.

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 4, 5, 8, 11, 12, 16, 24, 27, 28, 32, 51, 60, 63, 64, 64, 107, 131, 140, 143, 144, 128, 222, 282, 307, 316, 319, 320, 256, 457, 601, 666, 691, 700, 703, 704, 512, 935, 1270, 1432, 1498, 1523, 1532, 1535, 1536, 1024, 1904, 2665, 3057, 3224, 3290

Offset: 0

Robert G. Wilson v, Jun 28 2005

Evolution of 2^n into 2^(n-2)(n+3) as exhibited by A109433.

The forward differences of the last row of the table approache A058396.

			Triangle begins
   0  0
   1  1
   2  2
   4  5
   8 11 12
  16 24 27 28
  32 51 60 63 64

Cf. A109433, A109436.

T[n_, m_] := Length[ Select[ StringPosition[ #, ToString[(10^m - 1)/9]] & /@ Table[ ToString[ FromDigits[ IntegerDigits[i, 2]]], {i, 2^n, 2^(n + 1) - 1}], # != {} &]]; Join[{0, 0, 1, 1, 2}, Flatten[ Table[ T[n + i, i], {n, 0, 9}, {i, n + 1}]]]

New definition from R. J. Mathar, Nov 27 2007

cp's OEIS Frontend

A109435 Triangle read by rows: T(n,m) = number of binary numbers n digits long, which have m 0's as a substring.

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