A213629 In binary representation: T(n,k) = number of (possibly overlapping) occurrences of k in n, triangle read by rows, 1<=k<=n.

1, 1, 1, 2, 0, 1, 1, 1, 0, 1, 2, 1, 0, 0, 1, 2, 1, 1, 0, 0, 1, 3, 0, 2, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 2, 1, 0, 1, 0, 0, 0, 0, 1, 2, 2, 0, 0, 1, 0, 0, 0, 0, 1, 3, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 2, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 3, 1, 1, 0, 1, 1, 0, 0

Offset

1,4

Comments

The definition is based on the definition of pattern functions in the paper of Allouche and Shallit;

sum of n-th row = A029931(n);

T(n,1) = A000120(n);

T(n,2) = A033264(n) for n > 1;

T(n,3) = A014081(n) for n > 2;

T(n,4) = A056978(n) for n > 3;

T(n,5) = A056979(n) for n > 4;

T(n,6) = A056980(n) for n > 5;

T(n,7) = A014082(n) for n > 6;

T(n,k) = 0 for k with floor(n/2) < k < n;

T(n,n) = 1;

A122953(n) = sum (A057427(T(n,k): k=1..n);

A005811(n) = T(n,1) + T(n,2) - T(n,3);

A007302(n) = A000120(n) - sum (A213629(n,A136412(k))).

Links

Reinhard Zumkeller, Rows n = 1..150 of triangle, flattened

J.-P. Allouche, J. Shallit, The Ring of k-regular Sequences II, Example 4, p. 12

Index entries for sequences related to binary expansion of n

Example

The triangle begins:
.   1:                        1
.   2:                      1   1
.   3:                    2   0   1
.   4:                  1   1   0   1
.   5:                2   1   0   0   1
.   6:              2   1   1   0   0   1
.   7:            3   0   2   0   0   0   1
.   8:          1   1   0   1   0   0   0   1
.   9:        2   1   0   1   0   0   0   0   1
.  10:      2   2   0   0   1   0   0   0   0   1
.  11:    3   1   1   0   1   0   0   0   0   0   1
.  12:  2   1   1   1   0   1   0   0   0   0   0   1.

Mathematica

t[n_, k_] := (idn = IntegerDigits[n, 2]; idk = IntegerDigits[k, 2]; ln = Length[idn]; lk = Length[idk]; For[cnt = 0; i = 1, i <= ln - lk + 1, i++, If[idn[[i ;; i + lk - 1]] == idk, cnt++]]; cnt); Table[t[n, k], {n, 1, 14}, {k, 1, n}] // Flatten (* Jean-François Alcover, Oct 22 2012 *)

Prog

(Haskell)
import Data.List (inits, tails, isPrefixOf)
a213629 n k = a213629_tabl !! (n-1) !! (k-1)
a213629_row n = a213629_tabl !! (n-1)
a213629_tabl = map f $ tail $ inits $ tail $ map reverse a030308_tabf where
   f xss = map (\xs ->
           sum $ map (fromEnum . (xs `isPrefixOf`)) $ tails $ last xss) xss

Crossrefs

Cf. A030308, A007088.

Sequence in context: A262696 A344986 A117165 * A357380 A278522 A024363

Adjacent sequences: A213626 A213627 A213628 * A213630 A213631 A213632

Keyword

nonn,base,tabl

Author

Reinhard Zumkeller, Jun 17 2012

Status

approved