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A282714 Base-2 generalized Pascal triangle P_2 read by rows (see Comments for precise definition). 4
1, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 2, 0, 1, 1, 2, 1, 1, 0, 1, 1, 2, 2, 1, 0, 0, 1, 1, 3, 0, 3, 0, 0, 0, 1, 1, 1, 3, 0, 3, 0, 0, 0, 1, 1, 2, 2, 1, 1, 2, 0, 0, 0, 1, 1, 2, 3, 1, 1, 1, 1, 0, 0, 0, 1, 1, 3, 1, 3, 0, 2, 0, 1, 0, 0, 0, 1, 1, 2, 4, 1, 2, 0, 2, 0, 0 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,8

COMMENTS

List the binary numbers in their natural order as binary strings, beginning with the empty string epsilon, which represents 0. Row n of the triangle gives the number of times the k-th string occurs as a (scattered) substring of the n-th string.

Row n has sum n+1.

LINKS

Lars Blomberg, Table of n, a(n) for n = 0..10000

Julien Leroy, Michel Rigo, Manon Stipulanti, Counting the number of non-zero coefficients in rows of generalized Pascal triangles, Discrete Mathematics 340 (2017), 862-881.

Julien Leroy, Michel Rigo, Manon Stipulanti, Counting Subwords Occurrences in Base-b Expansions, arXiv:1705.10065 [math.CO], 2017.

Julien Leroy, Michel Rigo, Manon Stipulanti, Counting Subwords Occurrences in Base-b Expansions, Integers, Electronic Journal of Combinatorial Number Theory 18A (2018), #A13.

Manon Stipulanti, Convergence of Pascal-Like Triangles in Parry-Bertrand Numeration Systems, arXiv:1801.03287 [math.CO], 2018.

EXAMPLE

Triangle begins:

  1,

  1,1,

  1,1,1,

  1,2,0,1,

  1,1,2,0,1,

  1,2,1,1,0,1,

  1,2,2,1,0,0,1,

  1,3,0,3,0,0,0,1,

  1,1,3,0,3,0,0,0,1

  1,2,2,1,1,2,0,0,0,1

  1,2,3,1,1,1,1,0,0,0,1

  1,3,1,3,0,2,0,1,0,0,0,1

  1,2,4,1,2,0,2,0,0,0,0,0,1

  ...

The binary numbers are epsilon, 1, 10, 11, 100, 101, 110, 111, 1000, ...

The fifth number 101 contains

eps 1 10 11 100 101 respectively

.1..2..1..1...0...1 times, which is row 5 of the triangle.

MAPLE

Nscatsub := proc(subw, w)

    local lsubw, lw, N, wri, wr, i ;

    lsubw := nops(subw) ;

    lw := nops(w) ;

    N := 0 ;

    if lsubw = 0 then

        return 1 ;

    elif lsubw > lw then

        return 0 ;

    else

        for wri in combinat[choose](lw, lsubw) do

            wr := [] ;

            for i in wri do

                wr := [op(wr), op(i, w)] ;

            end do:

            if verify(subw, wr, 'sublist') then

                N := N+1 ;

            end if;

        end do:

    end if;

    return N ;

end proc:

P := proc(n, k, b)

    local n3, k3 ;

    n3 := convert(n, base, b) ;

    k3 := convert(k, base, b) ;

    Nscatsub(k3, n3) ;

end proc:

A282714 := proc(n, k)

    P(n, k, 2) ;

end proc: # R. J. Mathar, Mar 03 2017

MATHEMATICA

nmax = 12;

row[n_] := Module[{bb, ss}, bb = Table[IntegerDigits[k, 2], {k, 0, n}]; ss = Subsets[Last[bb]]; Prepend[Count[ss, #]& /@ bb // Rest, 1]];

Table[row[n], {n, 0, nmax}] // Flatten (* Jean-Fran├žois Alcover, Dec 14 2017 *)

CROSSREFS

A007306 gives (essentially) the number of nonzero entries in the rows.

Sequence in context: A226194 A053692 A286934 * A280634 A281491 A099494

Adjacent sequences:  A282711 A282712 A282713 * A282715 A282716 A282717

KEYWORD

nonn,tabl

AUTHOR

N. J. A. Sloane, Mar 02 2017

EXTENSIONS

More terms from Lars Blomberg, Mar 03 2017

STATUS

approved

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Last modified November 19 20:42 EST 2019. Contains 329323 sequences. (Running on oeis4.)