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-François Alcover, Dec 14 2017 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
N. J. A. Sloane, Mar 02 2017
EXTENSIONS
More terms from Lars Blomberg, Mar 03 2017
STATUS
approved