A206567 S(m,n) = (number of nonzero terms common to the base 3 expansions of m and n), a symmetric matrix read by antidiagonals.

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

Offset: 1

Author

Clark Kimberling, Feb 09 2012

Keywords

• nonn
• tabl
• base

Comments

Every nonnegative integer occurs infinitely many times in the matrix.

Examples

			Northwest corner:
1 0 0 1 0 0 1 0 0 1 0 0 1
0 1 0 0 1 0 0 1 0 0 1 0 0
0 0 1 1 1 0 0 0 0 0 0 1 1
1 0 1 2 1 0 1 0 0 1 0 1 2
0 1 1 1 2 0 0 1 0 0 1 1 1
0 0 0 0 0 1 1 1 0 0 0 0 0
1 0 0 1 0 1 2 1 0 1 0 0 1
0 1 0 0 1 1 1 2 0 0 1 0 0
0 0 0 0 0 0 0 0 1 1 1 1 1
1 0 0 1 0 0 1 0 1 2 1 1 2
0 1 0 0 1 0 0 1 1 1 2 1 1
0 0 1 1 1 0 0 0 1 1 1 2 2
1 0 1 2 1 0 1 0 1 2 1 2 3
4 = 3 + 1 and 13 = 3^2 + 3 + 1, so S(13,4)=2.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10011 (antidiagonals 1 to 141, flattened)

Crossrefs

Cf. A160384, A206479 (similar in base 2).

Programs

Maple

S:= proc(m,n) local M,N;
  M:= convert(m,base,3);
  N:= convert(n,base,3);
  convert(zip((s,t) -> `if`(s=t and s <> 0, 1, 0),M,N),`+`);
end proc:
seq(seq(S(k,n-k+1),k=1..n),n=1..30); # Robert Israel, Mar 19 2018

Mathematica

d[n_] := IntegerDigits[n, 3];
t[n_] := Reverse[Array[d, 100][[n]]]
s[n_, k_] := Position[t[n], k]
t[m_, n_] := Sum[Length[Intersection[s[m, k], s[n, k]]], {k, 1, 2}]
TableForm[Table[t[m, n], {m, 1, 24},
  {n, 1, 24}]]  (* A206567 as a matrix *)
Flatten[Table[t[i, n + 1 - i], {n, 1, 24},
  {i, 1, n}]]   (* A206567 as a sequence *)

PARI

d(n) = Vecrev(digits(n, 3));
T(n, k) = {my(dn = d(n), dk = d(k), nb = min(#dn, #dk)); sum(i=1, nb, dn[i] && (dn[i] == dk[i]));} \\ Michel Marcus, Mar 19 2018

Formula

Diagonal entries S(n,n) = A160384(n) since all nonzero digits match. - Robert Israel, Mar 18 2018

Extensions

Edited by Robert Israel, Mar 19 2018