A206479 -id:A206479 - cp's OEIS frontend

A206566 Triangular array: T(i,j) = number of terms common to the binary expansions of i+1 and j, for j=1,2,3,...,i; i=1,2,3,...

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

Offset: 1

Clark Kimberling, Feb 09 2012

Row n consists of the first n terms of column n+1 of A206479, and (n-th row sum)=A115478(n+1).

			First ten rows:
0
1 1
0 0 0
1 0 1 1
0 1 1 1 1
1 1 2 1 2 2
0 0 0 0 0 0 0
1 0 1 0 1 0 1 1
0 1 1 0 0 1 1 1 1
1 1 2 0 1 1 2 1 2 2

Cf. A206479, A115478.

d[n_] := IntegerDigits[n, 2];
t[n_] := Reverse[Array[d, 120][[n]]]
s[n_] := Position[t[n], 1]
t[m_, n_] := Length[Intersection[s[m], s[n]]]
TableForm[Table[t[m, n], {m, 1, 14},
  {n, 1, 14}]] (* A206479 as a matrix *)
Flatten[Table[t[i, n + 1 - i], {n, 1, 14},
  {i, 1, n}]]  (* A206479 as a sequence *)
u = Table[t[n - 1, m], {n, 3, 16}, {m, 1, n - 2}];
TableForm[u]   (* A206566 as a triangle *)
Flatten[u]     (* A206566 as a sequence *)
v[n_] := Table[t[k, n + 1], {k, 1, n}]
Table[Count[v[n], 0], {n, 1, 100}] (* A115478 *)

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

Original entry on oeis.org

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

Clark Kimberling, Feb 09 2012

Every nonnegative integer occurs infinitely many times in the matrix.

			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.

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

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

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

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 *)

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

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

Edited by Robert Israel, Mar 19 2018

cp's OEIS Frontend

A206566 Triangular array: T(i,j) = number of terms common to the binary expansions of i+1 and j, for j=1,2,3,...,i; i=1,2,3,...

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

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