A099027 -id:A099027 - cp's OEIS frontend

A099026 Array x AND NOT y, read by rising antidiagonals.

0, 1, 0, 2, 0, 0, 3, 2, 1, 0, 4, 2, 0, 0, 0, 5, 4, 1, 0, 1, 0, 6, 4, 4, 0, 2, 0, 0, 7, 6, 5, 4, 3, 2, 1, 0, 8, 6, 4, 4, 0, 2, 0, 0, 0, 9, 8, 5, 4, 1, 0, 1, 0, 1, 0, 10, 8, 8, 4, 2, 0, 0, 0, 2, 0, 0, 11, 10, 9, 8, 3, 2, 1, 0, 3, 2, 1, 0, 12, 10, 8, 8, 8, 2, 0, 0, 4, 2, 0, 0, 0, 13, 12, 9, 8, 9, 8, 1

Offset: 0

Ralf Stephan, Sep 26 2004

For n>0, the n-th row and the differences of the n-th column have period 2^floor(log_(n)+1).

			0,0,0,0,0,0,
1,0,1,0,1,0,
2,2,0,0,2,2,
3,2,1,0,3,2,
4,4,4,4,0,0,
5,4,5,4,1,0,

Rémy Sigrist, Table of n, a(n) for n = 0..10010
Rémy Sigrist, Colored representation of the table for n, k < 2^10 (the color at (x, y) is function of T(x, y))

Rows include A000004, A059841. Columns include A001477, A052928. Antidiagonal sums are in A099027.

Cf. A003985 (AND), A003986 (OR), A003987 (XOR).

Table[BitAnd[x - y, BitNot[y]], {x, 0, 15}, {y, 0, x}] (* Paolo Xausa, Sep 30 2024 *)

PARI
```
T(x,y)=bitnegimply(x,y)
```

T(x, y) = x AND NOT y. The AND NOT operation satisfies the bitwise truth table: (0, 0) = 0, (0, 1) = 0, (1, 0) = 1, (1, 1) = 0.

A375551 a(n) = Sum_{k=0..n} k XOR n-k, where XOR is the bitwise exclusive disjunction. Row sums of A003987.

Original entry on oeis.org

0, 2, 4, 12, 12, 22, 32, 56, 48, 58, 68, 100, 108, 142, 176, 240, 208, 210, 212, 252, 252, 294, 336, 424, 416, 458, 500, 596, 636, 734, 832, 992, 896, 866, 836, 876, 844, 886, 928, 1048, 1008, 1050, 1092, 1220, 1260, 1390, 1520, 1744, 1680, 1714, 1748, 1884, 1916

Offset: 0

Peter Luschny, Sep 27 2024

nonn

Paolo Xausa, Table of n, a(n) for n = 0..10000

Cf. A003986, A003987, A006582, A020522, A051933, A099027, A224915, A376585.

XOR := (n, k) -> Bits:-Xor(n, k):
a := n -> local k; add(XOR(k, n-k), k=0..n):
seq(a(n), n = 0..52);

(* Using definition *)
Table[Sum[BitXor[n - k, k], {k, 0, n}], {n, 0, 100}]
(* Using recurrence -- faster *)
a[0] = 0; a[n_] := a[n] = If[OddQ[n], 4*a[(n-1)/2] + n + 1, 2*(a[n/2] + a[n/2-1])];
Table[a[n], {n, 0, 100}] (* Paolo Xausa, Oct 01 2024 *)

a(n) = sum(k=0, n, bitxor(k, n-k)); \\ Michel Marcus, Sep 28 2024

a(n) = 2*A099027(n).
a(n) = 2*n + A006582(n).
a(2^n - 1) = 4^n - 2^n = A020522(n).
a(2^n) = 4^n - 2^n*(n - 1) = 2*A376585(n).
Recurrence: a(0) = 0; a(2*n) = 2*(a(n) + a(n-1)); a(2*n+1) = 2*(2*a(n) + n + 1). - Paolo Xausa, Oct 01 2024, derived from recurrence in A099027.

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A099026 Array x AND NOT y, read by rising antidiagonals.

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