A269158 Square array A(row,col) = F(row,(2*col)-1), where F(0,q) = F(1,q) = 0, F(2p,q) = F(p,q) XOR A003188(q), F(2p+1,q) = F(q mod 2p+1, 2p+1) XOR (2p+1 AND q). Array is read by descending antidiagonals as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ...

0, 0, 1, 0, 2, 1, 0, 7, 3, 0, 0, 4, 3, 0, 1, 0, 13, 3, 0, 2, 0, 0, 14, 1, 0, 5, 1, 1, 0, 11, 1, 0, 2, 4, 0, 1, 0, 8, 1, 0, 1, 7, 7, 2, 1, 0, 25, 3, 0, 1, 12, 7, 7, 0, 0, 0, 26, 3, 0, 6, 15, 5, 4, 0, 0, 1, 0, 31, 3, 0, 5, 10, 3, 13, 4, 2, 2, 1, 0, 28, 1, 0, 6, 11, 2, 14, 9, 6, 0, 3, 1, 0, 21, 1, 0, 1, 26, 7, 11, 4, 12, 0, 3, 0, 0

Offset

1,5

Comments

The array gives the values of bivariate function F(p,q) which is well-defined only when q is odd, thus while here its argument p obtains all integer values from 1 onward, argument q gets only odd numbers 1, 3, 5, 7, 9, ... as its values.

Any row n occurs also as row (4^k * n), for all k >= 0.

Links

Antti Karttunen, Table of n, a(n) for n = 1..32896; the first 256 antidiagonals of the array

Formula

A(row,col) = F(row,(2*col)-1), where function F is defined as: If p <= 1, F(p,q) = 0, otherwise if p is an odd number > 1, F(p,q) = F(q mod p, p) XOR (p AND q), otherwise [when p is an even number] F(p,q) = F(p/2,q) XOR A003188(q).

Example

The top left [1 .. 16] x [1 .. 25] section of the array:
0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0
1,  2,  7,  4, 13, 14, 11,  8, 25, 26, 31, 28, 21, 22, 19, 16
1,  3,  3,  3,  1,  1,  1,  3,  3,  3,  1,  1,  1,  3,  3,  3
0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0
1,  2,  5,  2,  1,  1,  6,  5,  6,  1,  5,  6,  1,  6,  5,  5
0,  1,  4,  7, 12, 15, 10, 11, 26, 25, 30, 29, 20, 21, 16, 19
1,  0,  7,  7,  5,  3,  2,  7,  2,  1,  5,  3,  1,  4,  5,  4
1,  2,  7,  4, 13, 14, 11,  8, 25, 26, 31, 28, 21, 22, 19, 16
1,  0,  0,  4,  9,  4,  9,  5, 12,  1,  0,  0, 12,  9,  4,  9
0,  0,  2,  6, 12, 15, 13, 13, 31, 27, 26, 26, 20, 16, 22, 21
1,  2,  0,  0, 13, 11,  7, 11, 14, 13, 14,  3,  8, 10, 10, 15
1,  3,  3,  3,  1,  1,  1,  3,  3,  3,  1,  1,  1,  3,  3,  3
1,  0,  3,  7,  0, 14, 13,  6,  1, 11, 14,  8,  8,  9, 12, 11
0,  2,  0,  3,  8, 13,  9, 15, 27, 27, 26, 31, 20, 18, 22, 20
1,  0,  0,  0, 12,  0, 11, 15,  9,  3, 14, 15,  4,  8,  2, 15
0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0
1,  2,  7,  3, 13, 15,  0,  8, 17,  8, 17, 11,  8, 14, 18, 10
0,  2,  7,  0,  4, 10,  2, 13, 21, 27, 31, 28, 25, 31, 23, 25
1,  0,  0,  2,  0, 14, 10,  0, 25, 19, 11, 19,  8,  9, 10, 16
1,  2,  5,  2,  1,  1,  6,  5,  6,  1,  5,  6,  1,  6,  5,  5
1,  0,  0,  0,  1, 15, 11, 11,  0, 26, 21, 10, 17, 15, 10, 15
0,  0,  7,  4,  0,  5, 12,  3, 23, 23, 17, 31, 29, 28, 25, 31
1,  2,  3,  4,  1,  0, 13,  8, 26,  0, 31, 23, 13, 19,  8, 11
0,  1,  4,  7, 12, 15, 10, 11, 26, 25, 30, 29, 20, 21, 16, 19
1,  0,  0,  0,  5,  1,  1, 13, 25, 25,  0, 28, 25, 12, 25, 13

Mathematica

F[p_, q_] := F[p, q] = Which[p <= 1, 0, p > 1 && OddQ[p], F[Mod[q, p], p] ~BitXor~ BitAnd[p, q], True, F[p/2, q] ~BitXor~ BitXor[q, Floor[q/2]]];
A[n_, k_] := F[n, 2 k - 1];
Table[A[n - k, k], {n, 1, 14}, {k, 1, n}] // Flatten (* Jean-François Alcover, Sep 11 2017 *)

Prog

(Scheme)
(define (A269158 n) (A269158auxbi (A002260 n) (+ -1 (* 2 (A004736 n)))))
;; A269158auxbi can be implemented either as a tail-recursive loop:
(define (A269158auxbi p q) (if (not (odd? q)) (error "A269158bi: the second argument should be odd: " p q) (let loop ((p p) (q q) (s 0)) (cond ((<= p 1) s) ((odd? p) (loop (modulo q p) p (A003987bi s (A004198bi p q)))) (else (loop (/ p 2) q (A003987bi s (A003987bi q (/ (- q 1) 2)))))))))
;; Or a recurrence (reflecting the given recursive formula):
(define (A269158auxbi p q) (cond ((<= p 1) 0) ((even? p) (A003987bi (A269158auxbi (/ p 2) q) (A003188 q))) (else (A003987bi (A269158auxbi (modulo q p) p) (A004198bi p q)))))

Crossrefs

Transpose: A269159.

Column 1: Seems to be 0 followed by A039982.

Column 32769: A268819.

Cf. A065621 (occurs as row 2, row 8, and in general, as any row 2^(2n+1) for n >= 0. Seems to be also present as a slanted diagonal F(2n+1,2n-1).)

Cf. A268816 (row 6, row 24, etc.).

Cf. arrays A268829 and A268728 (variants), and also A268931.

Cf. A003188, A003987, A004198.

Sequence in context: A187197 A174869 A330862 * A109971 A357585 A284797

Adjacent sequences: A269155 A269156 A269157 * A269159 A269160 A269161

Keyword

nonn,tabl

Author

Antti Karttunen, Feb 20 2016

Status

approved