A268931 -id:A268931 - cp's OEIS frontend

A268932 Transpose of A268931.

3, 2, 3, 0, 5, 3, 2, 7, 2, 1, 0, 0, 7, 1, 1, 2, 3, 0, 3, 6, 3, 0, 7, 7, 11, 2, 6, 3, 2, 0, 3, 14, 7, 2, 1, 1, 2, 3, 2, 15, 13, 14, 1, 6, 3, 0, 0, 4, 14, 0, 1, 13, 3, 5, 3, 0, 0, 0, 0, 10, 17, 11, 3, 5, 5, 1, 2, 7, 3, 3, 13, 25, 8, 8, 10, 4, 2, 3, 0, 0, 4, 4, 1, 26, 19, 11, 12, 15, 1, 1, 3, 2, 3, 7, 0, 6, 3, 0, 19, 18, 11, 1, 6, 2, 1

Offset: 1

Antti Karttunen, Feb 19 2016

See comments in A268931.

			The top left [1 .. 19] x [1 .. 19] section of the array:
3, 2, 0,  2,  0,  2,  0,  2,  2,  0,  0,  2,  0,  2,  2,  2,  0,  0,  2
3, 5, 7,  0,  3,  7,  0,  3,  0,  0,  7,  0,  3,  7,  3,  0,  0,  7,  0
3, 2, 7,  0,  7,  3,  2,  4,  0,  3,  4,  7,  0,  2,  0,  3,  2,  0,  0
1, 1, 3, 11, 14, 15, 14,  0,  3,  4,  0, 14, 15,  1,  4,  0, 15,  0,  1
1, 6, 2,  7, 13,  0, 10, 13,  1,  6,  7, 11,  0, 11,  0,  2,  9, 11, 10
3, 6, 2, 14,  1, 17, 25, 26,  3, 27,  3,  2, 12,  1, 25, 25, 31,  0,  3
3, 1, 1, 13, 11,  8, 19,  0, 27,  3, 27,  3,  1,  1,  3, 26,  0, 27, 27
1, 6, 3,  3,  8, 11, 19, 23, 29, 28, 31, 28,  6,  0,  3,  8, 15, 10, 28
3, 5, 5, 10, 12, 18, 10,  8, 29, 19, 19,  3, 22, 23, 16,  0,  3,  4, 12
3, 5, 4, 15, 11, 10, 16, 11, 14, 31, 19, 21,  3,  0, 31,  4, 31,  5,  9
1, 2, 1,  1,  2,  2, 26, 26, 22, 22, 37,  0, 54, 53,  0, 54, 54, 53, 53
3, 1, 6,  7,  2,  3,  2, 29, 10, 28, 33, 41, 61, 62,  3, 53,  1, 63, 63
3, 2, 3,  4,  9, 13,  2,  5, 31,  8, 23, 20, 43,  0, 60,  0, 50, 61,  3
1, 2, 5, 10,  6,  0,  2,  7, 26, 15, 16, 23, 43, 47, 57,  1, 61,  2, 57
3, 6, 5,  5,  5,  8, 18, 22,  5, 10, 37, 34, 29, 28, 53, 44, 47, 42, 35
1, 1, 0, 11, 11,  8,  9, 27, 25, 31, 23, 28, 43, 42, 29, 59, 38, 38,  1
1, 5, 7,  6,  4, 14, 17, 26, 30,  2, 19, 40, 27, 16, 26, 31, 61, 34, 38
3, 6, 3,  3, 10,  1, 24,  9,  5,  6, 52, 62, 62,  1, 43, 37, 35, 67,  0
1, 5, 7,  2, 15,  2, 24, 27,  9, 14, 48, 62,  0, 62, 38,  2, 35, 67, 71

Antti Karttunen, Table of n, a(n) for n = 1..4186; the first 91 antidiagonals of the array

Transpose: A268931.

Cf. A065091, A269158.

(define (A268932 n) (A269158auxbi (A065091 (A004736 n)) (A065091 (A002260 n)))) ;; Code for A269158auxbi given in A269158.

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), ...

Original entry on oeis.org

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

Antti Karttunen, Feb 20 2016

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.

			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

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

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.

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 *)

(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)))))

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).

cp's OEIS Frontend

A268932 Transpose of A268931.

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