A072732 -id:A072732 - cp's OEIS frontend

A072734 Simple triangle-stretching N X N -> N bijection, variant of A072732.

0, 1, 2, 3, 12, 4, 7, 17, 18, 5, 6, 23, 40, 24, 8, 11, 31, 49, 50, 25, 9, 10, 30, 59, 84, 60, 32, 13, 16, 39, 71, 97, 98, 61, 33, 14, 15, 38, 70, 111, 144, 112, 72, 41, 19, 22, 48, 83, 127, 161, 162, 113, 73, 42, 20, 21, 47, 82, 126, 179, 220, 180, 128, 85, 51, 26, 29, 58

Offset: 0

Antti Karttunen, Jun 12 2002

Index entries for sequences that are permutations of the natural numbers

Inverse: A072735, projections: A072740 & A072741, variant of the same theme: A072732. Used to construct the global arithmetic ranking scheme of plane binary trees presented in A072787/A072788. Cf. also A001477 and its projections A025581 & A002262.

(define (A072734 n) (packA072734 (A025581 n) (A002262 n)))
(define (packA001477 x y) (/ (+ (expt (+ x y) 2) x (* 3 y)) 2))
(define (packA072734 x y) (let ((x-y (- x y))) (cond ((negative? x-y) (packA001477 (+ (* 2 x) (modulo (1+ x-y) 2)) (+ (* 2 x) (floor->exact (/ (+ (- x-y) (modulo x-y 2)) 2))))) ((< x-y 3) (packA001477 (+ (* 2 y) x-y) (* 2 y))) (else (packA001477 (+ (* 2 y) (floor->exact (/ (1+ x-y) 2)) (modulo (1+ x-y) 2)) (+ (* 2 y) (modulo x-y 2)))))))

A072733 Simple triangle-stretching N X N -> N bijection: Inverse of A072732.

Original entry on oeis.org

0, 1, 5, 6, 2, 14, 15, 3, 9, 27, 28, 10, 4, 20, 44, 45, 21, 7, 13, 35, 65, 66, 36, 16, 8, 26, 54, 90, 91, 55, 29, 11, 19, 43, 77, 119, 120, 78, 46, 22, 12, 34, 64, 104, 152, 153, 105, 67, 37, 17, 25, 53, 89, 135, 189, 190, 136, 92, 56, 30, 18, 42, 76, 118, 170, 230, 231

Offset: 0

Antti Karttunen, Jun 12 2002

Index entries for sequences that are permutations of the natural numbers

Inverse: A072732, projections: A072738 & A072739, variant of the same theme: A072735. Cf. also A001477 and its projections A025581 & A002262.

(define (A072733 n) (packA072733 (A025581 n) (A002262 n)))
(define (packA001477 x y) (/ (+ (expt (+ x y) 2) x (* 3 y)) 2))
(define (packA072733 x y) (cond ((<= x y) (let ((half-x (floor->exact (/ x 2)))) (packA001477 half-x (+ half-x (* 2 (- y (* 2 half-x) (modulo x 2))) (modulo x 2))))) (else (let ((half-y (floor->exact (/ y 2)))) (packA001477 (+ 1 half-y (* 2 (- (-1+ x) (* 2 half-y) (modulo y 2))) (modulo y 2)) half-y)))))

A072736 X-projection of the tabular N X N -> N bijection A072732.

Original entry on oeis.org

0, 1, 0, 3, 0, 0, 5, 2, 0, 0, 7, 4, 1, 0, 0, 9, 6, 2, 1, 0, 0, 11, 8, 4, 1, 1, 0, 0, 13, 10, 6, 3, 1, 1, 0, 0, 15, 12, 8, 5, 2, 1, 1, 0, 0, 17, 14, 10, 7, 3, 2, 1, 1, 0, 0, 19, 16, 12, 9, 5, 2, 2, 1, 1, 0, 0, 21, 18, 14, 11, 7, 4, 2, 2, 1, 1, 0, 0, 23, 20, 16, 13, 9, 6, 3, 2, 2, 1, 1, 0, 0, 25, 22

Offset: 0

Antti Karttunen, Jun 12 2002

The Y-projection is A072737. Composition of A025581 & A072733. A072783(n) = A072740(n)-A072736(n).

(define (A072736 n) (A025581 (A072733 n)))

A072737 Y-projection of the tabular N X N -> N bijection A072732.

Original entry on oeis.org

0, 0, 2, 0, 1, 4, 0, 0, 3, 6, 0, 0, 1, 5, 8, 0, 0, 1, 3, 7, 10, 0, 0, 1, 2, 5, 9, 12, 0, 0, 1, 1, 4, 7, 11, 14, 0, 0, 1, 1, 2, 6, 9, 13, 16, 0, 0, 1, 1, 2, 4, 8, 11, 15, 18, 0, 0, 1, 1, 2, 3, 6, 10, 13, 17, 20, 0, 0, 1, 1, 2, 2, 5, 8, 12, 15, 19, 22, 0, 0, 1, 1, 2, 2, 3, 7, 10, 14, 17, 21, 24, 0, 0

Offset: 0

Antti Karttunen, Jun 12 2002

The X-projection is A072736. Composition of A002262 and A072733. A072784(n) = A072741(n)-A072737(n).

(define (A072737 n) (A002262 (A072733 n)))

A072738 X-projection of the tabular N X N -> N bijection A072733.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Jun 12 2002

The Y-projection is A072739. Composition of A025581 & A072732. A072785(n) = A072781(n)-A072738(n).

(define (A072738 n) (A025581 (A072732 n)))

A072739 Y-projection of the tabular N X N -> N bijection A072733.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Jun 12 2002

The X-projection is A072738. Composition of A002262 and A072732. A072786(n) = A072782(n)-A072739(n).

(define (A072739 n) (A002262 (A072732 n)))

cp's OEIS Frontend

A072734 Simple triangle-stretching N X N -> N bijection, variant of A072732.

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A072733 Simple triangle-stretching N X N -> N bijection: Inverse of A072732.

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A072736 X-projection of the tabular N X N -> N bijection A072732.

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A072737 Y-projection of the tabular N X N -> N bijection A072732.

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A072738 X-projection of the tabular N X N -> N bijection A072733.

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A072739 Y-projection of the tabular N X N -> N bijection A072733.

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