A072734 -id:A072734 - cp's OEIS frontend

A072735 Simple triangle-stretching N X N -> N bijection: Inverse of A072734, variant of A072733.

0, 1, 2, 3, 5, 9, 10, 6, 14, 20, 21, 15, 4, 27, 35, 36, 28, 7, 8, 44, 54, 55, 45, 11, 13, 19, 65, 77, 78, 66, 22, 16, 26, 34, 90, 104, 105, 91, 37, 29, 12, 43, 53, 119, 135, 136, 120, 56, 46, 17, 18, 64, 76, 152, 170, 171, 153, 79, 67, 23, 25, 33, 89, 103, 189, 209, 210

Offset: 0

Antti Karttunen, Jun 12 2002

Index entries for sequences that are permutations of the natural numbers

Inverse: A072734, projections: A072781 & A072782, variant of the same theme: A072733. Cf. also A001477 and its projections A025581 & A002262.

(define (A072735 n) (packA072735 (A025581 n) (A002262 n)))
(define (packA001477 x y) (/ (+ (expt (+ x y) 2) x (* 3 y)) 2))
(define (packA072735 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) (if (and (eq? x y) (even? x)) 0 -1))))) (else (let ((half-y (floor->exact (/ y 2)))) (packA001477 (+ half-y (* 2 (- (-1+ x) (* 2 half-y))) (modulo y 2) (if (and (eq? x (1+ y)) (even? y)) 1 0)) half-y)))))

A072740 X-projection of the tabular N X N -> N bijection A072734.

Original entry on oeis.org

0, 1, 0, 2, 0, 0, 4, 3, 0, 0, 6, 5, 1, 0, 0, 8, 7, 2, 1, 0, 0, 10, 9, 3, 1, 1, 0, 0, 12, 11, 5, 4, 1, 1, 0, 0, 14, 13, 7, 6, 2, 1, 1, 0, 0, 16, 15, 9, 8, 3, 2, 1, 1, 0, 0, 18, 17, 11, 10, 4, 2, 2, 1, 1, 0, 0, 20, 19, 13, 12, 6, 5, 2, 2, 1, 1, 0, 0, 22, 21, 15, 14, 8, 7, 3, 2, 2, 1, 1, 0, 0, 24, 23

Offset: 0

Antti Karttunen, Jun 12 2002

The Y-projection is A072741. Composition of A025581 and A072735. A072783(n) = A072740(n)-A072736(n).

(define (A072740 n) (A025581 (A072735 n)))

A072741 Y-projection of the tabular N X N -> N bijection A072734.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Jun 12 2002

The X-projection is A072740. Composition of A002262 & A072735. A072784(n) = A072741(n)-A072737(n).

(define (A072741 n) (A002262 (A072735 n)))

A072787 Permutation of natural numbers induced by reranking plane binary trees given in the standard lexicographic order (A014486) with an "arithmetic global ranking algorithm", using A072734 as the packing bijection N X N -> N.

Original entry on oeis.org

0, 1, 3, 2, 6, 5, 13, 8, 4, 14, 10, 36, 20, 9, 25, 19, 24, 11, 12, 18, 38, 16, 7, 44, 27, 209, 77, 21, 105, 66, 104, 28, 35, 65, 230, 54, 15, 34, 33, 75, 43, 26, 85, 50, 40, 37, 22, 31, 191, 67, 23, 51, 41, 69, 107, 68, 49, 92, 30, 29, 32, 56, 211, 46, 17, 299, 120, 5671

Offset: 0

Antti Karttunen, Jun 12 2002

nonn

This ranking scheme condenses the structures of the same size (cf. A072789) somewhat better than scheme presented in A072656 (which uses the N X N -> N bijection A072793). Compare the sequences A072790 and A072640 giving the max positions where the last structure with size n will occur in these orderings and the respective binary widths A072791 & A072642. However, by using the second or third power of the bijection A072734 one gets even better results in a certain range.

A. Karttunen, Gatomorphisms
Index entries for sequences that are permutations of the natural numbers

Inverse permutation: A072788. Cf. also A014486, A072734, A072789.

A072732 Simple triangle-stretching N X N -> N bijection: push terms in the middle twice as far down to make space for the terms obtained by bisecting the edges, which are thus contracted by the same factor. Do this for all "triangle-shells" successively contained inside each other in A001477.

Original entry on oeis.org

0, 1, 4, 7, 12, 2, 3, 17, 24, 8, 11, 31, 40, 18, 5, 6, 23, 49, 60, 32, 13, 16, 39, 71, 84, 50, 25, 9, 10, 30, 59, 97, 112, 72, 41, 19, 22, 48, 83, 127, 144, 98, 61, 33, 14, 15, 38, 70, 111, 161, 180, 128, 85, 51, 26, 29, 58, 96, 143, 199, 220, 162, 113, 73, 42, 20, 21, 47

Offset: 0

Antti Karttunen, Jun 12 2002

Index entries for sequences that are permutations of the natural numbers

Inverse: A072733, projections: A072736 & A072737, variant of the same theme: A072734. Cf. also A001477 and its projections A025581 & A002262.

(define (A072732 n) (packA072732 (A025581 n) (A002262 n)))
(define (packA001477 x y) (/ (+ (expt (+ x y) 2) x (* 3 y)) 2))
(define (packA072732 x y) (let ((x-y (- x y))) (cond ((<= x-y 0) (packA001477 (+ (* 2 x) (modulo x-y 2)) (+ (* 2 x) (floor->exact (/ (1+ (- x-y)) 2))))) (else (packA001477 (+ (* 2 (1+ y)) (floor->exact (/ (- x-y 2) 2))) (+ (* 2 y) (modulo (1+ x-y) 2)))))))

A072781 X-projection of the tabular N X N -> N bijection A072735.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Jun 12 2002

The Y-projection is A072782. Composition of A025581 & A072734. A072785(n) = A072781(n)-A072738(n).

(define (A072781 n) (A025581 (A072734 n)))

A072782 Y-projection of the tabular N X N -> N bijection A072735.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Jun 12 2002

The X-projection is A072781. Composition of A002262 and A072734. A072786(n) = A072782(n)-A072739(n).

(define (A072782 n) (A002262 (A072734 n)))

cp's OEIS Frontend

A072735 Simple triangle-stretching N X N -> N bijection: Inverse of A072734, variant of A072733.

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A072740 X-projection of the tabular N X N -> N bijection A072734.

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A072741 Y-projection of the tabular N X N -> N bijection A072734.

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A072787 Permutation of natural numbers induced by reranking plane binary trees given in the standard lexicographic order (A014486) with an "arithmetic global ranking algorithm", using A072734 as the packing bijection N X N -> N.

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A072732 Simple triangle-stretching N X N -> N bijection: push terms in the middle twice as far down to make space for the terms obtained by bisecting the edges, which are thus contracted by the same factor. Do this for all "triangle-shells" successively contained inside each other in A001477.

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A072781 X-projection of the tabular N X N -> N bijection A072735.

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A072782 Y-projection of the tabular N X N -> N bijection A072735.

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