A072781 -id:A072781

Search: a072781 -id:a072781

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A072785

Differences between A072781 and A072738.

+20
4

0, 0, -1, 0, 0, 1, 0, 0, -1, -1, 0, 0, 0, 1, 1, 0, 0, 0, -1, -1, -1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, -1, -1, -1, -1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, -1, -1, -1, -1, -1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, -1, -1, -1, -1, -1, -1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, -1, -1, -1, -1, -1, -1, -1, 0, 0, 0, 0, 0, 0, 0

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,1

LINKS

Table of n, a(n) for n=0..111.

Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions, 2012, arXiv:1212.2732 [math.CO], 2012.

FORMULA

From Boris Putievskiy, Jan 16 2013: (Start)

a(n) = floor((2*A000027(n)-A003056(n)^2-A003056(n))/(A003056(n)+3))*(-1)^A003056(n).

a(n) = floor((2*n-t*t-t)/(t+3))*(-1)^t where t=floor((-1+sqrt(8*n-7))/2).

(End)

PROG

(Scheme) (define (A072785 n) (- (A072781 n) (A072738 n)))

CROSSREFS

Cf. A130517, A162630.

KEYWORD

sign,tabl

AUTHOR

Antti Karttunen, Jun 12 2002

STATUS

approved

A072735

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

+10
7

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

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

Table of n, a(n) for n=0..66.

Index entries for sequences that are permutations of the natural numbers

PROG

(Scheme) (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)))))

CROSSREFS

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

KEYWORD

nonn,tabl

AUTHOR

Antti Karttunen, Jun 12 2002

STATUS

approved

A072738

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

+10
5

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

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,4

LINKS

Table of n, a(n) for n=0..97.

PROG

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

CROSSREFS

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

KEYWORD

nonn,tabl

AUTHOR

Antti Karttunen, Jun 12 2002

STATUS

approved

A072782

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

+10
5

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

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,5

LINKS

Table of n, a(n) for n=0..98.

PROG

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

CROSSREFS

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

KEYWORD

nonn,tabl

AUTHOR

Antti Karttunen, Jun 12 2002

STATUS

approved

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