cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A154472 a(n) = A126309(A154471(n)).

Original entry on oeis.org

42, 218, 359, 875, 1763, 2359496, 926357642, 1431707647759, 239418125921492, 9138627621456887, 5265474933763866437, 36640566669911088560059, 1544449741807406472977531, 12634727265105076809694418
Offset: 0

Views

Author

Antti Karttunen, Jan 11 2009

Keywords

Comments

This sequence essentially gives the iterated S-expressions of the sequence A154471, with ()'s removed. See A154473.

Links

Crossrefs

a(n) = A080300(A154473(n)).

A125985 Signature-permutation of Vaillé's 1997 bijection on 'bridges' (Dyck paths).

Original entry on oeis.org

0, 1, 3, 2, 8, 7, 5, 6, 4, 22, 21, 18, 20, 17, 13, 12, 19, 15, 16, 10, 11, 14, 9, 64, 63, 59, 62, 58, 50, 49, 61, 55, 57, 46, 48, 54, 45, 36, 35, 32, 34, 31, 60, 56, 41, 52, 40, 47, 53, 43, 44, 27, 26, 33, 29, 30, 51, 38, 39, 42, 24, 25, 28, 37, 23, 196, 195, 190, 194, 189
Offset: 0

Views

Author

Antti Karttunen, Jan 02 2007

Keywords

Comments

Vaillé shows in 1997 paper that this automorphism transforms a 'derivation' of a Dyck path to its 'compression', i.e., in OEIS terms, A125985(A126310(n)) = A126309(A125985(n)) holds for all n. He also proves that A057515(A125985(n)) = A126307(n) and A057514(A125985(n)) = A072643(n) - A057514(n) + 1 (the latter identity for all n >= 1).

Links

Crossrefs

Inverse: A125986. The number of cycles, maximum cycle sizes and LCM's of all cycle sizes in range [A014137(n-1)..A014138(n-1)] of this permutation are given by A126291, A126292 and A126293. The fixed points are given by A126300/A126301.

Programs

  • Scheme
    (define (A125985 n) (A080300 (rising-list->binexp (A125985-aux2 (A014486 n)))))
(define (A125985-aux2 n) (let loop ((lists (A125985-aux1 n)) (z (list)) (m 1)) (if (null? lists) z (loop (cdr lists) (m-join z (car lists) m) (+ m 1)))))
(define (A125985-aux1 n) (if (zero? n) (list) (let ((begin_from (<< 1 (- (- (A000523 n) (A090996 n)) 1)))) (let loop ((s (A090996 n)) (t 0) (nth_list 1) (p begin_from) (b (if (= 0 (A004198bi n begin_from)) 0 1)) (lists (list (list)))) (cond ((< s 1) (cond ((< p 1) (reverse! lists)) (else (loop (- t (- 1 b)) b (+ 1 nth_list) (>> p 1) (if (= 0 (A004198bi n (>> p 1))) 0 1) (cons (list (+ b 1 nth_list)) lists))))) (else (loop (- s (- 1 b)) (+ t b) nth_list (>> p 1) (if (= 0 (A004198bi n (>> p 1))) 0 1) (cons (cons (+ b nth_list) (car lists)) (cdr lists)))))))))
(define (A125985-aux2 n) (let loop ((lists (A125985-aux1 n)) (z (list)) (m 1)) (if (null? lists) z (loop (cdr lists) (m-join z (car lists) m) (+ m 1)))))
(define (m-join a b m) (let loop ((a a) (b b) (c (list))) (cond ((and (not (pair? a)) (not (pair? b))) (reverse! c)) ((not (pair? a)) (loop a (cdr b) (cons (car b) c))) ((not (pair? b)) (loop (cdr a) b (cons (car a) c))) ((equal? (car a) (car b)) (loop (cdr a) (cdr b) (cons (car a) c))) ((> (car b) m) (loop a (cdr b) (cons (car b) c))) (else (loop (cdr a) b (cons (car a) c))))))
(define (rising-list->binexp rising-list) (let loop ((s 0) (i 0) (h 0) (fs rising-list)) (cond ((null? fs) (+ s (<< (- (<< 1 h) 1) i))) ((> (car fs) h) (loop s (+ i 1) (car fs) (cdr fs))) (else (loop (+ s (<< (- (<< 1 (+ 1 (- h (car fs)))) 1) i)) (+ i 2 (- h (car fs))) (car fs) (cdr fs))))))
(define (>> n i) (if (zero? i) n (>> (floor->exact (/ n 2)) (- i 1))))
(define (<< n i) (if (<= i 0) (>> n (- i)) (<< (+ n n) (- i 1))))

A126310 A014486-index for the Dyck path "derived" from the n-th Dyck path encoded by A014486(n).

Original entry on oeis.org

0, 0, 1, 0, 2, 1, 1, 1, 0, 4, 2, 2, 2, 1, 2, 1, 2, 3, 1, 1, 1, 1, 0, 9, 4, 4, 4, 2, 4, 2, 4, 5, 2, 2, 2, 2, 1, 4, 2, 2, 2, 1, 4, 2, 6, 7, 3, 2, 2, 3, 1, 2, 1, 2, 3, 1, 2, 3, 3, 1, 1, 1, 1, 1, 0, 23, 9, 9, 9, 4, 9, 4, 9, 10, 4, 4, 4, 4, 2, 9, 4, 4, 4, 2, 9, 4, 11, 12, 5, 4, 4, 5, 2, 4, 2, 4, 5, 2, 4, 5, 5
Offset: 0

Views

Author

Antti Karttunen, Jan 02 2007

Keywords

Comments

According to Vaillé, the concept of "dérivation des ponts" is defined by Kreweras, in "Sur les éventails de segments" paper.

Links

Crossrefs

Cf. A125986, A126309, A125985.

Formula

a(n) = A125986(A126309(A125985(n))).

A126308 Delete '10'-substrings in the binary expansion of n.

Original entry on oeis.org

0, 1, 0, 3, 0, 1, 1, 7, 0, 1, 0, 3, 2, 3, 3, 15, 0, 1, 0, 3, 0, 1, 1, 7, 4, 5, 1, 7, 6, 7, 7, 31, 0, 1, 0, 3, 0, 1, 1, 7, 0, 1, 0, 3, 2, 3, 3, 15, 8, 9, 2, 11, 2, 3, 3, 15, 12, 13, 3, 15, 14, 15, 15, 63, 0, 1, 0, 3, 0, 1, 1, 7, 0, 1, 0, 3, 2, 3, 3, 15, 0, 1, 0, 3, 0, 1, 1, 7, 4, 5, 1, 7, 6, 7, 7, 31
Offset: 0

Views

Author

Antti Karttunen, Jan 02 2007

Keywords

Examples

			10 is 1010 in binary, thus it is rewritten to empty string, thus a(10)=0. 12 is 1100 in binary, thus it is rewritten to '10', so a(12)=2. 27 is 11011 in binary and when '10' is deleted, results 111, 7 in decimal, thus a(27)=7.

Crossrefs

A126309 = A080300(a(A014486(n))). Cf. A080303.

Programs

  • Scheme
    (define (A126308 n) (cond ((zero? n) 0) ((= 2 (modulo n 4)) (A126308 (/ (- n 2) 4))) (else (+ (modulo n 2) (* 2 (A126308 (floor->exact (/ n 2))))))))
Showing 1-4 of 4 results.

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