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.

A080300 Global ranking function for totally balanced binary sequences.

Original entry on oeis.org

0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 5, 0, 0, 0, 0, 0, 6, 0, 7, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
Offset: 0

Views

Author

Antti Karttunen, Feb 21 2003

Keywords

Comments

Note: the next nonzero value occurs at a(170)=9, as 170 = 10101010 is the lexicographically earliest totally balanced binary sequence of length 2*4.

Links

Crossrefs

Inverse function of A014486, i.e. a(A014486(n)) = n for all n. Cf. A080116, A215406, A213704, A209640.

Programs

  • Maple
    A080300 := n -> A080116(n)*A215406(n); # Untested (as of Aug 19 2012)
A080300 := n -> `if`((0 = n) or (0 = A080116(n)),0, A014137(((A000523(n)+1)/2)-1)+A080301(n));
  • Mathematica
    A080116[n_] := Module[{lev = 0, c = n}, While[c > 0, lev = lev + (-1)^c; c = Floor[c/2]; If[lev<0, Return[0]]]; If[lev>0, Return[0], Return[1]]];
A215406[n_] := Module[{m, d, a, y, t, x, u, v}, m = Quotient[Length[d = IntegerDigits[n, 2]], 2]; a = FromDigits[Reverse[d], 2]; y = 0; t = 1; For[x = 0, x <= 2*m - 2, x++, If[Mod[a, 2] == 1, y++, u = 2*m - x; v = m - Quotient[x + y, 2] - 1; t = t - Binomial[u - 1, v - 1] + Binomial[u - 1, v]; y--]; a = Quotient[a, 2]]; (1 - I*Sqrt[3])/2 - 4^(m + 1)*Gamma[m + 3/2]*Hypergeometric2F1[1, m + 3/2, m + 3, 4]/(Sqrt[Pi]*Gamma[m + 3]) -t];
a[n_] := A080116[n]*A215406[n] // Simplify;
Table[a[n], {n, 0, 170}] (* Jean-François Alcover, Mar 05 2016 *)

Formula

a(n) = A080116(n)*A215406(n).
a(n) = 0 if n=0 or (A080116(n)=0), otherwise a(n) = A014137(((A000523(n)+1)/2)-1)+A080301(n)

A209861 Permutation of nonnegative integers which maps A209641 to A209642.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 11, 13, 14, 15, 16, 17, 18, 21, 19, 22, 24, 27, 20, 23, 25, 28, 26, 29, 30, 31, 32, 33, 34, 38, 35, 39, 42, 48, 36, 40, 43, 49, 45, 51, 54, 58, 37, 41, 44, 50, 46, 52, 55, 59, 47, 53, 56, 60, 57, 61, 62, 63, 64, 65, 66, 71, 67, 72, 76, 86, 68, 73, 77, 87, 80, 90, 96, 106, 69, 74, 78, 88, 81, 91, 97, 107, 83
Offset: 0

Views

Author

Antti Karttunen, Mar 24 2012

Keywords

Comments

Conjecture: For all n, A209861(A054429(n)) = A054429(A209861(n)), i.e. A054429 acts as an homomorphism (automorphism) of the cyclic group generated by this permutation. This implies also a weaker conjecture given in A209860.
The scatterplot graph of the sequence has a nice texture and interesting pattern.

Links

Crossrefs

Inverse permutation: A209862. Cf. A209860, A209863, A209864, A209865, A209866, A209867, A209868.

Formula

a(n) = A209640(A209642(n)).

A071163 A014486-indices for rooted binary trees with height equal to number of internal vertices. (Binary trees where at each internal vertex at least the other child is leaf.)

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 7, 8, 9, 10, 12, 13, 17, 18, 21, 22, 23, 24, 26, 27, 31, 32, 35, 36, 45, 46, 49, 50, 58, 59, 63, 64, 65, 66, 68, 69, 73, 74, 77, 78, 87, 88, 91, 92, 100, 101, 105, 106, 129, 130, 133, 134, 142, 143, 147, 148, 170, 171, 175, 176, 189, 190, 195, 196, 197
Offset: 0

Views

Author

Antti Karttunen, May 14 2002

Keywords

Comments

This subset of integers is closed by the actions of A069770, A057163, A069767, A069768, A122353, A122354, A122301, A122302, etc. (meaning, e.g., that A069767(a(n)) is a member from this sequence for all n), that is, by any Catalan bijection which is an image of some element of the automorphism group of infinite binary tree (the latter in a sense given by Grigorchuk, et al., being isomorphic to an infinitely iterated wreath product of cyclic groups of two elements). See the comments about the isomorphism "psi" given at A153141.
a(n) could be probably computed directly from the binary expansion of n by using a (somewhat) similar ranking function as given in A209640, but utilizing A009766 instead of A007318.

Links

Formula

a(n) = A080300(A071162(n)).

A209641 A014486-codes for rooted plane trees where non-leaf branches can occur only as the leftmost branch of any level, but nowhere else. Sorted into ascending order.

Original entry on oeis.org

0, 2, 10, 12, 42, 50, 52, 56, 170, 202, 210, 212, 226, 228, 232, 240, 682, 810, 842, 850, 852, 906, 914, 916, 930, 932, 936, 962, 964, 968, 976, 992, 2730, 3242, 3370, 3402, 3410, 3412, 3626, 3658, 3666, 3668, 3722, 3730, 3732, 3746, 3748, 3752, 3850, 3858, 3860, 3874, 3876, 3880, 3906, 3908, 3912, 3920, 3970, 3972
Offset: 0

Views

Author

Antti Karttunen, Mar 24 2012

Keywords

Links

Crossrefs

A209640(a(n)) = n for all n. a(n) = A014486(A209643(n)). Sequence A209642 sorted into ascending order with permutation A209862, i.e. a(n) = A209642(A209862(n)).

Programs

  • Scheme
    (define (member_of_A209641? n) (let loop ((n n) (lev 0)) (cond ((zero? n) (zero? lev)) ((< lev 0) #f) ((even? n) (loop (/ n 2) (+ lev 1))) ((and (odd? (/ (-1+ n) 2)) (even? (/ (-1+ (/ (-1+ n) 2)) 2)) (not (zero? (/ (-1+ (/ (-1+ n) 2)) 2)))) #f) (else (loop (/ (- n 1) 2) (- lev 1))))))
(define A209641 (MATCHING-POS 0 0 member_of_A209641?)) ;; MATCHING-POS in AK's Scheme-SeqFun-Transform package.
Showing 1-4 of 4 results.

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