Fwd: A025581vs A122200 - very few differences (and very consistent pattern in those differences)

[Alexander Povolotsky]

Forwarding message from Antti Karttunen
---------- Forwarded message ----------
From: Antti Karttunen <antti.karttunen at gmail.com>
Date: Sun, May 11, 2008 at 7:28 AM
Subject: Re: A025581vs A122200 - very few differences (and very
consistent pattern in those differences)
To: Alexander Povolotsky <apovolot at gmail.com>, davidwwilson at comcast.net
(I cannot mail this directly to SeqFan-list, as I am not currently its
member. Maybe you can forward this there, if the topic is still
relevant?)
On Sat, May 10, 2008 at 9:57 PM, Alexander Povolotsky
<apovolot at gmail.com> wrote:
> It looks that A025581 & A122200 have very few differences (and very
> consistent pattern exists in those differences) ?
>
> Anybody has an explanation for that ?
Yes. The table A025581 consists of just rows 0,1,2,3,4,5,6,7,...
and the transformation applied in A122200 to the rows of A089840
is of such nature, that the corresponding automorphisms fix the
most small binary trees (of a few nodes), so the resulting sequences
look like A001477 in their initial terms.

Yours,

Antti Karttunen
>
> Alex P.
>
> A025581
> Triangle T(n,k) = n-k, n >= 0, 0<=k<=k. Integers m to 0 followed by
> integers m+1 to 0 etc.
>
> 0, 1, 0, 2, 1, 0, 3, 2, 1, 0, 4, 3, 2, 1, 0, 5, 4, 3, 2, 1, 0, 6, 5,
> 4, 3, 2, 1, 0, 7, 6, 5, 4, 3, 2, 1, 0, 8, 7, 6, 5, 4, 3, 2, 1, 0, 9,
> 8, 7, 6, 5, 4, 3, 2, 1, 0, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 11, 10,
> 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1,
> 0, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3 (list; table; graph; listen)
> OFFSET
>
> 0,4
>
> FORMULA
>
> a(n) = (((trinv(n)-1)*(((1/2)*trinv(n))+1))-n) # Cf. A002262
>
> G.f.: y / [(1-x)^2 * (1-xy) ]. - R. Stephan, Jan 25 2005
>
> MAPLE
>
> A025581 := n -> binomial(1+floor((1/2)+sqrt(2*(1+n))), 2) - (n+1);
>
> PROGRAM
>
> (PARI) a(n)=binomial(1+floor(1/2+sqrt(2+2*n)), 2)-(n+1) /* produces a(n) */
>
> (PARI) t1(n)=binomial(floor(3/2+sqrt(2+2*n)), 2)-(n+1) /* A025581 */
>
> (PARI) t2(n)=n-binomial(floor(1/2+sqrt(2+2*n)), 2) /* A002262 */
>
> CROSSREFS
>
> A004736(n+1)=1+A025581(n)
>
> AUTHOR
>
> David W. Wilson (davidwwilson(AT)comcast.net)
>
> A122200
> Signature permutations of RIBS-transformations of non-recursive
> Catalan automorphisms in table A089840.
>
> 0, 1, 0, 2, 1, 0, 3, 2, 1, 0, 4, 3, 2, 1, 0, 5, 4, 3, 2, 1, 0, 6, 5,
> 4, 3, 2, 1, 0, 7, 6, 5, 4, 3, 2, 1, 0, 8, 8, 6, 5, 4, 3, 2, 1, 0, 9,
> 7, 7, 6, 5, 4, 3, 2, 1, 0, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 11, 10,
> 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1,
> 0, 13, 13, 11, 10, 9, 8 (list; table; graph; listen)
> OFFSET
>
> 0,4
>
> CROSSREFS
>
> Row 0 (identity permutation): A001477, row 1: A122282. See also tables
> A089840, A122201-A122204, A122283-A122284, A122285-A122288,
> A122289-A122290.
>
> Adjacent sequences: A122197 A122198 A122199 this_sequence A122201
> A122202 A122203
>
> Sequence in context: A025677 A025651 A025670 this_sequence A025646
> A025661 A025671
> AUTHOR
>
> Antti Karttunen (His-Firstname.His-Surname(AT)gmail.com), Sep 01 2006