[seqfan] Fwd: Quest of a trinagular decomposition of A081113
Alexander P-sky
apovolot at gmail.com
Tue Jan 4 00:34:52 CET 2011
FYI
Alexander R. Povolotsky
---------- Forwarded message ----------
From: David Scambler <dscambler at bmm.com>
Date: Mon, 3 Jan 2011 17:25:35 -0600
Subject: RE: [seqfan] Quest of a trinagular decomposition of A081113
To: Alexander P-sky <apovolot at gmail.com>
Cc: "mathar at strw.leidenuniv.nl" <mathar at strw.leidenuniv.nl>
Alexander,
Yes, that recurrence seems to work. I had no success with Superseeker
finding a recurrence or a g.f. That prompted my question to the list
(http://list.seqfan.eu/pipermail/seqfan/2010-August/005605.html) "Run
Superseeker on existing sequence?"
The original sequence was generated by a simple brute-force computer
algorithm. The relationship to the Motzkin triangle A026300 was
observed while playing around with the paths and their counts.
I cannot add anything that would help Richard's correspondent.
Regards
dave
-----Original Message-----
From: Alexander P-sky [mailto:apovolot at gmail.com]
Sent: Tuesday, 4 January 2011 8:13 AM
To: Sequence Fanatics Discussion list
Cc: mathar at strw.leidenuniv.nl; David Scambler
Subject: Re: [seqfan] Quest of a trinagular decomposition of A081113
WolframAlpha - given the partial set of A081113 terms as Input :
{1, 4, 17, 68, 259, 950, 3387, 11814, 40503, 136946, 457795, 1515926,
4979777, 16246924, 52694573, 170028792, 546148863, ...}
correctly guesses the rest of A081113 terms
Possible continuation:
1,4,17,68,259,950,3387,11814,40503,136946,457795,1515926,4979777,16246924,52694573,170028792,546148863,1747255194,5569898331,17698806798,56076828573,177208108824,558658899825,1757365514652,5517064515933,17288356984560,54082946719557,...
And suggests recurrent formula
Possible recurrence relation:
a_(n+4) = (27 n a_n)\/(n+3)+(27 a_(n+1))\/(n+3)-(9 (2 n+5)
a_(n+2))\/(n+3)+((8 n+21) a_(n+3))\/(n+3) (for all terms given)
Alexander R. Povolotsky
===============================================================
On 1/3/11, Richard Mathar <mathar at strw.leidenuniv.nl> wrote:
>
> I received a request to provide a formula for a triangular decomposition of
> a
> sum sequence (which is A081113 in the OEIS, but not in the partitioning
> which is obvious from the formula there), and another one which isn't
> (perhaps
> only in a first difference or partial sum format).
> If anyone has seen these before, please contact the inventor of these
> numbers.
> My attempts to receive further information on how these were generated,
> failed.
>
>
> ad> Delivered-To: mathar
> ad> Date: Tue, 30 Nov 2010 06:17:12 -0800 (PST)
> ad> From: adesola dauda adesoladaudaATyahooDOTcom
> ad> Subject: ON RESEARCH ASSISTANCE
> ad>
> ad> Good day Sir. I was the one that wrote some months(May/June) ago on
> Ph.D. Research Assistance on Semigroups of Sequences: 1, 4, 17, 68, 259,950,
> . . . , .
> ad> In your reply:
> ad>
> ad> "The description is rather simple and based on geometry. You take a
> standard
> ad> chess board with variable edge length. Take the king of the chess game
> ad> (which can only move to one of the 8 adjacent squares), place it
> initially
> ad> at one side, and count the different paths. So for n=2 we have . . . "
> .
> ad>
> ad> I did as directed but I could not get the formular even the reference
> you made to sequence A081113 was followed but no results.
> ad>
> ad>
> ad> Equally, I also sent Idempotent sequence: 1, 3, 8, 21, 56, 149, . . .
> to you but what I found on Encyclopedia of Integer sequences is: 1, 3, 8,
> 21, 56, 150, . . . . What could be the problem as I have been working on
> this problems for years without edge way.
> ad>
> ad> Sir, I am not only requesting you to send the formulars to me but also
> to please re-direct me on how to get the problems solved and equally get me
> materials(if available) on a similar problems on Combinatorics Mathematics.
> ad>
>
> ad> Date: Wed, 29 Dec 2010 02:04:13 -0800 (PST)
> ad> From: adesola dauda adesoladaudaATyahooDOTcom
> ad> Subject: ELEMENTS GENERATED
> ad>
> ad>
> ad> Dear Sir,
> ad> Once again thanks for the assistance you have been rendering. The
> triangular array you sent to me is just a guide to the sequences: 1, 4, 17,
> 68, 259, 950, . . . and 1, 3, 8, 21, 56, 149, . . . . while the triangular
> array for the two sequences as it appears in my research work are:
> ad>
> ad> n/k 1 2 3 4 5 6
> ad> 1 1
> ad> 2 2 2
> ad> 3 3 12 2
> ad> 4 4 42 20 2
> ad> 5 5 120 108 24 2
> ad> 6 6 310 448 156 28 2
> ad>
> ad> and
> ad>
> ad> n/k 1 2 3 4 5 6
> ad> 1 1
> ad> 2 2 1
> ad> 3 3 4 1
> ad> 4 4 12 4 1
> ad> 5 5 32 14 4 1
> ad> 6 6 80 44 14 4 1
>
>
> The standard decomposition of A081113 as sum_{k=1..n) k*(n-k+1)*A026300(.,.)
> is
> +1 = 1.
> +2 +2 = 4.
> +3 +8 +6 = 17.
> +4 +18 +30 +16 = 68.
> +5 +32 +81 +96 +45 = 259.
> +6 +50 +168 +300 +300 +126 = 950.
> +7 +72 +300 +704 +1035 +912 +357 = 3387.
> and therefore different from the first triangle shown above.
>
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