EDITED A015455

Sun Jun 15 21:01:38 CEST 2008

Further good work.  Here are some more sequences. The first
two are A001906 and the anagram A006190, but these don't
have negative terms and are not mentioned as being Gaussian
Markov numbers.  I hope I've got the signs `right' in the
others :-

..., -144, -55, -21, -8, -3, -1, 0, 1, 3, 8, 21, 55, 144, ...

  ..., -360, 109, -33, 10, -3, 1, 0, 1, 3, 10, 33, 109, 360, ...

           ..., -12141, 505, -21, 1, 3, 73, 1755, 42193, ...

           ..., -29637, 989, -33, 1, 3, -91, 2727, -81719, ...

              -, +, -, 3466, -55, 1, 8, 505, 31823, ...

        -, +, -, 97921, -989, 10, 1, 109, 10792, ...

and there are dozens of others!     R.

On Sun, 15 Jun 2008, Alexander Povolotsky wrote:

> FYI,
> AP
> ============================================================
> From:	The On-Line Encyclopedia of Integer Sequences <oeis at research.att.com>
> To 	: 	njas at research.att.com
> Cc	:	pevnev at juno.com
> Subject : 	COMMENT Alexander R. Povolotsky A015455
> Date 	: 	Fri, Jun 13, 2008 10:01 PM
>
> %I A015455
> %F A015455 a(n) = round(1/2*(9/2-1/2*sqrt(85))^n+
> 7/170*sqrt(85)*(9/2-1/2*sqrt(85))^n-
> 7/170*sqrt(85)*(9/2 +1/2*sqrt(85))^n+1/2*(9/2+1/2*sqrt(85))^n)
> %o A015455 (PARI) gp > a(n) =
> round(1/2*(9/2-1/2*sqrt(85))^n+7/170*sqrt(85)*(9/2-1/2*sqrt(85))^n-7/170*sqrt(8
> 5)*(9/2 +1/2*sqrt(85))^n+1/2*(9/2+1/2*sqrt(85))^n)
> gp > for (n=1, 30, print1(" "a(n)", "))
> 1,  10,  91,  829,  7552,  68797,  626725,  5709322,  52010623,
> 473804929,  4316254984,
> 39320099785,  358197153049,  3263094477226,  29726047448083,
> 270797521509973,
> 2466903741037840,  22472931190850533,  204723284458692637,
> 1864982491319084266,
> 16989565706330451031,  154771073848293143545,
> 1409929230340968742936,  12844134146917011829969,  117007136552594075212657,
> 1065908363120263688743882,  9710182404634967273907595,
> 88457550004834969153912237,
> 805828132448149689659117728,
> 7340910742038182176085971789,
> %O A015455 0
> %K A015455 ,nonn,
> %A A015455 Alexander R. Povolotsky (pevnev at juno.com), Jun 13 2008
> RH
> RA 192.20.225.32
> RU
> RI
> ===================================================
> On Sun, Jun 15, 2008 at 11:32 AM, Richard Guy <rkg at cpsc.ucalgary.ca> wrote:
>> Thankyou, Max.  It would be good to add the comment that these
>> are Gaussian Markov numbers.  Perhaps that will have to wait
>> until Alex Fink & I produce our paper?  And there are several
>> similar sequences which fall into the same category.  Some
>> examples are in Friday's message.       R.
>>
>> On Sun, 15 Jun 2008, Maximilian Hasler wrote:
>>
>>> old title "a(0) = 1, a(2) = 1; thereafter a(n) = 9 a(n-1) + a(n-2)"
>>> has a typo (a(2)), I suggest new title, give g.f., PARI & cite R.K.G's
>>> idea of extending to the left side.
>>>
>>> %I A015455
>>> %S A015455 1,1,10,91,829,7552,68797,626725,5709322,52010623,473804929,
>>> %T A015455
>>> 4316254984,39320099785,358197153049,3263094477226,29726047448083,
>>> %U A015455 270797521509973,2466903741037840,22472931190850533
>>> %N A015455 a(n) = 9 a(n-1) + a(n-2) for n>1; a(0) = a(1) = 1.
>>> %C A015455 Generalized Fibonacci numbers.
>>> %C A015455 As R.K.Guy suggested on the SeqFan list, the sequence could
>>> be extended "to the left side" by ..., 10, 1, 1, -8, 73, -665, 6058,
>>> -55187, 502741, -4579856, 41721445, ... by using the recurrence
>>> equation to get a(n-2) = a(n) - 9 a(n-1). The sequence 1,-8,73,...
>>> would have g.f. (1+x)/(1+9x-x^2).
>>> %H A015455 R. K. Guy, "A further family of sequences", SeqFan mailing
>>> list (www.seqfan.eu), Jun 13 2008
>>> %H A015455 Tanya Khovanova, <a
>>>
>>> href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive
>>> Sequences</a>
>>> %F A015455 g.f. = (1 - 8x)/(1 - 9x - x^2). - M.F.Hasler, Jun 14 2008
>>> %F A015455 a(n)=Sum_{k, 0<=k<=n} 8^k*A055830(n,k) .
>>>               - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 18 2006
>>> %o A015455 (PARI) A015455(n) =
>>> polcoeff((1-(O(x^n)+8)*x)/(1-9*x-x^2),n) \\ - M.F.Hasler, Jun 14 2008
>>> %Y A015455 Sequence in context: A079928 A002452 A096261 this_sequence
>>> A110410 A051789 A015467
>>> %Y A015455 Adjacent sequences: A015452 A015453 A015454 this_sequence
>>> A015456 A015457 A015458
>>> %K A015455 nonn,easy
>>> %O A015455 0,3
>>> %A A015455 Olivier Gerard (ogerard(AT)ext.jussieu.fr)
>>> %E A015455 Edited by M.F.Hasler (www.univ-ag.fr/~mhasler), Jun 14 2008
>>
>>
>
>
>

rg> From seqfan-owner at ext.jussieu.fr  Sun Jun 15 21:03:06 2008
rg> From: Richard Guy <rkg at cpsc.ucalgary.ca>
rg> To: Alexander Povolotsky <apovolot at gmail.com>
rg> cc: Maximilian Hasler <maximilian.hasler at gmail.com>,
rg>         "Neil J. A. Sloane" <njas at research.att.com>, seqfans at seqfan.net,
rg>         Sequence Fans <seqfan at ext.jussieu.fr>
rg> Subject: Re: EDITED A015455
rg> 
rg> Further good work.  Here are some more sequences. The first
rg> two are A001906 and the anagram A006190, but these don't
rg> have negative terms and are not mentioned as being Gaussian
rg> Markov numbers.  I hope I've got the signs `right' in the
rg> others :-
rg> 
rg> ..., -144, -55, -21, -8, -3, -1, 0, 1, 3, 8, 21, 55, 144, ...
rg> 
rg>   ..., -360, 109, -33, 10, -3, 1, 0, 1, 3, 10, 33, 109, 360, ...
rg> 
rg>            ..., -12141, 505, -21, 1, 3, 73, 1755, 42193, ...

The sequence 1,3,73,1755,... with offset 0 is classified as
a(n)= 24a(n-1)+a(n-2);
o.g.f. (-1+21*x)/(-1+24*x+x^2)
continues
1,3,73,1755,42193,1014387,24387481,586313931,14095921825,338888437731,8147418427369,195876930694587,4709193755097457,113216527053033555,2721905843027902777,65438956759722700203,1573256868076372707649,37823603790592667683779
a(n)=
1/290*145^(1/2)*(-9*(-12+145^(1/2))^(-n)+(-12+145^(1/2))^(-n)*145^(1/2)+9*(-1)
^n*(12+145^(1/2))^(-n)+(-1)^n*(12+145^(1/2))^(-n)*145^(1/2))

rg>            ..., -29637, 989, -33, 1, 3, -91, 2727, -81719, ...
The sequence 1,3,-91,2727 etc could be classified
a(n)=-30a(n-1)-a(n-2),
o.g.f. (1+33*x)/(1+30*x+x^2),
continues
1,3,-91,2727,-81719,2448843,-73383571,2199058287,-65898365039,1974751892883,-59176658421451,1773325000750647,-53140573364097959,1592443875922188123,-47720175704301545731,1430012827253124183807,-42852664641889423968479,1284149926429429594870563,-38481645128240998422148411,1153165203920800523069581767
a(n)=1/56*14^(1/2)*(9*(-1)^n*(15+4*14^(1/2))^(-n)+2*(-1)^n*(15+4*14^(1/2))^(-n)*14^
(1/2)-9*(-15+4*14^(1/2))^(-n)+2*(-15+4*14^(1/2))^(-n)*14^(1/2))

rg>              -, +, -, 3466, -55, 1, 8, 505, 31823, ...
The sequence 1,8,505,...  appears to be
a(n)=63a(n-1)+a(n-2),
o.g.f. (-1+55*x)/(-1+63*x+x^2);
continues
1,8,505,31823,2005354,126369125,7963260229,501811763552,31622104364005,1992694386695867,125571368466203626,7912988907757524305,498643872557190234841,31422476960010742319288,1980114692353233956349985,
a(n)=1/7946*3973^(1/2)*2^n*(47*(-1)^n*(63+3973^(1/2))^(-n)+(-1)^n*(63+3973^(1/2))^(
-n)*3973^(1/2)-47*(-63+3973^(1/2))^(-n)+(-63+3973^(1/2))^(-n)*3973^(1/2))

rg>        -, +, -, 97921, -989, 10, 1, 109, 10792, ...
1,109,10792 with offset 0 seems to be
a(n)=99a(n-1)+a(n-2),
o.g.f. (-1-10*x)/(-1+99*x+x^2);
continues
1,109,10792,1068517,105793975,10474672042,1037098326133,102683208959209,10166674785287824,1006603486952453785,99663911883078212539,9867733879911695495146,977005318023140932231993,96733394218170863986462453,
a(n)= 1/19610*9805^(1/2)*2^n*(119*(-99+9805^(1/2))^(-n)+(-99+9805^(1/2))^(-n)*9805^(
1/2)+119*(-1)^(n+1)*(99+9805^(1/2))^(-n)+(-1)^n*(99+9805^(1/2))^(-n)*9805^(1/2
))