Superseeker error?
Creighton Dement
crowdog at crowdog.de
Tue Mar 1 19:47:29 CET 2005
Dear Seqfans,
Well, I've had a bit of a fever over the last few days and am not fully
recovered (so if I'm overlooking something obvious here I'd like to say
it's because of that..).
Superseeker reports:
Report on [ 0,1,3,10,36,133,495,1846,6888,25705,95931,358018,1336140]:
Transformation T020 ( u[j+3]-3*u[j+2]-3*u[j+1]+u[j] ) gave a match with:
%I A003500 M1278
%S A003500
2,4,14,52,194,724,2702,10084,37634,140452,524174,1956244,7300802,
%N A003500 a(n) = 4a(n-1) - a(n-2).
%C A003500 a(n) gives values of x satisfying x^2 - 3*y^2 = 4;
corresponding y values are given by 2*A001353(n).
%C A003500 If M is any given term of the sequence, then the next one is
2M + sqrt(3M^2 - 12). - Lekraj Beedassy (boodhiman(AT)yahoo.com), Feb 18
2002
However, for my sequence below I get:
u[j+3]-3*u[j+2]-3*u[j+1]+u[j] = -2 forall j
What is it I've overlooked?
Sincerely,
Creighton
%I A000001
%S A000001 1, 3, 10, 36, 133, 495, 1846, 6888, 25705, 95931, 358018,
1336140, 4986541, 18610023, 69453550, 259204176, 967363153, 3610248435,
13473630586, 50284273908, 187663465045, 700369586271, 2613814880038,
%N A000001 a(n) = a(n-3) - 5a(n-2) + 5a(n-1), a(0) = 1, a(1) = 3, a(2) =
10
%C A000001 A floretion-generated sequence resulting from a particular
transform of the periodic sequence (-1,1).
%H A000001 C. Dement, <a
href="http://www.crowdog.de/20801/home.html">Force
Transforms</a>.
%F A000001 2a(n) - A001834(n) = (-1)^(n+1) ;
a(n) = 4a(n-1) - a(n) - 1 ;
G.f. x(2x-1)/((x-1)(x^2-4x+1))
Superseeker results:
a(n+2) - 2a(n+1) + a(n) = A001834(n+1) (from this and the first relation
involving A001834 it follows that 4a(n+1) - a(n+2) - a(n) = (-1)^n as
well as
the recurrence relation given for A001834 ) ;
a(n+1) - a(n) = A001075(n+1) (Chebyshev's T(n,x) polynomials evaluated
at x=2) ; a(n+2) - a(n) = A082841(n+1)
%o A000001 Floretion Algebra Multiplication Program, FAMP Code:
.5em[J*]forseq[ .25( 'i + 'j + 'k + i' + j' + k' + 'ii' + 'jj' + 'kk' +
'ij' + 'ik' + 'ji' + 'jk' + 'ki' + 'kj' + e ) ], em[J]forseq = A001834,
vesforseq = (1,-1,1,-1). ForType 1A. Identity used: em[J]forseq +
em[J*]forseq = vesforseq.
%Y A000001 Cf. A001834, A001075, A082841.
%O A000001 0
%K A000001 ,nonn,
%A A000001 Creighton Dement (crowdog at crowdog.de), Mar 01 2005
-- Sonja is bigger than me and I am bigger than Sonja! (my 3-year-old
daughter expressing that she and her friend Sonja are exactly the same
height)
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