A094943

[Gottfried Helms]

Another surprising(?) coincidence ...

I'm currently looking at sequences, which arise from rules
like

   (a + b)^n   + (a - b)^n
  -----------------------------
              2

where a is usually an integer, or a fraction and b may be an irrational
or even an imaginary number. Such rules produce known sequences
of recurrence-order two, interesting patterns of prime-factorizations,
and I thought to try some generalization.

My current generalization is, to view +b and -b in the above
terms as b*w0 and b*w1, where w0 and w1 are the two complex square-roots
of 1  (=1 and -1),

and to consider cases of higher complex roots. An extended case of order 3
could be ( w = (-1 + sqrt(2)+i)/2,       (w0=w^0,w1=w, w2 = w^2 )

   (a + b + c)^n   + (a + b*w1 + c*w2)^n  + (a + b*w2 + c*w4)^n
  --------------------------------------------------------------
                        3

which leads to real numbers for all n>0, and for proper selection of a,b,c
it also produces an integer sequence, like in the case of order 2 the
known Fibonacci and Lucas-sequences.

The generalization looks like (for order p)

    _p-1    (   _ p-1              ) n
    \       (   \                  )
     )      (    ) (a_i * w ^ij)   )
    /_      (   /_                 )
    j=0     (   i=0                )

Anyway - the numbers I got for the first parameters (a,b,c)=(1,2,3)
of order 3 are just the numbers of the sequence A094943 . Maybe that is of
interest; for me this connection seems to be useful, since to compute
the coefficients of the resulting real terms is a bit tedious.

If the matrix-method given in the example below can be extended to higher
order, then this would be a much more handy rule to compute the coefficients
than to employ a symbolic algebra program and evaluate the nominator for
higher orders ...


Gottfried Helms


ID Number: A094943
URL:       http://www.research.att.com/projects/OEIS?Anum=A094943
Sequence:  1,13,72,429,2601,15534,93339,559845,3359232,20155473,
           120932109,725594598,4353563943,26121388761,156728328192,
           940369966965,5642219821473,33853318876350,203119913356515,
           1218719480001309
Name:      A sequence generated from a semi-magic square.
Comments:  The 3 rows: 1 3 2, 2 1 3, and 3 2 1 form a semi-magic square; with rows,
              columns and a diagonal having a sum of 6. a(n)/a(n-1) tends to 6, an
              eigenvalue of the matrix. E.g.: a(7)/a(6) = 93339/15534 = 6.0086...
              A094944 uses the same format and operations but has different terms.
Formula:   Let [1 3 2 / 2 1 3 / 3 2 1] = the 3 X 3 matrix M. Take M^n * [1 0 0] = [p q r]; then a(n) =
              p.
Example:   a(4) = 429 since M^4 * [1 0 0] = [429 q r]
Math'ca:   a[n_] := (MatrixPower[{{1, 3, 2}, {2, 1, 3}, {3, 2, 1}}, n].{{1}, {0},
              {0}})[[1, 1]]; Table[ a[n], {n, 10}] (from Robert G. Wilson v May 29
              2004)
See also:  Cf. A094944.
           Adjacent sequences: A094940 A094941 A094942 this_sequence A094944
              A094945 A094946
           Sequence in context: A096913 A026916 A106173 this_sequence A000470
              A107141 A097460
Keywords:  nonn
Offset:    1
Author(s): Gary W. Adamson (qntmpkt(AT)yahoo.com), May 25 2004
Extension: Edited and extended by Robert G. Wilson v (rgwv(AT)rgwv.com), May 29
              2004
S