Fibbinary sequence

[Gottfried Helms]

A courious result by chance:

A003714	Fibbinary numbers:
       if n = F_i1+F_i2+...+F_ik is the Zeckendorf representation of n
       (i.e. write n in Fibonacci number system)
       then
         a(n) = 2^{i1-2}+2^{i2-2}+...+2^{ik-2}.

	0, 1, 2, 4, 5, 8, 9, 10, 16, 17, 18, 20, 21, 32, 33, 34, 36, 37,
      40, 41, 42, 64, 65, 66, 68, 69, 72, 73, 74, 80, 81, 82, 84, 85,
     128, 129, 130, 132, 133, 136, 137, 138, 144, 145, 146, 148, 149,
     160, 161, 162, 164, 165, 168, 169, 170, 256, 257, 258, 260

I was checking some variants of the Pascal-matrix and got the
following
Let Px =
      1       .    .     .     .    .  .  .
      1      -1    .     .     .    .  .  .
    3/2      -3    1     .     .    .  .  .
      4      -9    6    -1     .    .  .  .
     15     -35   30   -10     1    .  .  .
  137/2  -345/2  165   -75    15   -1  .  .
 1491/4 -2009/2 1050  -560 315/2  -21  1  .
   2374   -6720 7588 -4515  1540 -294 28 -1

then the index r of the rows, r=1..inf(??) , which contain only integers, is A003714
      1, 2, 4, 5, 8, 9, 10, 16, 17, 18, 20, 21, 32, 33, 34, 36, 37,
     40, 41, 42, 64,

(although I checked this only up to N=64)


Explanation: -----------------------------------------------------------

The matrix Px is constructed as a variant of Pj
Pj:
 1  .  .   .  .   . .  .
 1 -1  .   .  .   . .  .
 1 -2  1   .  .   . .  .
 1 -3  3  -1  .   . .  .
 1 -4  6  -4  1   . .  .
 1 -5 10 -10  5  -1 .  .
 1 -6 15 -20 15  -6 1  .
 1 -7 21 -35 35 -21 7 -1

which is the column-signed version of the Pascalmatrix P
   Pj = P*J
where J is diagonal unit with alternating signs (diag(1,-1,1,-1,...))

The matrix S2 of the stirling-numbers of 2'nd kind
 1   .   .    .    .   .  . .
 1   1   .    .    .   .  . .
 1   3   1    .    .   .  . .
 1   7   6    1    .   .  . .
 1  15  25   10    1   .  . .
 1  31  90   65   15   1  . .
 1  63 301  350  140  21  1 .
 1 127 966 1701 1050 266 28 1

and S1 of the stirling numbers of first kind (where also S1 = S2^-1)
     1     .      .    .     .   .   . .
    -1     1      .    .     .   .   . .
     2    -3      1    .     .   .   . .
    -6    11     -6    1     .   .   . .
    24   -50     35  -10     1   .   . .
  -120   274   -225   85   -15   1   . .
   720 -1764   1624 -735   175 -21   1 .
 -5040 13068 -13132 6769 -1960 322 -28 1

with the eigenvalues Z(1) = diag(1,1/2,1/3,1/4) such that

   G = S2 * Z(1) * S2^-1

G:
     1     .    .     .   .    .   .   .
   1/2   1/2    .     .   .    .   .   .
   1/6   1/2  1/3     .   .    .   .   .
     0   1/4  1/2   1/4   .    .   .   .
 -1/30     0  1/3   1/2 1/5    .   .   .
     0 -1/12    0  5/12 1/2  1/6   .   .
  1/42     0 -1/6     0 1/2  1/2 1/7   .
     0  1/12    0 -7/24   0 7/12 1/2 1/8

form an eigensystem of G.
G ist interesting as it contains the bernoulli-
numbers, and coefficients of the integrals of the
bernoulli-functions.



Also G is Eigenmatrix of the signed Pascal-matrix Pj,
with the eigenvalues J = diag(1,-1,1,-1,...)

  Pj = G * J * G^-1

and Pj is used in divergent Euler-summation, Hasse's
zeta-summation and elsewhere.




Now I tried a variant of P by varying G, where the eigenvalues
are not Z(1), but are V(1/2) = diag(1,1/2,1/4,1/8,....) for some
analyses about summing zeta series, since Z(1) and V(1/2) are
closely related in this subject and complementary/mutually supplementary
in certain manners.

Now with the G-variant

  Gx  = S2 * V(1/2) * S2^-1
Gx:
     1      .       .        .      .      .    .     .
   1/2    1/2       .        .      .      .    .     .
     0    3/4     1/4        .      .      .    .     .
  -1/4    3/8     3/4      1/8      .      .    .     .
     0   -5/8   15/16      5/8   1/16      .    .     .
   1/2 -15/16  -15/32    45/32  15/32   1/32    .     .
     0    7/4 -105/32    35/64 105/64  21/64 1/64     .
 -17/8 147/32    7/32 -735/128  35/16 105/64 7/32 1/128

I come to

  Px =         Gx               * J *       Gx^-1
     =  ( S2 * V(1/2) * S2^-1 ) * J * (S2 * V(2) * S2^-1)

(while Pj is
  Pj =         G                * J *       G ^-1
     =  ( S2 * Z(-1) * S2^-1 )  * J * (S2 * Z(1) * S2^-1)
  )

and as shown above:

Px:
      1       .    .     .     .    .  .  .
      1      -1    .     .     .    .  .  .
    3/2      -3    1     .     .    .  .  .
      4      -9    6    -1     .    .  .  .
     15     -35   30   -10     1    .  .  .
  137/2  -345/2  165   -75    15   -1  .  .
 1491/4 -2009/2 1050  -560 315/2  -21  1  .
   2374   -6720 7588 -4515  1540 -294 28 -1

and the rows containing only integers, are uniquely
indicated by A003714, up to r=64, where the rownumber r
starts with 1.

Don't have an idea, how this coincidence could be interesting -
I thought I'd just sentd the observation..

Regards -

Gottfried Helms