Thank you, Joerg. That was interesting. Does your
generating function hypothesis hold when I extend the sequence by
another 5 values, to a(n) = 0, 0, 1, 1, 3, 1, 7, 4,<br>
14, 17, 28, 54, 66, 143, 182, 350, 687, 987, 2611, 4298 ? Again,
assuming that I got the elementary algebra and arithmetic correct.<br>
<br>
If so, I should submit this sequence as by the two us!<br>
<br>
Eisenstein-Fibonacci sequences. Cube root of unity analogues of square-root of unity A014291 Imaginary Rabbits.<br>
<br>
[update of 1 Nov 06]<br>
<br>
Let b(0) = w, b(1) = w^2, b(n) = w*b(n-1) + b(n-2),<br>
where w = omega = (-1 + i*sqrt(3))/2 and w^2 = omega^2<br>
= (-1 - i*sqrt(3))/2.<br>
<br>
There are various integer sequences derived from this<br>
complex sequence (such as the real part, the imaginary<br>
part, the polynomial coefficient is of 1, w, and<br>
w^2)Unless my arithmetic by hand is in error again,<br>
is:<br>
<br>
n b(n)<br>
0 w<br>
1 w^2<br>
2 w(w^2) + w = w^3 + w = 1 + w.<br>
3 w(1 + w) + w^2 = w + w^2 + w^2 = 1 + 2w^2.<br>
4 w(1 + 2w^2) + 1 + w = w + 2w^3 + 1 + w = 3 + 2w.<br>
5 w(3 + 2w) + 1 + 2w^2 = 3w + 2w^2 + 1 + 2w^2.<br>
= 1 + 3w + 4w^2.<br>
6 w(1 + 3w + 4w^2) + 3 + 2w = w + 3w^2 + 4w^3 + 3 +<br>
2w<br>
= 7 + 3w + 3w^2.<br>
7 w(7 + 3w + 3w^2) + 1 + 3w + 4w^2 = 7w + 3w^2 + 3w^3<br>
<br>
+ 1 + 3w + 4w^2 = 4 + 10w + 7w^2.<br>
8 w(4 + 10w + 7w^2) + 7 + 3w + 3w^2 = 4w + 10w^2 +<br>
7w^3 + 7 + 3w + 3w^2 = 14 + 7w + 13w^2.<br>
9 w(14 + 7w + 13w^2) + 4 + 10w + 7w^2 = 14w + 7w^2 +<br>
13w^3 + 4 + 10w + 7w^2 = 17 + 24w + 14w^2.<br>
10 w(17 + 24w + 14w^2) + 14 + 7w + 13w^2 = 17w + 24w^2<br>
+ 14w^3 + 14 + 7w + 13w^2 = 28 + 24w + 37w^2.<br>
11 w(28 + 24w + 37w^2) + 17 + 24w + 14w^2 = 28w +<br>
24w^2 + 37w^3 + 17 + 24w + 14w^2 = 54 + 52w +<br>
38w^2.<br>
12 w(54 + 52w + 38w^2) + 28 + 24w + 37w^2 = 54w +<br>
52w^2 + 38w^3 + 28 + 24w + 37w^2 = 66 + 78w +<br>
89w^2.<br>
13 w(66 + 78w + 89w^2) + 54 + 52w + 38w^2 =<br>
66w + 78w^2 + 89w^3 + 54 + 52w + 38w^2 =<br>
143 + 118w + 116w^2.<br>
14 w(143 + 118w + 116w^2) + 66 + 78w + 89w^2 =<br>
143w + 118w^2 + 116w^3 + 66 + 78w + 89w^2 =<br>
182 + 221w + 207w^2.<br>
15 w(182 + 221w + 207w^2) + 143 + 118w + 116w^2 =<br>
182w + 221w^2 + 207w^3 + 143 + 118w + 116w^2 =<br>
350 + 300w + 337w^2.<br>
<br>
16 w(350 + 300w + 337w^2) + 350 + 300w + 337w^2 =<br>
350w + 300w^2 + 337w^3 + 350 + 300w + 337w^2 =<br>
687 + 650w + 637w^2.<br>
<br>
17 w(687 + 650w + 637w^2) + 350 + 300w + 337w^2 =<br>
687w + 650w^2 + 637w^3 + 350 + 300w + 337w^2 =<br>
987 + 987w + 987w^2.<br>
<br>
18 w(987 + 987w + 987w^2) + 687 + 650w + 637w^2 =<br>
987w + 987w^2 + 987w^3 + 687 + 650w + 637w^2 =<br>
1674 + 1637w + 1624w^2.<br>
<br>
19 w(1674 + 1637w + 1624w^2) + 987 + 987w + 987w^2 =<br>
1674w + 1637w^2 + 1624w^3 + 987 + 987w + 987w^2<br>
=<br>
2611 + 2661w + 2624w^2.<br>
<br>
20 w(2611 + 2661w + 2624w^2) + 1674 + 1637w + 1624w^2<br>
=<br>
2611w + 2661w^2 + 2624w^3 + 1674 + 1637w +<br>
1624w^2 =<br>
4298 + 4248w + 4285w^2.<br>
<br>
<br>
The coefficients of 1 = a(n) = 0, 0, 1, 1, 3, 1, 7, 4,<br>
14, 17, 28, 54, 66, 143, 182, 350, 687, 987, 2611,<br>
4298<br>
<br>
which does not seem to be in OEIS.<br>
<br>
To excerpt my comment in A107890 (and some other<br>
seqs):<br>
<br>
"Eisenstein integers are of the form a + b*omega,<br>
where a and b are ordinary integers, and omega = (-1 +<br>
i*sqrt(3))/2 is a cube root of 1, the other cube roots<br>
of 1 being 1 and omega^2 = (-1 - i*sqrt(3))/2.<br>
Eisenstein integers are complex numbers that are also<br>
members of the imaginary quadratic field Q(sqrt -3) =<br>
Z[omega]. The sums, differences, and products of<br>
Eisenstein integers are other Eisenstein integers."<br>
<br>
Other sequences come from:<br>
b(0) = w, b(1) = w^2, b(n) = b(n-1) + w*b(n-2),<br>
and from the O, J notation of Dörrie (1965) in<br>
Weisstein, Eric W. "Eisenstein Integer." From<br>
MathWorld--A Wolfram Web Resource.<br>
<a onclick="return top.js.OpenExtLink(window,event,this)" href="http://mathworld.wolfram.com/EisensteinInteger.html" target="_blank">http://mathworld.wolfram.com/EisensteinInteger.html</a><br>
<br>
<br>
Eisenstein-Fib.doc<br>
<br>
=== end 2nd email ===<br><br><div><span class="gmail_quote">On 11/1/06, <b class="gmail_sendername">Joerg Arndt</b> <<a href="mailto:arndt@jjj.de">arndt@jjj.de</a>> wrote:</span><blockquote class="gmail_quote" style="border-left: 1px solid rgb(204, 204, 204); margin: 0pt 0pt 0pt 0.8ex; padding-left: 1ex;">
* Jonathan Post <<a href="mailto:jvospost3@gmail.com">jvospost3@gmail.com</a>> [Nov 01. 2006 10:14]:<br>> Eisenstein-Fibonacci sequences. Cube root of unity<br>> analogues of square-root of unity A014291 Imaginary
<br>> Rabbits.<br>><br>> Let b(0) = w, b(1) = w^2, b(n) = w*b(n-1) + b(n-2),<br>> where w = omega = (-1 + i*sqrt(3))/2 and w^2 = omega^2<br>> = (-1 - i*sqrt(3))/2.<br>><br>> [...]<br>> The coefficients of 1 = a(n) = 0, 0, 1, 1, 3, 1, 7, 4,
<br>> 14, 17, 28, 54, 66, 143, 182, 350, ...<br>><br>> which does not seem to be in OEIS.<br><br>Ralf Stephan's ggf() says that the OGF may be:<br><br>? g=ggf([1, 1, 3, 1, 7, 4,14, 17, 28, 54, 66, 143, 182, 350])
<br>(x^5 - 3*x^3 + x + 1)/(-x^6 + 3*x^4 - x^3 - 3*x^2 + 1)<br>? factor(g)<br><br>[x - 1 1]<br><br>[x^4 + x^3 - 2*x^2 - 2*x - 1 1]<br><br>[x^2 + x - 1 -1]<br><br>[x^4 - x^3 - x^2 + x + 1 -1]<br><br><br>> [...]<br><br><br>
</blockquote></div><br>