[seqfan] Re: plastic number base

Wed Jul 10 22:09:48 CEST 2013

Dear Seqfans,

My follow-up question apparently got mangled. It's no fault of the
moderator, I simply have a medical condition that often causes me to blank
out in the middle of performing some task. Anyway, here is the question
that I meant to ask (along with a little more thrown in).

. Suppose you take the sequence of exponents and reinterpret then as
indices of Padovan numbers. Then there are 9 offsets that seem to work to
give you a Zeckendorf or Martin Bunder-style representation:(-21,-8,
-7,-2,0,3,5,6,7). Is this peculiar, or is there some reason to expect this?

With the lowest offset being -21 and the highest being 7, one might guess
that something is going on with a cycle of length 28. So further below, I
show some examples of all the offsets from -21 to 7, including the ones
that "don't work." Clearly there's a pattern here, but I have no
explanation.

1 [0]
    -21 a_{-21} [1]
    -8 a_{-8} [1]
    -7 a_{-7} [1]
    -2 a_{-2} [1]
    0 a_{0} [1]
    3 a_{3} [1]
    5 a_{5} [1]
    6 a_{6} [1]
    7 a_{7} [1]

2 [2, -5]
    -21 a_{-26}+a_{-19} [-14, 16]
    -8 a_{-13}+a_{-6} [-2, 4]
    -7 a_{-12}+a_{-5} [2, 0]
    -2 a_{-7}+a_{0} [1, 1]
    0 a_{-5}+a_{2} [0, 2]
    3 a_{-2}+a_{5} [1, 1]
    5 a_{0}+a_{7} [1, 1]
    6 a_{1}+a_{8} [2, 0]
    7 a_{2}+a_{9} [2, 0]

3 [3, -2, -8]
    -21 a_{-29}+a_{-23}+a_{-18} [11, 25, -33]
    -8 a_{-16}+a_{-10}+a_{-5} [2, 4, -3]
    -7 a_{-15}+a_{-9}+a_{-4} [-1, -3, 7]
    -2 a_{-10}+a_{-4}+a_{1} [0, -1, 4]
    0 a_{-8}+a_{-2}+a_{3} [1, 1, 1]
    3 a_{-5}+a_{1}+a_{6} [1, 0, 2]
    5 a_{-3}+a_{3}+a_{8} [2, 1, 0]
    6 a_{-2}+a_{4}+a_{9} [2, 0, 1]
    7 a_{-1}+a_{5}+a_{10} [3, 1, -1]

4 [4, -1, -7, -13]
    -21 a_{-34}+a_{-28}+a_{-22}+a_{-17} [-4, -15, 49, -26]
    -8 a_{-21}+a_{-15}+a_{-9}+a_{-4} [-1, -3, 7, 1]
    -7 a_{-20}+a_{-14}+a_{-8}+a_{-3} [0, 1, -7, 10]
    -2 a_{-15}+a_{-9}+a_{-3}+a_{2} [0, 0, -3, 7]
    0 a_{-13}+a_{-7}+a_{-1}+a_{4} [0, -1, 1, 4]
    3 a_{-10}+a_{-4}+a_{2}+a_{7} [1, 0, -1, 4]
    5 a_{-8}+a_{-2}+a_{4}+a_{9} [2, 0, 1, 1]
    6 a_{-7}+a_{-1}+a_{5}+a_{10} [3, 1, -1, 1]
    7 a_{-6}+a_{0}+a_{6}+a_{11} [4, 1, 1, -2]

5 [5, -1, -7, -13]
    -21 a_{-34}+a_{-28}+a_{-22}+a_{-16} [-3, -15, 49, -26]
    -8 a_{-21}+a_{-15}+a_{-9}+a_{-3} [0, -3, 7, 1]
    -7 a_{-20}+a_{-14}+a_{-8}+a_{-2} [1, 1, -7, 10]
    -2 a_{-15}+a_{-9}+a_{-3}+a_{3} [1, 0, -3, 7]
    0 a_{-13}+a_{-7}+a_{-1}+a_{5} [1, -1, 1, 4]
    3 a_{-10}+a_{-4}+a_{2}+a_{8} [2, 0, -1, 4]
    5 a_{-8}+a_{-2}+a_{4}+a_{10} [3, 0, 1, 1]
    6 a_{-7}+a_{-1}+a_{5}+a_{11} [4, 1, -1, 1]
    7 a_{-6}+a_{0}+a_{6}+a_{12} [5, 1, 1, -2]

6 [6, -2, -13]
    -21 a_{-34}+a_{-23}+a_{-15} [7, 25, -26]
    -8 a_{-21}+a_{-10}+a_{-2} [1, 4, 1]
    -7 a_{-20}+a_{-9}+a_{-1} [-1, -3, 10]
    -2 a_{-15}+a_{-4}+a_{4} [0, -1, 7]
    0 a_{-13}+a_{-2}+a_{6} [1, 1, 4]
    3 a_{-10}+a_{1}+a_{9} [2, 0, 4]
    5 a_{-8}+a_{3}+a_{11} [4, 1, 1]
    6 a_{-7}+a_{4}+a_{12} [5, 0, 1]
    7 a_{-6}+a_{5}+a_{13} [7, 1, -2]

7 [6, 1, -5, -13]
    -21 a_{-34}+a_{-26}+a_{-20}+a_{-15} [7, 10, 16, -26]
    -8 a_{-21}+a_{-13}+a_{-7}+a_{-2} [1, 1, 4, 1]
    -7 a_{-20}+a_{-12}+a_{-6}+a_{-1} [-1, -2, 0, 10]
    -2 a_{-15}+a_{-7}+a_{-1}+a_{4} [0, -1, 1, 7]
    0 a_{-13}+a_{-5}+a_{1}+a_{6} [1, 0, 2, 4]
    3 a_{-10}+a_{-2}+a_{4}+a_{9} [2, 0, 1, 4]
    5 a_{-8}+a_{0}+a_{6}+a_{11} [4, 1, 1, 1]
    6 a_{-7}+a_{1}+a_{7}+a_{12} [5, 1, 0, 1]
    7 a_{-6}+a_{2}+a_{8}+a_{13} [7, 2, 0, -2]

8 [7, -1, -9, -18]
    -21 a_{-39}+a_{-30}+a_{-22}+a_{-14} [-7, -15, -7, 37]
    -8 a_{-26}+a_{-17}+a_{-9}+a_{-1} [-1, -3, -4, 16]
    -7 a_{-25}+a_{-16}+a_{-8}+a_{0} [1, 1, -3, 9]
    -2 a_{-20}+a_{-11}+a_{-3}+a_{5} [1, 0, -3, 10]
    0 a_{-18}+a_{-9}+a_{-1}+a_{7} [1, -1, -3, 11]
    3 a_{-15}+a_{-6}+a_{2}+a_{10} [3, 0, -2, 7]
    5 a_{-13}+a_{-4}+a_{4}+a_{12} [5, 0, -1, 4]
    6 a_{-12}+a_{-3}+a_{5}+a_{13} [7, 1, 0, 0]
    7 a_{-11}+a_{-2}+a_{6}+a_{14} [9, 1, 1, -3]

      All offsets from -21 to 7:

1 [0]
    -21 a_{-21} [1] 1
    -20 a_{-20} [10] 10
    -19 a_{-19} [-14] -14
    -18 a_{-18} [11] 11
    -17 a_{-17} [-4] -4
    -16 a_{-16} [-3] -3
    -15 a_{-15} [7] 7
    -14 a_{-14} [-7] -7
    -13 a_{-13} [4] 4
    -12 a_{-12} [0] 0
    -11 a_{-11} [-3] -3
    -10 a_{-10} [4] 4
    -9 a_{-9} [-3] -3
    -8 a_{-8} [1] 1
    -7 a_{-7} [1] 1
    -6 a_{-6} [-2] -2
    -5 a_{-5} [2] 2
    -4 a_{-4} [-1] -1
    -3 a_{-3} [0] 0
    -2 a_{-2} [1] 1
    -1 a_{-1} [-1] -1
    0 a_{0} [1] 1
    1 a_{1} [0] 0
    2 a_{2} [0] 0
    3 a_{3} [1] 1
    4 a_{4} [0] 0
    5 a_{5} [1] 1
    6 a_{6} [1] 1
    7 a_{7} [1] 1

2 [2, -5]
    -21 a_{-26}+a_{-19} [-14, 16] 2
    -20 a_{-25}+a_{-18} [11, 9] 20
    -19 a_{-24}+a_{-17} [-4, -24] -28
    -18 a_{-23}+a_{-16} [-3, 25] 22
    -17 a_{-22}+a_{-15} [7, -15] -8
    -16 a_{-21}+a_{-14} [-7, 1] -6
    -15 a_{-20}+a_{-13} [4, 10] 14
    -14 a_{-19}+a_{-12} [0, -14] -14
    -13 a_{-18}+a_{-11} [-3, 11] 8
    -12 a_{-17}+a_{-10} [4, -4] 0
    -11 a_{-16}+a_{-9} [-3, -3] -6
    -10 a_{-15}+a_{-8} [1, 7] 8
    -9 a_{-14}+a_{-7} [1, -7] -6
    -8 a_{-13}+a_{-6} [-2, 4] 2
    -7 a_{-12}+a_{-5} [2, 0] 2
    -6 a_{-11}+a_{-4} [-1, -3] -4
    -5 a_{-10}+a_{-3} [0, 4] 4
    -4 a_{-9}+a_{-2} [1, -3] -2
    -3 a_{-8}+a_{-1} [-1, 1] 0
    -2 a_{-7}+a_{0} [1, 1] 2
    -1 a_{-6}+a_{1} [0, -2] -2
    0 a_{-5}+a_{2} [0, 2] 2
    1 a_{-4}+a_{3} [1, -1] 0
    2 a_{-3}+a_{4} [0, 0] 0
    3 a_{-2}+a_{5} [1, 1] 2
    4 a_{-1}+a_{6} [1, -1] 0
    5 a_{0}+a_{7} [1, 1] 2
    6 a_{1}+a_{8} [2, 0] 2
    7 a_{2}+a_{9} [2, 0] 2

3 [3, -2, -8]
    -21 a_{-29}+a_{-23}+a_{-18} [11, 25, -33] 3
    -20 a_{-28}+a_{-22}+a_{-17} [-4, -15, 49] 30
    -19 a_{-27}+a_{-21}+a_{-16} [-3, 1, -40] -42
    -18 a_{-26}+a_{-20}+a_{-15} [7, 10, 16] 33
    -17 a_{-25}+a_{-19}+a_{-14} [-7, -14, 9] -12
    -16 a_{-24}+a_{-18}+a_{-13} [4, 11, -24] -9
    -15 a_{-23}+a_{-17}+a_{-12} [0, -4, 25] 21
    -14 a_{-22}+a_{-16}+a_{-11} [-3, -3, -15] -21
    -13 a_{-21}+a_{-15}+a_{-10} [4, 7, 1] 12
    -12 a_{-20}+a_{-14}+a_{-9} [-3, -7, 10] 0
    -11 a_{-19}+a_{-13}+a_{-8} [1, 4, -14] -9
    -10 a_{-18}+a_{-12}+a_{-7} [1, 0, 11] 12
    -9 a_{-17}+a_{-11}+a_{-6} [-2, -3, -4] -9
    -8 a_{-16}+a_{-10}+a_{-5} [2, 4, -3] 3
    -7 a_{-15}+a_{-9}+a_{-4} [-1, -3, 7] 3
    -6 a_{-14}+a_{-8}+a_{-3} [0, 1, -7] -6
    -5 a_{-13}+a_{-7}+a_{-2} [1, 1, 4] 6
    -4 a_{-12}+a_{-6}+a_{-1} [-1, -2, 0] -3
    -3 a_{-11}+a_{-5}+a_{0} [1, 2, -3] 0
    -2 a_{-10}+a_{-4}+a_{1} [0, -1, 4] 3
    -1 a_{-9}+a_{-3}+a_{2} [0, 0, -3] -3
    0 a_{-8}+a_{-2}+a_{3} [1, 1, 1] 3
    1 a_{-7}+a_{-1}+a_{4} [0, -1, 1] 0
    2 a_{-6}+a_{0}+a_{5} [1, 1, -2] 0
    3 a_{-5}+a_{1}+a_{6} [1, 0, 2] 3
    4 a_{-4}+a_{2}+a_{7} [1, 0, -1] 0
    5 a_{-3}+a_{3}+a_{8} [2, 1, 0] 3
    6 a_{-2}+a_{4}+a_{9} [2, 0, 1] 3
    7 a_{-1}+a_{5}+a_{10} [3, 1, -1] 3

4 [4, -1, -7, -13]
    -21 a_{-34}+a_{-28}+a_{-22}+a_{-17} [-4, -15, 49, -26] 4
    -20 a_{-33}+a_{-27}+a_{-21}+a_{-16} [-3, 1, -40, 82] 40
    -19 a_{-32}+a_{-26}+a_{-20}+a_{-15} [7, 10, 16, -89] -56
    -18 a_{-31}+a_{-25}+a_{-19}+a_{-14} [-7, -14, 9, 56] 44
    -17 a_{-30}+a_{-24}+a_{-18}+a_{-13} [4, 11, -24, -7] -16
    -16 a_{-29}+a_{-23}+a_{-17}+a_{-12} [0, -4, 25, -33] -12
    -15 a_{-28}+a_{-22}+a_{-16}+a_{-11} [-3, -3, -15, 49] 28
    -14 a_{-27}+a_{-21}+a_{-15}+a_{-10} [4, 7, 1, -40] -28
    -13 a_{-26}+a_{-20}+a_{-14}+a_{-9} [-3, -7, 10, 16] 16
    -12 a_{-25}+a_{-19}+a_{-13}+a_{-8} [1, 4, -14, 9] 0
    -11 a_{-24}+a_{-18}+a_{-12}+a_{-7} [1, 0, 11, -24] -12
    -10 a_{-23}+a_{-17}+a_{-11}+a_{-6} [-2, -3, -4, 25] 16
    -9 a_{-22}+a_{-16}+a_{-10}+a_{-5} [2, 4, -3, -15] -12
    -8 a_{-21}+a_{-15}+a_{-9}+a_{-4} [-1, -3, 7, 1] 4
    -7 a_{-20}+a_{-14}+a_{-8}+a_{-3} [0, 1, -7, 10] 4
    -6 a_{-19}+a_{-13}+a_{-7}+a_{-2} [1, 1, 4, -14] -8
    -5 a_{-18}+a_{-12}+a_{-6}+a_{-1} [-1, -2, 0, 11] 8
    -4 a_{-17}+a_{-11}+a_{-5}+a_{0} [1, 2, -3, -4] -4
    -3 a_{-16}+a_{-10}+a_{-4}+a_{1} [0, -1, 4, -3] 0
    -2 a_{-15}+a_{-9}+a_{-3}+a_{2} [0, 0, -3, 7] 4
    -1 a_{-14}+a_{-8}+a_{-2}+a_{3} [1, 1, 1, -7] -4
    0 a_{-13}+a_{-7}+a_{-1}+a_{4} [0, -1, 1, 4] 4
    1 a_{-12}+a_{-6}+a_{0}+a_{5} [1, 1, -2, 0] 0
    2 a_{-11}+a_{-5}+a_{1}+a_{6} [1, 0, 2, -3] 0
    3 a_{-10}+a_{-4}+a_{2}+a_{7} [1, 0, -1, 4] 4
    4 a_{-9}+a_{-3}+a_{3}+a_{8} [2, 1, 0, -3] 0
    5 a_{-8}+a_{-2}+a_{4}+a_{9} [2, 0, 1, 1] 4
    6 a_{-7}+a_{-1}+a_{5}+a_{10} [3, 1, -1, 1] 4
    7 a_{-6}+a_{0}+a_{6}+a_{11} [4, 1, 1, -2] 4

5 [5, -1, -7, -13]
    -21 a_{-34}+a_{-28}+a_{-22}+a_{-16} [-3, -15, 49, -26] 5
    -20 a_{-33}+a_{-27}+a_{-21}+a_{-15} [7, 1, -40, 82] 50
    -19 a_{-32}+a_{-26}+a_{-20}+a_{-14} [-7, 10, 16, -89] -70
    -18 a_{-31}+a_{-25}+a_{-19}+a_{-13} [4, -14, 9, 56] 55
    -17 a_{-30}+a_{-24}+a_{-18}+a_{-12} [0, 11, -24, -7] -20
    -16 a_{-29}+a_{-23}+a_{-17}+a_{-11} [-3, -4, 25, -33] -15
    -15 a_{-28}+a_{-22}+a_{-16}+a_{-10} [4, -3, -15, 49] 35
    -14 a_{-27}+a_{-21}+a_{-15}+a_{-9} [-3, 7, 1, -40] -35
    -13 a_{-26}+a_{-20}+a_{-14}+a_{-8} [1, -7, 10, 16] 20
    -12 a_{-25}+a_{-19}+a_{-13}+a_{-7} [1, 4, -14, 9] 0
    -11 a_{-24}+a_{-18}+a_{-12}+a_{-6} [-2, 0, 11, -24] -15
    -10 a_{-23}+a_{-17}+a_{-11}+a_{-5} [2, -3, -4, 25] 20
    -9 a_{-22}+a_{-16}+a_{-10}+a_{-4} [-1, 4, -3, -15] -15
    -8 a_{-21}+a_{-15}+a_{-9}+a_{-3} [0, -3, 7, 1] 5
    -7 a_{-20}+a_{-14}+a_{-8}+a_{-2} [1, 1, -7, 10] 5
    -6 a_{-19}+a_{-13}+a_{-7}+a_{-1} [-1, 1, 4, -14] -10
    -5 a_{-18}+a_{-12}+a_{-6}+a_{0} [1, -2, 0, 11] 10
    -4 a_{-17}+a_{-11}+a_{-5}+a_{1} [0, 2, -3, -4] -5
    -3 a_{-16}+a_{-10}+a_{-4}+a_{2} [0, -1, 4, -3] 0
    -2 a_{-15}+a_{-9}+a_{-3}+a_{3} [1, 0, -3, 7] 5
    -1 a_{-14}+a_{-8}+a_{-2}+a_{4} [0, 1, 1, -7] -5
    0 a_{-13}+a_{-7}+a_{-1}+a_{5} [1, -1, 1, 4] 5
    1 a_{-12}+a_{-6}+a_{0}+a_{6} [1, 1, -2, 0] 0
    2 a_{-11}+a_{-5}+a_{1}+a_{7} [1, 0, 2, -3] 0
    3 a_{-10}+a_{-4}+a_{2}+a_{8} [2, 0, -1, 4] 5
    4 a_{-9}+a_{-3}+a_{3}+a_{9} [2, 1, 0, -3] 0
    5 a_{-8}+a_{-2}+a_{4}+a_{10} [3, 0, 1, 1] 5
    6 a_{-7}+a_{-1}+a_{5}+a_{11} [4, 1, -1, 1] 5
    7 a_{-6}+a_{0}+a_{6}+a_{12} [5, 1, 1, -2] 5

On Wed, Jul 10, 2013 at 6:15 AM, Graeme McRae <graememcrae at gmail.com> wrote:

> Yes, there is a relation between A205579 and the numbers that have a
> "sparse" representation in "base plastic", where the plastic number is
> p=1.324717957...  A205579 lists 10717, for example, because 10717 is
> ever-so-slightly larger than p^33, so the next digit in the "base plastic"
> representation of 10717 is a tiny number -- p^-18.
>
> The "base plastic" representation of an integer is always sparse, because
> the "standard form" of such a representation always has at least four zeros
> between every pair of ones.  That's because
> 010001(base p) = 100000 (base p),
> 0100100000 (base p) = 1000000001 (base p),
> 0101000 (base p) = 1000001 (base p),
> 0011 (base p) = 1000 (base p),
> 00200000 (base p) = 10000001 (base p)
> The right-hand-side of each of these identities has the 1's more "spread
> out" than the left side, so in each case where the 1's are separated by
> fewer than 4 zeros, one of these identities can be used to spread out the
> ones, until the number is in "standard form".
>
> --Graeme McRae
> Palmdale, CA
>
>
> On Tue, Jul 9, 2013 at 12:31 PM, L. Edson Jeffery <lejeffery2 at gmail.com
> >wrote:
>
> > >Is there any signicance to these numbers?
> >
> > Cf. A205579 and A109377 which contain some of your terms.
> >
> > Ed Jeffery
> >
> > _______________________________________________
> >
> > Seqfan Mailing list - http://list.seqfan.eu/
> >
>
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