[seqfan] More and more commas
bk263401 at skynet.be
Tue May 12 19:38:52 CEST 2020
[Brussels, May 12 2020, 19:40]
We want to read the sum a(n) + a(n+1) around the comma coming immediately after a(n+1).
This is easy to build from a(1) = 1 on and might produce a nice sequence.
S = 1,11,23,46,91,374,6506,8801,53076,18777,18533,...
a(1)=1 + a(2)=11 is 12 and 12 is visible around the comma following 11:
S = 1,11,23,
a(2)=11 + a(3)=23 is 34 and 34 is visible around the comma following 23:
S = 1,11,23,46,
a(3)=23 + a(4)=46 is 69 and 69 is visible around the comma following 46:
S = 1,11,23,46,91,
a(4)=46 + a(5)=91 is 137 and 137 is visible around the comma following 91:
S = 1,11,23,46,91,374,
The rule here is that the rightmost digit of a(n+1) is the first digit of the sum a(n) + a(n+1), the other digits of the said sum being put after the comma in order to start a(n+2).
As no digit 0 (zero) can start a term, one will have to backtrack sometimes – and pick another term for a(n+1), compatible with the above rule.
This is always possible. We will try though to extend the sequence S as far as possible with the smallest available integer not yet present in S.
The nice part of this idea is that S is not monotonically increasing (look at the last three terms of the sequence that opens the page): who knows how it will behave in the future?
More can be said about S, but enough for today!
(some color and highlights here, on my personal blog: https://bit.ly/3brFiSz)
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