T-seqs

[Russell Walsmith]

Thanks to Max and Dean for the feedback on the "Perrin-type sequence"
thread. I see now that many sequences share the property "n divides a(n) if
n is prime" (from Max) and that the sequence discussed has many small
pseudoprimes (from Dean).

Max's question, "What is special about this sequence except that property?"
was a good one, and I'm still trying to figure that out. What follows is
sort of a progress report…

Dean pointed out that the sequence S =
0,3,5,3,15,13,105,117,175,387,825,3397,2145,7347,7735…
is the absolute value of the sequence a(n) defined by a(1)=0, a(2)=3,
a(3)=5, and a(n)=3a(n-1)-6a(n-2)+8a(n-3). More explicitly, a(n) = (2^(n-1) -
((1+sqrt(-15))/2)^n - ((1-sqrt(-15))/2)^n)/3

Way cool! I'm curious as to how that was derived, and the extent to which it
can be extended to related sequences. To elaborate, we construct S2 on
a(1)=0, a(2)=6, a(3)=10 and/or a(n) = 2*(2^(n-1) - ((1+sqrt(-15))/2)^n -
((1-sqrt(-15))/2)^n)/3.

This gives S2 =  0, –6, –10, 6, 30, –26, –210, –234, 350, 774, –1650, –6794…
which may be split up into two other sequences, S2a and S2b, where S2a – S2b
= S2.
S2a = 1, –1, –1, 11, 31, 19, –41, 11, 431, 899, 199, –1349… and
S2b = 1, 5, 9, 5, 1, 45, 169, 245, 81, 125, 1849, 5445…

So given 1, –1, –1 and 1, 5, 9 as initial
terms,a(n)=3a(n-1)-6a(n-2)+8a(n-3) generates S2a
and S2b. May S2a and S2b be represented in the more explicit form as well?

Note that S2a + S2b = 2, 4, 8, 16, 32, 64, 128, 256, 512… In fact, every
power sequence of the type a^n may be represented as the sum of two
sequences in countless ways. For more 2^n examples:

A1 = 0, 2, 2, 6, 10, 22, 42, 86, 170, 342, 682, 1366, 2730, 5462, 10922,
21846, 43690, 87382, 174762, 349526...
A2 = 2, 2, 6, 10, 22, 42, 86, 170, 342, 682, 1366, 2730, 5462, 10922, 21846,
43690, 87382, 174762, 349526, 699050…

B1 = 1, 1, 2, 5, 11, 22, 43, 85, 170, 341, 683, 1366, 2731, 5461, 10922,
21845, 43691, 87382, 174763, 349525…
B1 = 1, 3, 6, 11, 21, 42, 85, 171, 342, 683, 1365, 2730, 5461, 10923, 21846,
43691, 87381, 174762, 349525, 699051…

C1 = 2, 2, -4, -26, -68, -74, 212, 1430, 4364, 7702, 772, -49802, -202564,
-457546, -404044, 1603030, 9273196, 25937110, 38967140, -25429066…
C1 = 0, 2, 12, 42, 100, 138, -84, -1174, -3852, -6678, 1276, 53898, 210756,
473930, 436812, -1537494, -9142124, -25674966, -38442852, 26477642…

D1 = 1, -3, -4, 29, 81, -74, -545, 165, 4364, 3977, -25035, -49802, 131587,
492489, -404044, -3802315, -875703, 25937110, 32460247, -148313091…
D2 = 1, 7, 12, -13, -49, 138, 673, 91, -3852, -2953, 27083, 53898, -123395,
-476105, 436812, 3867851, 1006775, -25674966, -31935959, 149361667…

Do these sequences also have the kinds of representations cited above? What
else might they have in common?

Russell
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