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A010925
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Pisot sequence T(5,21), a(n) = floor( a(n-1)^2/a(n-2) ).
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13
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5, 21, 88, 368, 1538, 6427, 26857, 112229, 468978, 1959746, 8189306, 34221135, 143001871, 597570335, 2497102330, 10434788478, 43604464772, 182212543365, 761422279419, 3181800093939, 13295975323332, 55560674643076, 232174661258332, 970201922073653, 4054239874815929, 16941690784755705, 70795240417122019
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OFFSET
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0,1
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COMMENTS
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Comments from Pab Ter (pabrlos(AT)yahoo.com), May 23 2004, with updates from N. J. A. Sloane, Aug 05 2016: (Start)
Different from A019992. The two sequences differ from n=26 on (A010925(26) = 70795240417122019 != 70795240417122020 = A019992(26)).
From Boyd's paper "Linear recurrence relations for some generalized Pisot sequences", T(5,21) satisfies the rational generating function F(x)/(1+x-x*F(x)), with F(x) = 5 + x - x^2 - x^4 - x^26 - x^2048, a 2049th-order recurrence; and not the A019992 generating function: F(x)/(1+x-x*F(x)), with F(x) = 5 + x - x^2 - x^4, which gives the 5th-order recurrence for A019992.
The g.f. F(x)/(1+x-x*F(x)) with F(x) = 5 + x - x^2 - x^4 - x^26 - x^2048 is not in lowest terms, however, and a factor of 1+x can be canceled. The lowest-order recurrence satisfied by this sequence has order 2048.
This and other examples show that it is essential to reject conjectured generating functions for Pisot sequences until a proof or reference is provided. (End)
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LINKS
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FORMULA
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G.f.: F(x)/(1+x-x*F(x)), with F(x) = 5 + x - x^2 - x^4 - x^26 - x^2048 (D. W. Boyd). - Pab Ter (pabrlos(AT)yahoo.com), May 23 2004
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MATHEMATICA
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nxt[{a_, b_}]:={b, Floor[b^2/a]}; NestList[nxt, {5, 21}, 30][[All, 1]] (* Harvey P. Dale, May 15 2017 *)
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PROG
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(PARI) pisotT(nmax, a1, a2) = {
a=vector(nmax); a[1]=a1; a[2]=a2;
for(n=3, nmax, a[n] = floor(a[n-1]^2/a[n-2]));
a
}
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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More terms from Pab Ter (pabrlos(AT)yahoo.com), May 23 2004
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STATUS
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approved
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