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%I #16 Aug 12 2016 21:23:58
%S 1,2,6,36,2048
%N Order of homogeneous linear recurrence satisfied by the Pisot sequence T(n, n^2-n+1).
%C Degree of denominator of minimal g.f. for T(n, n^2-n+1).
%C Conjecture: a(6) = 6852224. The conjectured generating function for T(6,31) is A(x)/(1+x - x*A(x)) where A(x) = 6 + x - x^2 - x^4 - x^22 - x^1130 - x^6852224 (and as usual there is a common factor of (1+x) in numerator and denominator). - _David Boyd_, Aug 12 2016.
%D David Boyd, Email communication to _N. J. A. Sloane_, Aug 06 2016
%H D. W. Boyd, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa32/aa32110.pdf">Pisot sequences which satisfy no linear recurrences</a>, Acta Arith. 32 (1) (1977) 89-98
%H D. W. Boyd, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa34/aa3444.pdf">Some integer sequences related to the Pisot sequences</a>, Acta Arithmetica, 34 (1979), 295-305
%H D. W. Boyd, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa47/aa4712.pdf">On linear recurrence relations satisfied by Pisot sequences</a>, Acta Arithm. 47 (1) (1986) 13
%H D. W. Boyd, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa48/aa4825.pdf">Pisot sequences which satisfy no linear recurrences. II</a>, Acta Arithm. 48 (1987) 191
%H D. W. Boyd, <a href="https://www.researchgate.net/profile/David_Boyd7/publication/262181133_Linear_recurrence_relations_for_some_generalized_Pisot_sequences_-_annotated_with_corrections_and_additions/links/00b7d536d49781037f000000.pdf">Linear recurrence relations for some generalized Pisot sequences</a>, in Advances in Number Theory (Kingston ON, 1991), pp. 333-340, Oxford Univ. Press, New York, 1993; with updates from 1996 and 1999.
%e T(1,1) is the all-ones sequence, with g.f. 1/(1-x).
%e T(2,3) is 2,3,4,5,6,... with g.f. (2-x)/(1-2*x+x^2).
%e T(3,7) is A020746, with a linear recurrence of order 6.
%e T(4,13) is A010919, with a linear recurrence of order 36.
%e T(5,21) is A010925, with a linear recurrence of order 2048.
%Y Cf. A008776, A020746, A010919, A010925.
%K nonn,more
%O 1,2
%A _N. J. A. Sloane_, Aug 11 2016
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