A010900 - OEIS

%I #46 Jun 24 2018 12:24:51

%S 4,13,42,136,440,1424,4609,14918,48285,156284,505844,1637264,5299328,

%T 17152321,55516872,179691313,581606398,1882483892,6093030640,

%U 19721296176,63831867233,206604436042,668716032329,2164431415224,7005609443657,22675037578854

%N Pisot sequence E(4,13): a(n) = floor( a(n-1)^2/a(n-2) + 1/2 ).

%C According to David Boyd his last use (as of April, 2006) of his Pisot number finding program was to prove that in fact this sequence does not satisfy a linear recurrence. He remarks "This took a couple of years in background on various Sun workstations." - _Gene Ward Smith_, Apr 11 2006

%C Satisfies a linear recurrence of order 6 just for n <= 23 (see A274952). - _N. J. A. Sloane_, Aug 07 2016

%D Cantor, D. G. "Investigation of T-numbers and E-sequences." In Computers in Number Theory, ed. AOL Atkin and BJ Birch, Acad. Press, NY (1971); pp. 137-140.

%D Cantor, D. G. (1976). On families of Pisot E-sequences. In Annales scientifiques de l'École Normale Supérieure (Vol. 9, No. 2, pp. 283-308).

%H Colin Barker, <a href="/A010900/b010900.txt">Table of n, a(n) for n = 0..1000</a>

%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="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>, Advances in Number Theory ( Kingston ON, 1991) 333-340, Oxford Sci. Publ., Oxford Univ. Press, New York, 1993.

%H David Cantor, <a href="/A010900/a010900.pdf">Investigation of T-numbers and E-sequences</a>, In Computers in Number Theory, ed. A. O. L. Atkin and B. J. Birch, Acad. Press, NY (1971); pp. 137-140. [Annotated scanned copy]

%H C. Pisot, <a href="http://www.numdam.org/item?id=ASNSP_1938_2_7_3-4_205_0">La répartition modulo 1 et les nombres algébriques</a>, Ann. Scuola Norm. Sup. Pisa, 7 (1938), 205-248.

%F It is known that this does not satisfy any linear recurrence [Boyd].

%t nxt[{a_,b_}]:={b,Floor[b^2/a+1/2]}; NestList[nxt,{4,13},30][[All,1]] (* _Harvey P. Dale_, Jun 24 2018 *)

%o (PARI) pisotE(nmax, a1, a2) = {

%o   a=vector(nmax); a[1]=a1; a[2]=a2;

%o   for(n=3, nmax, a[n] = floor(a[n-1]^2/a[n-2]+1/2));

%o   a

%o }

%o pisotE(50, 4, 13) \\ _Colin Barker_, Jul 28 2016

%Y Cf. A007698, A007699, A010916, A274952.

%Y See A008776 for definitions of Pisot sequences.

%K nonn

%O 0,1

%A _Simon Plouffe_