

A010900


Pisot sequence E(4,13): a(n) = floor( a(n1)^2/a(n2) + 1/2 ).


3



4, 13, 42, 136, 440, 1424, 4609, 14918, 48285, 156284, 505844, 1637264, 5299328, 17152321, 55516872, 179691313, 581606398, 1882483892, 6093030640, 19721296176, 63831867233, 206604436042, 668716032329, 2164431415224, 7005609443657, 22675037578854
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OFFSET

0,1


COMMENTS

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
Satisfies a linear recurrence of order 6 just for n <= 23 (see A274952).  N. J. A. Sloane, Aug 07 2016


REFERENCES

Cantor, D. G. "Investigation of Tnumbers and Esequences." In Computers in Number Theory, ed. AOL Atkin and BJ Birch, Acad. Press, NY (1971); pp. 137140.
Cantor, D. G. (1976). On families of Pisot Esequences. In Annales scientifiques de l'École Normale Supérieure (Vol. 9, No. 2, pp. 283308).


LINKS

Colin Barker, Table of n, a(n) for n = 0..1000
D. W. Boyd, Some integer sequences related to the Pisot sequences, Acta Arithmetica, 34 (1979), 295305
D. W. Boyd, Linear recurrence relations for some generalized Pisot sequences, Advances in Number Theory ( Kingston ON, 1991) 333340, Oxford Sci. Publ., Oxford Univ. Press, New York, 1993.
David Cantor, Investigation of Tnumbers and Esequences, In Computers in Number Theory, ed. A. O. L. Atkin and B. J. Birch, Acad. Press, NY (1971); pp. 137140. [Annotated scanned copy]
C. Pisot, La répartition modulo 1 et les nombres algébriques, Ann. Scuola Norm. Sup. Pisa, 7 (1938), 205248.


FORMULA

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


MATHEMATICA

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


PROG

(PARI) pisotE(nmax, a1, a2) = {
a=vector(nmax); a[1]=a1; a[2]=a2;
for(n=3, nmax, a[n] = floor(a[n1]^2/a[n2]+1/2));
a
}
pisotE(50, 4, 13) \\ Colin Barker, Jul 28 2016


CROSSREFS

Cf. A007698, A007699, A010916, A274952.
See A008776 for definitions of Pisot sequences.
Sequence in context: A010919 A277667 A274952 * A175005 A070031 A082989
Adjacent sequences: A010897 A010898 A010899 * A010901 A010902 A010903


KEYWORD

nonn


AUTHOR

Simon Plouffe


STATUS

approved



