A007699 Pisot sequence E(10,219): a(n) = nearest integer to a(n-1)^2 / a(n-2), starting 10, 219, ... Deviates from A007698 at 1403rd term. (Formerly M4747)

10, 219, 4796, 105030, 2300104, 50371117, 1103102046, 24157378203, 529034393290, 11585586272312, 253718493496142, 5556306986017175, 121680319386464850, 2664737596978110299, 58356408797678883616, 1277975907130111287030, 27987027523701766535844

Offset

1,1

Comments

a(n+1)/a(n) -> 21.8994954189323... which is very near to a root of 11*x^4 - 18*x^3 + 3*x^2 - 22*x + 1. This is only an approximation since the sequence does not satisfy any known recurrence. The difference between the root of the equation and the real value is 1.1357748460267988*10^(-1877). - Simon Plouffe, Feb 26 2012

References

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

J. Wroblewski, personal communication.

Links

Colin Barker, Table of n, a(n) for n = 1..700

David Boyd (originator), Pisot sequence. Encyclopedia of Mathematics.

D. W. Boyd, Linear recurrence relations for some generalized Pisot sequences, Advances in Number Theory ( Kingston ON, 1991) 333-340, Oxford Sci. Publ., Oxford Univ. Press, New York, 1993

S. B. Ekhad, N. J. A. Sloane and D. Zeilberger, Automated proofs (or disproofs) of linear recurrences satisfied by Pisot Sequences, arXiv:1609.05570 [math.NT], 2016.

J. Wroblewski, Email to N. J. A. Sloane, Jun. 1994

Index entries for sequences which agree for a long time but are different

Example

a(1403) is
1943708471314943308059445452657010940487450311864066842732596790939279068191\
    168021439671095304800683519756645143142801766345115405789059172602192426\
    024357604507643919310528104572431148473422703387902120314696316682603735\
    267692111685622339243356242260056059336217912799059786079481997806631913\
    955493134941095358770263918313025848373581726054928149011342047774528154\
    248287433782463237576416857026309254788755903742777139477594456385042020\
    381315538604379941789590322666368814892780385046811477655985825537894431\
    894143994712043942268394043823543450207513886190799409707531632679517052\
    869104335940723488960240770470438470434329535343866330429132657179201894\
    810776495469936998716229270764904917198741365340242782600909003168195629\
    553831589770365472687705483796661474238920271726070390505179067208859490\
    817765494636249793643314197295308500154814706778732034270622318621910522\
    030142040283435992446877395852252468365235219657327211742475429216859612\
    898009146799397834207588995393930733511691021384920256724554594857336855\
    550714963221355049079118765001875374835520434138927516201876958496564958\
    805765202364476313555615826884516631224599151532590504446541236893625713\
    832620042439077419006777861484860386048975978762433100742439296700782881\
    889486380714070148887484098410694218233687263042755465493793927981497199\
    521026920386200848153568287674310343346371498689283968784694184354766679\
    111870702565268681491357079215569781219694309328629243757829281537544222\
    305623084962270299300645420182502879046175714261919397771509700298570157\
    891004711917373029290386303109701959096841328964650889891682871446978568\
    692922345060182670103628056600403977432916893829069098732545636174794446\
    362475483205590674696119315488543667867514676786440758126850754300452964\
    368265133082563202580908171650074203739290735941387946242005524276316413\
    356912394816492851593842390985938520048268384592849898513622096090183587\
    01821
- from N. J. A. Sloane, Jul 27 2016
A007698(1403) = 22*a(1402) - 3*a(1401) + 18*a(1400) - 11*a(1399) = a(1403) + 1. - M. F. Hasler, Feb 09 2014. This is one more than the number displayed above.

Maple

a := proc(n) options remember; if n = 1 then RETURN(10); elif n = 2 then RETURN(219); else RETURN(round(a(n-1)^2/a(n-2))); fi; end:

Mathematica

a = {10, 219}; Do[AppendTo[a, Round[a[[k - 1]]^2/a[[k - 2]]]], {k, 3, 17}]; a (* Michael De Vlieger, Feb 08 2016 *)
nxt[{a_, b_}]:={b, Round[b^2/a]}; NestList[nxt, {10, 219}, 20][[All, 1]] (* Harvey P. Dale, Jan 01 2022 *)

Prog

(PARI) A007699(n, a=10, b=100/219)=for(k=2, n, a=(a^2+b\2)\(b+0*b=a)); a \\ M. F. Hasler, Feb 09 2014
(PARI) pisotE(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]+1/2));
  a
}
pisotE(50, 10, 219) \\ Colin Barker, Jul 27 2016

Crossrefs

See A008776 for definitions of Pisot sequences.

Cf. A007698.

Sequence in context: A259189 A326208 A007698 * A024291 A024292 A094420

Adjacent sequences: A007696 A007697 A007698 * A007700 A007701 A007702

Keyword

nonn

Author

N. J. A. Sloane and J. H. Conway

Status

approved