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
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
KEYWORD
nonn
AUTHOR
STATUS
approved