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%I M4747 #65 Jan 01 2022 14:05:42
%S 10,219,4796,105030,2300104,50371117,1103102046,24157378203,
%T 529034393290,11585586272312,253718493496142,5556306986017175,
%U 121680319386464850,2664737596978110299,58356408797678883616,1277975907130111287030,27987027523701766535844
%N 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.
%C 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
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D J. Wroblewski, personal communication.
%H Colin Barker, <a href="/A007699/b007699.txt">Table of n, a(n) for n = 1..700</a>
%H David Boyd (originator), <a href="https://www.encyclopediaofmath.org/index.php/Pisot_sequence">Pisot sequence</a>. Encyclopedia of Mathematics.
%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 S. B. Ekhad, N. J. A. Sloane and D. Zeilberger, <a href="http://arxiv.org/abs/1609.05570">Automated proofs (or disproofs) of linear recurrences satisfied by Pisot Sequences</a>, arXiv:1609.05570 [math.NT], 2016.
%H J. Wroblewski, <a href="/A007698/a007698.pdf">Email to N. J. A. Sloane, Jun. 1994</a>
%H <a href="/index/Se#sequences_which_agree_for_a_long_time">Index entries for sequences which agree for a long time but are different</a>
%e a(1403) is
%e 1943708471314943308059445452657010940487450311864066842732596790939279068191\
%e 168021439671095304800683519756645143142801766345115405789059172602192426\
%e 024357604507643919310528104572431148473422703387902120314696316682603735\
%e 267692111685622339243356242260056059336217912799059786079481997806631913\
%e 955493134941095358770263918313025848373581726054928149011342047774528154\
%e 248287433782463237576416857026309254788755903742777139477594456385042020\
%e 381315538604379941789590322666368814892780385046811477655985825537894431\
%e 894143994712043942268394043823543450207513886190799409707531632679517052\
%e 869104335940723488960240770470438470434329535343866330429132657179201894\
%e 810776495469936998716229270764904917198741365340242782600909003168195629\
%e 553831589770365472687705483796661474238920271726070390505179067208859490\
%e 817765494636249793643314197295308500154814706778732034270622318621910522\
%e 030142040283435992446877395852252468365235219657327211742475429216859612\
%e 898009146799397834207588995393930733511691021384920256724554594857336855\
%e 550714963221355049079118765001875374835520434138927516201876958496564958\
%e 805765202364476313555615826884516631224599151532590504446541236893625713\
%e 832620042439077419006777861484860386048975978762433100742439296700782881\
%e 889486380714070148887484098410694218233687263042755465493793927981497199\
%e 521026920386200848153568287674310343346371498689283968784694184354766679\
%e 111870702565268681491357079215569781219694309328629243757829281537544222\
%e 305623084962270299300645420182502879046175714261919397771509700298570157\
%e 891004711917373029290386303109701959096841328964650889891682871446978568\
%e 692922345060182670103628056600403977432916893829069098732545636174794446\
%e 362475483205590674696119315488543667867514676786440758126850754300452964\
%e 368265133082563202580908171650074203739290735941387946242005524276316413\
%e 356912394816492851593842390985938520048268384592849898513622096090183587\
%e 01821
%e - from _N. J. A. Sloane_, Jul 27 2016
%e 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.
%p 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:
%t a = {10, 219}; Do[AppendTo[a, Round[a[[k - 1]]^2/a[[k - 2]]]], {k, 3, 17}]; a (* _Michael De Vlieger_, Feb 08 2016 *)
%t nxt[{a_,b_}]:={b,Round[b^2/a]}; NestList[nxt,{10,219},20][[All,1]] (* _Harvey P. Dale_, Jan 01 2022 *)
%o (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
%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, 10, 219) \\ _Colin Barker_, Jul 27 2016
%Y See A008776 for definitions of Pisot sequences.
%Y Cf. A007698.
%K nonn
%O 1,1
%A _N. J. A. Sloane_ and _J. H. Conway_
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