%I A001834 M3890 N1598
%S A001834 1,5,19,71,265,989,3691,13775,51409,191861,716035,2672279,9973081,37220045,
%T A001834 138907099,518408351,1934726305,7220496869,26947261171,100568547815,375326930089,
%U A001834 1400739172541,5227629760075,19509779867759,72811489710961,271736178976085
%N A001834 a(0) = 1, a(1) = 5, a(n) = 4a(n-1) - a(n-2).
%C A001834 Sequence also gives values of x satisfying 3*y^2 - x^2 = 2, the corresponding
y being given by A001835(n+1). Moreover, quadruples(p, q, r, s) satisfying
p^2 + q^2 + r^2 = s^2, where p=q and r is either p+1 or p-1, are
termed nearly isosceles Pythagorean and are given by p={x + (-1)^n}/
3, r=p-(-1)^n, s=y for n>1. - Lekraj Beedassy (blekraj(AT)yahoo.com),
Jul 19 2002
%C A001834 a(n) = L(n,-4)*(-1)^n, where L is defined as in A108299; see also A001835
for L(n,+4). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com),
Jun 01 2005
%C A001834 a(n)= A002531(1+2*n) - Anton Vrba (antonvrba(AT)yahoo.com), Feb 14 2007
%C A001834 361 written in base A001835(n+1)-1 is the square of a(n). E.g. a(12)=2672279,
A001835(13)-1=1542840. We have 361_(1542840)=3*1542840+6*1542840+1=2672279^2
- Richard Choulet (richardchoulet(AT)yahoo.fr), Oct 04 2007
%C A001834 The lower principal convergents to 3^(1/2), beginning with 1/1, 5/3,
19/11, 71/41, comprise a strictly increasing sequence; numerators=A001834,
denominators=A001835. - Clark Kimberling (ck6(AT)evansville.edu),
Aug 27 2008
%C A001834 General recurrence is a(n)=(a(1)-1)*a(n-1)-a(n-2), a(1)>=4, lim n->infinity
a(n)= x*(k*x+1)^n, k =(a(1)-3), x=(1+sqrt((a(1)+1)/(a(1)-3)))/2.
Examples in OEIS: a(1)=4 gives A002878, primes in it A121534. a(1)=5
gives A001834, primes in it A086386. a(1)=6 gives A030221, primes
in it not in OEIS {29,139,3191,...}. a(1)=7 gives A002315, primes
in it A088165. a(1)=8 gives A033890, primes in it not in OEIS (does
there exist any ?). a(1)=9 gives A057080, primes in it not in OEIS
{71,34649,16908641,...}. a(1)=10 gives A057081, primes in it not
in OEIS {389806471,192097408520951,...}. [From Ctibor O. Zizka (ctibor.zizka(AT)seznam.cz),
Sep 02 2008]
%C A001834 Inverse binomial transform of A030192. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr),
Nov 19 2009]
%D A001834 L. Euler, (E388) Vollstaendige Anleitung zur Algebra, Zweiter Theil,
reprinted in: Opera Omnia. Teubner, Leipzig, 1911, Series (1), Vol.
1, p. 375.
%D A001834 Clark Kimberling, "Best lower and upper approximates to irrational numbers,
" Elemente der Mathematik, 52 (1997) 122-126.
%D A001834 Serge Lang, Introduction to Diophantine Approximations, Addison-Wesley,
New York, 1966.
%D A001834 W. Lang, On polynomials related to powers of the generating function
of Catalan's numbers, Fib. Quart. 38,5 (2000) 408-419; Eq. (44) rhs,
m=6.
%D A001834 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973
(includes this sequence).
%D A001834 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences,
Academic Press, 1995 (includes this sequence).
%D A001834 P.-F. Teilhet, Reply to Query 2094, L'Interm\'{e}diaire des Math\'{e}maticiens,
10 (1903), 235-238.
%D A001834 F. V. Waugh and M. W. Maxfield, Side-and-diagonal numbers, Math. Mag.,
40 (1967), 74-83.
%H A001834 T. D. Noe, <a href="b001834.txt">Table of n, a(n) for n=0..200</a>
%H A001834 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to
linear recurrences with constant coefficients</a>
%H A001834 L. Euler, <a href="http://www.mathematik.uni-bielefeld.de/~sieben/euler/
euler_2.djvu">Vollstaendige Anleitung zur Algebra, Zweiter Teil</
a>.
%H A001834 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/
RecursiveSequences.html">Recursive Sequences</a>
%H A001834 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">
Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</
a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al,
1992.
%H A001834 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">
1031 Generating Functions and Conjectures</a>, Universit\'{e} du
Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%H A001834 <a href="Sindx_Ch.html#Cheby">Index entries for sequences related to
Chebyshev polynomials.</a>
%F A001834 a(n) = ((1+sqrt(3))^(2*n+1)+(1-sqrt(3))^(2*n+1))/2^(n+1). - njas, Nov
10 2009
%F A001834 a(n) = (1/2) * ((1+sqrt(3))*(2+sqrt(3))^n + (1-sqrt(3))*(2-sqrt(3))^n).
- Dean Hickerson (dean.hickerson(AT)yahoo.com), Dec 01 2002
%F A001834 With a=2+sqrt(3), b=2-sqrt(3): a(n)=(1/sqrt(2))(a^(n+1/2)-b^(n+1/2)).
a(n)-a(n-1)=A003500(n). a(n)=sqrt(1+12*A061278(n)+12*A061278(n)^2).
- Mario Catalani (mario.catalani(AT)unito.it), Apr 11 2003
%F A001834 a(n)=((1+sqrt[3])^(2*n+1)+(1-sqrt[3])^(2*n+1))/2^(n+1) - Anton Vrba (antonvrba(AT)yahoo.com),
Feb 14 2007
%F A001834 G.f.: (1+x)/((1-4*x+x^2)). a(n)= S(2*n, sqrt(6)) = S(n, 4)+S(n-1, 4);
S(n, x) := U(n, x/2), Chebyshev polynomials of 2nd kind, A049310.
S(n, 4)= A001353(n).
%F A001834 For all members x of the sequence, 3*x^2 + 6 is a square. Lim. as n ->
Inf. a(n)/a(n-1) = 2 + Sqrt(3). - Gregory V. Richardson (omomom(AT)hotmail.com),
Oct 10 2002
%F A001834 a(n)=2*A001571(n)+1 - Bruce Corrigan (scentman(AT)myfamily.com), Nov
04 2002
%F A001834 Let q(n, x)=sum(i=0, n, x^(n-i)*binomial(2*n-i, i)); then (-1)^n*q(n,
-6)=a(n) - Benoit Cloitre (benoit7848c(AT)orange.fr), Nov 10 2002
%F A001834 a(n) = 2^(-n)*Sum{k>=0} binomial(2*n+1, 2*k)*3^k; see A091042 . - DELEHAM
Philippe (kolotoko(AT)wanadoo.fr), Mar 01 2004
%F A001834 a(n) = floor(sqrt(3)*A001835(n+1)). - DELEHAM Philippe (kolotoko(AT)wanadoo.fr),
Mar 03 2004
%F A001834 a(n+1) - 2*a(n) = 3*A001835(n+1). Using the known relation A001835(n+1)
= sqrt((a(n)^2 + 2)/3) it follows that a(n+1) - 2*a(n) = sqrt(3*(a(n)^2+2)).
Therefore a(n+1)^2 + a(n)^2 - 4*a(n+1)*a(n) - 6 = 0. - Creighton
Dement (creighton.k.dement(AT)uni-oldenburg.de), Apr 18 2005
%F A001834 a(n)=Jacobi_P(n,1/2,-1/2,2)/Jacobi_P(n,-1/2,1/2,1); - Paul Barry (pbarry(AT)wit.ie),
Feb 03 2006
%F A001834 Equals binomial transform of A026150 starting (1, 4, 10, 28, 76,...)
and double binomial transform of (1, 3, 3, 9, 9, 27, 27, 81, 81,...).
- Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 30 2007
%F A001834 Sequence satisfies 6 = f(a(n), a(n+1)) where f(u, v) = u^2 + v^2 - 4*u*v.
- Michael Somos Sep 19 2008
%F A001834 a(-1-n) = -a(n). - Michael Somos Sep 19 2008
%p A001834 A001834:=(1+z)/(1-4*z+z**2); [S. Plouffe in his 1992 dissertation.]
%p A001834 f:=n->((1+sqrt(3))^(2*n+1)+(1-sqrt(3))^(2*n+1))/2^(n+1); [njas, Nov 10
2009]
%t A001834 a[0] = 1; a[1] = 5; a[n_] := a[n] = 4a[n - 1] - a[n - 2]; Table[ a[n],
{n, 0, 25}] (from Robert G. Wilson v Apr 24 2004)
%t A001834 Table[Expand[((1+Sqrt[3])^(2*n+1)+(1+Sqrt[3])^(2*n+1))/2^(n+1)],{n, 0,
20}] - Anton Vrba (antonvrba(AT)yahoo.com), Feb 14 2007
%t A001834 q=24;s=0;lst={};Do[s+=n;If[Sqrt[q*s+1]==Floor[Sqrt[q*s+1]],AppendTo[lst,
Sqrt[q*s+1]]],{n,0,8!}];lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com),
Apr 02 2009]
%o A001834 Floretion Algebra Multiplication Program, FAMP Code: A001834 = (4/3)vesseq[
- .25'i + 1.25'j - .25'k - .25i' + 1.25j' - .25k' + 1.25'ii' + .25'jj'
- .75'kk' + .75'ij' + .25'ik' + .75'ji' - .25'jk' + .25'ki' - .25'kj'
+ .25e], apart from initial term
%o A001834 (PARI) {a(n) = real( (2 + quadgen(12))^n * (1 + quadgen(12)) )} /* Michael
Somos Sep 19 2008 */
%o A001834 (PARI) {a(n) = subst( polchebyshev(n-1, 2) + polchebyshev(n, 2), x, 2)}
/* Michael Somos Sep 19 2008 */
%o A001834 (Other) sage: [(lucas_number2(n,4,1)-lucas_number2(n-1,4,1))/2 for n
in xrange(1, 27)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com),
Nov 10 2009]
%Y A001834 A bisection of sequence A002531.
%Y A001834 Cf. A001352, A001835.
%Y A001834 Cf. A026150.
%Y A001834 Sequence in context: A026590 A095073 A128349 this_sequence A099393 A083588
A149759
%Y A001834 Adjacent sequences: A001831 A001832 A001833 this_sequence A001835 A001836
A001837
%K A001834 nonn,easy,nice,new
%O A001834 0,2
%A A001834 N. J. A. Sloane (njas(AT)research.att.com).
%E A001834 More terms from Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de),
Aug 07 2000
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