Search: id:A001834 Results 1-1 of 1 results found. %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, Table of n, a(n) for n=0..200 %H A001834 Index entries for sequences related to linear recurrences with constant coefficients %H A001834 L. Euler, Vollstaendige Anleitung zur Algebra, Zweiter Teil. %H A001834 Tanya Khovanova, Recursive Sequences %H A001834 S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992. %H A001834 S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992. %H A001834 Index entries for sequences related to Chebyshev polynomials. %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 Search completed in 0.002 seconds