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 A001834 a(0) = 1, a(1) = 5, a(n) = 4a(n-1) - a(n-2). (Formerly M3890 N1598) 63

%I M3890 N1598

%S 1,5,19,71,265,989,3691,13775,51409,191861,716035,2672279,9973081,

%T 37220045,138907099,518408351,1934726305,7220496869,26947261171,

%U 100568547815,375326930089,1400739172541,5227629760075,19509779867759,72811489710961,271736178976085

%N a(0) = 1, a(1) = 5, a(n) = 4a(n-1) - a(n-2).

%C 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, Jul 19 2002

%C a(n) = L(n,-4)*(-1)^n, where L is defined as in A108299; see also A001835 for L(n,+4). - _Reinhard Zumkeller_, Jun 01 2005

%C a(n)= A002531(1+2*n). - Anton Vrba (antonvrba(AT)yahoo.com), Feb 14 2007

%C 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_, Oct 04 2007

%C 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_, Aug 27 2008

%C 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 (do there exist any?). a(1)=9 gives A057080, primes in {71, 34649, 16908641,...}. a(1)=10 gives A057081, primes in it {389806471, 192097408520951, ...}. - _Ctibor O. Zizka_, Sep 02 2008]

%C Inverse binomial transform of A030192. - _Philippe Deléham_, Nov 19 2009

%C For positive n, a(n) equals the permanent of the (2n)X(2n) tridiagonal matrix with sqrt(6)'s along the main diagonal, and i's along the superdiagonal and the subdiagonal (i is the imaginary unit). - _John M. Campbell_, Jul 08 2011

%C x-values in the solution to 3x^2 + 6 = y^2 (see A082841 for the y-values). - _Sture Sjöstedt_, Nov 25 2011

%C Pisano period lengths: 1, 1, 2, 4, 3, 2, 8, 4, 6, 3, 10, 4, 12, 8, 6, 8, 18, 6, 5, 12, ... - _R. J. Mathar_, Aug 10 2012

%C The aerated sequence (b(n))n>=1 = [1, 0, 5, 0, 19, 0, 71, 0, ...] is a fourth-order linear divisibility sequence; that is, if n | m then b(n) | b(m). It is the case P1 = 0, P2 = -2, Q = -1 of the 3-parameter family of divisibility sequences found by Williams and Guy. See A100047 for a connection with Chebyshev polynomials. - _Peter Bala_, Mar 22 2015

%C Yong Hao Ng has shown that for any n, a(n) is coprime with any member of A001835 and with any member of A001075. - _René Gy_, Feb 26 2018

%D Bastida, Julio R. Quadratic properties of a linearly recurrent sequence. Proceedings of the Tenth Southeastern Conference on Combinatorics, Graph Theory and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1979), pp. 163--166, Congress. Numer., XXIII-XXIV, Utilitas Math., Winnipeg, Man., 1979. MR0561042 (81e:10009)

%D L. Euler, (E388) Vollstaendige Anleitung zur Algebra, Zweiter Theil, reprinted in: Opera Omnia. Teubner, Leipzig, 1911, Series (1), Vol. 1, p. 375.

%D Serge Lang, Introduction to Diophantine Approximations, Addison-Wesley, New York, 1966.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

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

%D P.-F. Teilhet, Reply to Query 2094, L'Intermédiaire des Mathématiciens, 10 (1903), 235-238.

%H T. D. Noe, <a href="/A001834/b001834.txt">Table of n, a(n) for n=0..200</a>

%H Marco Abrate, Stefano Barbero, Umberto Cerruti, Nadir Murru, <a href="https://www.emis.de/journals/INTEGERS/papers/p38/p38.Abstract.html">Polynomial sequences on quadratic curves</a>, Integers, Vol. 15, 2015, #A38.

%H J. B. Cosgrave and K. Dilcher, <a href="https://doi.org/10.1090/mcom/3111">A role for generalized Fermat numbers</a>, Math. Comp. 86 (2017), 899-933; see also <a href="http://johnbcosgrave.com/publications.php">Paper #10</a>.

%H Bruno Deschamps, <a href="http://dx.doi.org/10.1016/j.jnt.2010.06.006">Sur les bonnes valeurs initiales de la suite de Lucas-Lehmer</a>, Journal of Number Theory, Volume 130, Issue 12, December 2010, Pages 2658-2670.

%H L. Euler, <a href="http://www.mathematik.uni-bielefeld.de/~sieben/euler/euler_2.djvu">Vollstaendige Anleitung zur Algebra, Zweiter Teil</a>.

%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>

%H Seong Ju Kim, R. Stees, L. Taalman, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Stees/stees4.html">Sequences of Spiral Knot Determinants</a>, Journal of Integer Sequences, Vol. 19 (2016), # 16.1.4

%H Clark Kimberling, <a href="http://dx.doi.org/10.1007/s000170050020">Best lower and upper approximates to irrational numbers</a>, Elemente der Mathematik, 52 (1997) 122-126.

%H W. Lang, <a href="http://www.fq.math.ca/Scanned/38-5/lang.pdf">On polynomials related to powers of the generating function of Catalan's numbers</a>, Fib. Quart. 38 (2000) 408-419. Eq. (44) rhs, m=6.

%H Donatella Merlini and Renzo Sprugnoli, <a href="https://doi.org/10.1016/j.disc.2016.08.017">Arithmetic into geometric progressions through Riordan arrays</a>, Discrete Mathematics 340.2 (2017): 160-174.

%H Yong Hao Ng, <a href="https://math.stackexchange.com/a/2664328/130022">A partition in three classes of the set of all prime numbers?</a>, Math StackExchange.

%H S. Northshield, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Northshield/north4.html">An Analogue of Stern's Sequence for Z[sqrt(2)]</a>, Journal of Integer Sequences, 18 (2015), #15.11.6.

%H Simon Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992.

%H Simon Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">1031 Generating Functions and Conjectures</a>, Universite du Quebec a Montreal, 1992.

%H Ryan Stees, <a href="http://commons.lib.jmu.edu/cgi/viewcontent.cgi?article=1043&amp;context=honors201019">Sequences of Spiral Knot Determinants</a>, Senior Honors Projects. Paper 84. James Madison Univ., May 2016.

%H F. V. Waugh and M. W. Maxfield, <a href="http://www.jstor.org/stable/2688511">Side-and-diagonal numbers</a>, Math. Mag., 40 (1967), 74-83.

%H H. C. Williams and R. K. Guy, <a href="http://dx.doi.org/10.1142/S1793042111004587">Some fourth-order linear divisibility sequences</a>, Intl. J. Number Theory 7 (5) (2011) 1255-1277.

%H H. C. Williams and R. K. Guy, <a href="http://www.emis.de/journals/INTEGERS/papers/a17self/a17self.Abstract.html">Some Monoapparitic Fourth Order Linear Divisibility Sequences</a>, Integers, Volume 12A (2012) The John Selfridge Memorial Volume

%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4,-1)

%F a(n) = ((1+sqrt(3))^(2*n+1)+(1-sqrt(3))^(2*n+1))/2^(n+1). - _N. J. A. Sloane_, Nov 10 2009

%F a(n) = (1/2) * ((1+sqrt(3))*(2+sqrt(3))^n + (1-sqrt(3))*(2-sqrt(3))^n). - _Dean Hickerson_, Dec 01 2002

%F 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, Apr 11 2003

%F a(n) = ((1+sqrt[3])^(2*n+1) + (1-sqrt[3])^(2*n+1))/2^(n+1). - Anton Vrba, Feb 14 2007

%F G.f.: (1+x)/((1-4*x+x^2)). _Simon Plouffe_ in his 1992 dissertation.

%F 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 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_, Oct 10 2002

%F a(n) = 2*A001571(n) + 1. - Bruce Corrigan (scentman(AT)myfamily.com), Nov 04 2002

%F 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_, Nov 10 2002

%F a(n) = 2^(-n)*Sum_{k>=0} binomial(2*n+1, 2*k)*3^k; see A091042. - _Philippe Deléham_, Mar 01 2004

%F a(n) = floor(sqrt(3)*A001835(n+1)). - _Philippe Deléham_, Mar 03 2004

%F 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_, Apr 18 2005

%F a(n) = Jacobi_P(n,1/2,-1/2,2)/Jacobi_P(n,-1/2,1/2,1). - _Paul Barry_, Feb 03 2006

%F 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_, Nov 30 2007

%F 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 a(-1-n) = -a(n). - _Michael Somos_, Sep 19 2008

%e G.f. = 1 + 5*x + 19*x^2 + 71*x^3 + 265*x^4 + 989*x^5 + 3691*x^6 + ...

%p f:=n->((1+sqrt(3))^(2*n+1)+(1-sqrt(3))^(2*n+1))/2^(n+1); # _N. J. A. Sloane_, Nov 10 2009

%t a[0] = 1; a[1] = 5; a[n_] := a[n] = 4a[n - 1] - a[n - 2]; Table[ a[n], {n, 0, 25}] (* _Robert G. Wilson v_, Apr 24 2004 *)

%t Table[Expand[((1+Sqrt[3])^(2*n+1)+(1+Sqrt[3])^(2*n+1))/2^(n+1)],{n, 0, 20}] (* Anton Vrba, Feb 14 2007 *)

%t LinearRecurrence[{4, -1}, {1, 5}, 50] (* _Sture Sjöstedt_, Nov 27 2011 *)

%t a[c_, n_] := Module[{},

%t p := Length[ContinuedFraction[ Sqrt[ c]][[2]]];

%t d := Numerator[Convergents[Sqrt[c], n p]];

%t t := Table[d[[1 + i]], {i, 0, Length[d] - 1, p}];

%t Return[t];

%t ] (* Complement of A002531 *)

%t a[3, 20] (* _Gerry Martens_, Jun 07 2015 *)

%t Round@Table[LucasL[2n+1, Sqrt[2]]/Sqrt[2], {n, 0, 20}] (* _Vladimir Reshetnikov_, Sep 15 2016 *)

%o 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 (PARI) {a(n) = real( (2 + quadgen(12))^n * (1 + quadgen(12)) )}; /* _Michael Somos_, Sep 19 2008 */

%o (PARI) {a(n) = subst( polchebyshev(n-1, 2) + polchebyshev(n, 2), x, 2)}; /* _Michael Somos_, Sep 19 2008 */

%o (Sage) [(lucas_number2(n,4,1)-lucas_number2(n-1,4,1))/2 for n in xrange(1, 27)] # _Zerinvary Lajos_, Nov 10 2009

%o a001834 n = a001834_list !! (n-1)

%o a001834_list = 1 : 5 : zipWith (-) (map (* 4) \$ tail a001834_list) a001834_list

%o -- _Reinhard Zumkeller_, Jan 23 2012

%o (MAGMA) I:=[1,5]; [n le 2 select I[n] else 4*Self(n-1)-Self(n-2): n in [1..30]]; // _Vincenzo Librandi_, Mar 22 2015

%Y A bisection of sequence A002531.

%Y Cf. A001352, A001835, A086386 (prime members).

%Y Cf. A026150.

%Y Cf. A082841, A100047.

%Y Cf. A002531.

%Y a(n)^2+1 = A094347(n+1).

%K nonn,easy,nice,changed

%O 0,2

%A _N. J. A. Sloane_

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Last modified March 17 07:22 EDT 2018. Contains 300557 sequences. (Running on oeis4.)