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A001834 a(0) = 1, a(1) = 5, a(n) = 4a(n-1) - a(n-2).
(Formerly M3890 N1598)
59
1, 5, 19, 71, 265, 989, 3691, 13775, 51409, 191861, 716035, 2672279, 9973081, 37220045, 138907099, 518408351, 1934726305, 7220496869, 26947261171, 100568547815, 375326930089, 1400739172541, 5227629760075, 19509779867759, 72811489710961, 271736178976085 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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

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

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

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

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

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]

Inverse binomial transform of A030192. - Philippe Deléham, Nov 19 2009

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

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

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

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

REFERENCES

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)

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

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

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

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

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

LINKS

T. D. Noe, Table of n, a(n) for n=0..200

Marco Abrate, Stefano Barbero, Umberto Cerruti, Nadir Murru, Polynomial sequences on quadratic curves, Integers, Vol. 15, 2015, #A38.

J. B. Cosgrave and K. Dilcher, A role for generalized Fermat numbers, Math. Comp. 86 (2017), 899-933; see also Paper #10.

Bruno Deschamps, Sur les bonnes valeurs initiales de la suite de Lucas-Lehmer, Journal of Number Theory, Volume 130, Issue 12, December 2010, Pages 2658-2670.

L. Euler, Vollstaendige Anleitung zur Algebra, Zweiter Teil.

Tanya Khovanova, Recursive Sequences

Seong Ju Kim, R. Stees, L. Taalman, Sequences of Spiral Knot Determinants, Journal of Integer Sequences, Vol. 19 (2016), # 16.1.4

Clark Kimberling, Best lower and upper approximates to irrational numbers, Elemente der Mathematik, 52 (1997) 122-126.

W. Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fib. Quart. 38 (2000) 408-419. Eq. (44) rhs, m=6.

Donatella Merlini and Renzo Sprugnoli, Arithmetic into geometric progressions through Riordan arrays, Discrete Mathematics 340.2 (2017): 160-174.

S. Northshield, An Analogue of Stern's Sequence for Z[sqrt(2)], Journal of Integer Sequences, 18 (2015), #15.11.6.

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Universite du Quebec a Montreal, 1992.

Ryan Stees, Sequences of Spiral Knot Determinants, Senior Honors Projects. Paper 84. James Madison Univ., May 2016.

F. V. Waugh and M. W. Maxfield, Side-and-diagonal numbers, Math. Mag., 40 (1967), 74-83.

H. C. Williams and R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory 7 (5) (2011) 1255-1277.

H. C. Williams and R. K. Guy, Some Monoapparitic Fourth Order Linear Divisibility Sequences, Integers, Volume 12A (2012) The John Selfridge Memorial Volume

Index entries for sequences related to Chebyshev polynomials.

Index entries for linear recurrences with constant coefficients, signature (4,-1)

FORMULA

a(n) = ((1+sqrt(3))^(2*n+1)+(1-sqrt(3))^(2*n+1))/2^(n+1). - N. J. A. Sloane, Nov 10 2009

a(n) = (1/2) * ((1+sqrt(3))*(2+sqrt(3))^n + (1-sqrt(3))*(2-sqrt(3))^n). - Dean Hickerson, Dec 01 2002

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

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

G.f.: (1+x)/((1-4*x+x^2)). Simon Plouffe in his 1992 dissertation.

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).

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

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

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

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

a(n) = floor(sqrt(3)*A001835(n+1)). - Philippe Deléham, Mar 03 2004

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

a(n) = Jacobi_P(n,1/2,-1/2,2)/Jacobi_P(n,-1/2,1/2,1). - Paul Barry, Feb 03 2006

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

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

a(-1-n) = -a(n). - Michael Somos, Sep 19 2008

EXAMPLE

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

MAPLE

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

MATHEMATICA

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 *)

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 *)

LinearRecurrence[{4, -1}, {1, 5}, 50] (* Sture Sjöstedt, Nov 27 2011 *)

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

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

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

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

   Return[t];

] (* Complement of A002531 *)

a[3, 20] (* Gerry Martens, Jun 07 2015 *)

Round@Table[LucasL[2n+1, Sqrt[2]]/Sqrt[2], {n, 0, 20}] (* Vladimir Reshetnikov, Sep 15 2016 *)

PROG

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

(PARI) {a(n) = real( (2 + quadgen(12))^n * (1 + quadgen(12)) )}; /* Michael Somos, Sep 19 2008 */

(PARI) {a(n) = subst( polchebyshev(n-1, 2) + polchebyshev(n, 2), x, 2)}; /* Michael Somos, Sep 19 2008 */

(Sage) [(lucas_number2(n, 4, 1)-lucas_number2(n-1, 4, 1))/2 for n in xrange(1, 27)] # Zerinvary Lajos, Nov 10 2009

(Haskell)

a001834 n = a001834_list !! (n-1)

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

-- Reinhard Zumkeller, Jan 23 2012

(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

CROSSREFS

A bisection of sequence A002531.

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

Cf. A026150.

Cf. A082841, A100047.

Cf. A002531.

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

Sequence in context: A095073 A128349 A255449 * A099393 A083588 A149759

Adjacent sequences:  A001831 A001832 A001833 * A001835 A001836 A001837

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified August 18 14:09 EDT 2017. Contains 290720 sequences.