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%I M3890 N1598 #200 Mar 12 2024 14:51:29
%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) = 4*a(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 (2*n) X (2*n) 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
%C From _Wolfdieter Lang_, Oct 15 2020: (Start)
%C ((-1)^n)*a(n) = X(n) = (-1)^n*(S(n, 4) + S(n-1, 4) and Y(n) = X(n-1) gives all integer solutions (modulo sign flip between X and Y) of X^2 + Y^2 + 4*X*Y = +6, for n = -oo..+oo, with Chebyshev S polynomials (see A049310), with S(-1, x) = 0, and S(-|n|, x) = - S(|n|-2, x), for |n| >= 2.
%C This binary indefinite quadratic form of discriminant 12, representing 6, has only this family of proper solutions (modulo sign flip), and no improper ones.
%C This comment is inspired by a paper by Robert K. Moniot (private communication). See his Oct 04 2020 comment in A027941 related to the case of x^2 + y^2 - 3*x*y = -1 (special Markov solutions). (End)
%C 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
%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, and 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 K. Andersen, L. Carbone, and D. Penta, <a href="https://pdfs.semanticscholar.org/8f0c/c3e68d388185129a56ed73b5d21224659300.pdf">Kac-Moody Fibonacci sequences, hyperbolic golden ratios, and real quadratic fields</a>, Journal of Number Theory and Combinatorics, Vol 2, No. 3 pp 245-278, 2011. See Section 9.
%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 Alex Fink, Richard K. Guy, and Mark Krusemeyer, <a href="https://doi.org/10.11575/cdm.v3i2.61940">Partitions with parts occurring at most thrice</a>, Contributions to Discrete Mathematics, Vol 3, No 2 (2008), pp. 76-114. See Section 13.
%H Taras Goy and Mark Shattuck, <a href="https://doi.org/10.2478/amsil-2023-0027">Determinants of Toeplitz-Hessenberg Matrices with Generalized Leonardo Number Entries</a>, Ann. Math. Silesianae (2023). See p. 17.
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H Seong Ju Kim, R. Stees, and 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 Wolfdieter 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 Ioana-Claudia Lazăr, <a href="https://arxiv.org/abs/1904.06555">Lucas sequences in t-uniform simplicial complexes</a>, arXiv:1904.06555 [math.GR], 2019.
%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="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions and Conjectures</a>, Universite du Quebec a Montreal, 1992.
%H Ryan Stees, <a href="https://commons.lib.jmu.edu/honors201019/84">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 From Mario Catalani, Apr 11 2003: (Start)
%F With a = 2 + sqrt(3), b = 2 - sqrt(3): a(n) = (1/sqrt(2))(a^(n + 1/2) - b^(n + 1/2)).
%F a(n) - a(n-1) = A003500(n).
%F a(n) = sqrt(1 + 12*A061278(n) + 12*A061278(n)^2). (End)
%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. Limit_{n->infinity} 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
%F From _Franck Maminirina Ramaharo_, Nov 11 2018: (Start)
%F a(n) = (-1)^n*(5*A125905(n) + A125905(n+1)).
%F E.g.f.: exp(2*x)*(cosh(sqrt(3)*x) + sqrt(3)*sinh(sqrt(3)*x)). (End)
%F a(n) = A061278(n+1) - A061278(n-1) for n>=2. - _John P. McSorley_, Jun 20 2020
%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 (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 (SageMath) [(lucas_number2(n,4,1)-lucas_number2(n-1,4,1))/2 for n in range(1, 27)] # _Zerinvary Lajos_, Nov 10 2009
%o (Haskell)
%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, A001353, A049310.
%Y a(n)^2+1 = A094347(n+1).
%K nonn,easy,nice
%O 0,2
%A _N. J. A. Sloane_
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