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%I A001076 M3538 N1434 #327 Aug 10 2024 21:38:40
%S A001076 0,1,4,17,72,305,1292,5473,23184,98209,416020,1762289,7465176,
%T A001076 31622993,133957148,567451585,2403763488,10182505537,43133785636,
%U A001076 182717648081,774004377960,3278735159921,13888945017644,58834515230497,249227005939632,1055742538989025
%N A001076 Denominators of continued fraction convergents to sqrt(5).
%C A001076 a(2*n+1) with b(2*n+1) := A001077(2*n+1), n >= 0, give all (positive integer) solutions to Pell equation b^2 - 5*a^2 = -1, a(2*n) with b(2*n) := A001077(2*n), n >= 1, give all (positive integer) solutions to Pell equation b^2 - 5*a^2 = +1 (cf. Emerson reference).
%C A001076 Bisection: a(2*n+1) = T(2*n+1, sqrt(5))/sqrt(5) = A007805(n), n >= 0 and a(2*n) = 4*S(n-1,18), n >= 0, with T(n,x), resp. S(n,x), Chebyshev's polynomials of the first, resp. second kind. S(-1,x)=0. See A053120, resp. A049310. S(n,18)=A049660(n+1). - _Wolfdieter Lang_, Jan 10 2003
%C A001076 Apart from initial terms, this is the Pisot sequence E(4,17), a(n) = floor(a(n-1)^2/a(n-2) + 1/2).
%C A001076 This is also the Horadam sequence (0,1,1,4), having the recurrence relation a(n) = s*a(n-1) + r*a(n-2); for n > 1, where a(0) = 0, a(1) = 1, s = 4, r = 1. a(n) / a(n-1) converges to 5^1/2 + 2 as n approaches infinity. 5^(1/2) + 2 can also be written as (2 * Phi) + 1 and Phi^2 + Phi. - _Ross La Haye_, Aug 18 2003
%C A001076 Numerators of continued fraction [4, 4, 4, ...], where the convergents to [4, 4, 4, ...] = (4/1, 17/4, 72/17, ...). Let X = the 2 X 2 matrix [0, 1; 1, 4]; then X^n = [a(n-1), a(n); a(n), a(n+1)]; e.g., X^3 = [4, 17; 17, 72]. Let C = the limit of a(n)/a(n-1) = 2 + sqrt(5) = 4.236067977...; then C^n = a(n+1) + (1/C)*a(n), where (1/C) = 0.236067977... . Example: C^3 = 76.01315556..., = 72 + 17*(0.2360679...). - _Gary W. Adamson_, Dec 15 2007, corrected by _Greg Dresden_, Sep 16 2019, corrected by _Alex Mark_, Jul 21 2020
%C A001076 Sqrt(5) = 4/2 + 4/17 + 4/(17*305) + 4/(305*5473) + 4/(5473*98209) + ... . - _Gary W. Adamson_, Dec 15 2007
%C A001076 a(p) == 20^((p-1)/2) (mod p) for odd primes p. - _Gary W. Adamson_, Feb 22 2009
%C A001076 a(n) = A167808(3*n). - _Reinhard Zumkeller_, Nov 12 2009
%C A001076 For n >= 2, a(n) equals the permanent of the (n-1) X (n-1) tridiagonal matrix with 4's along the main diagonal and 1's along the superdiagonal and the subdiagonal. - _John M. Campbell_, Jul 08 2011
%C A001076 Moreover, a(n) is the second binomial transform of (0,1,0,5,0,25,...) (see also A033887). This fact can be proved similarly like the proof of _Paul Barry_'s remark in A033887 by using the following scaling identity for delta-Fibonacci numbers: y^n b(n;x/y) = Sum_{k=0..n} binomial(n,k) (y-1)^(n-k) b(k;x) and the fact that b(n;2) = (1-(-1)^n) 5^floor(n/2). - _Roman Witula_, Jul 12 2012
%C A001076 Binomial transform of 0, 1, 2, 8, 24, 80, 256, ... (A063727 with offset 1). - _R. J. Mathar_, Feb 05 2014
%C A001076 For n >= 1, a(n) equals the number of words of length n-1 on alphabet {0,1,...,4} avoiding runs of zeros of odd lengths. - _Milan Janjic_, Jan 28 2015
%C A001076 With offset 1 is the INVERT transform of A006190: (1, 3, 10, 33, 109, 360, ...). - _Gary W. Adamson_, Jul 24 2015
%C A001076 From _Rogério Serôdio_, Mar 30 2018: (Start)
%C A001076 This is a divisibility sequence (i.e., if n|m then a(n)|a(m)).
%C A001076 gcd(a(n),a(n+k)) = a(gcd(n, k)) for all positive integers n and k. (End)
%C A001076 The initial 0 of this sequence is in contradiction with the fact that 0 is no valid denominator and according to all standard references, the first convergent of a continued fraction is p(0)/q(0) = b(0)/1 where b(0) is the first term of the continued fraction, given by the integer part of the number. One may artificially define q(-1) = 0 to have a recurrent relation q(n) = b(n)*q(n-1) + q(n-2), n >= 1, but then its index should be -1. - _M. F. Hasler_, Nov 01 2019
%C A001076 Number of 4-compositions of n restricted to odd parts (and allowed zeros); see Hopkins & Ouvry reference. - _Brian Hopkins_, Aug 17 2020
%C A001076 From _Michael A. Allen_, Feb 15 2023: (Start)
%C A001076 Also called the 4-metallonacci sequence; the g.f. 1/(1-k*x-x^2) gives the k-metallonacci sequence.
%C A001076 a(n+1) is the number of tilings of an n-board (a board with dimensions n X 1) using unit squares and dominoes (with dimensions 2 X 1) if there are 4 kinds of squares available. (End)
%C A001076 a(n) is the smallest nonnegative integer that is the sum of n, but no fewer, Fibonacci numbers including negative-index Fibonacci numbers (A039834), with that sum being a(n) = Sum_{i=0..n-1} A000045(3*i+1). a(n) is also the smallest nonnegative integer that is the sum of n, but no fewer, terms each of which is either a Fibonacci number or the negative of a Fibonacci number. (See A027941 for negatives disallowed.) - _Mike Speciner_, Oct 08 2023
%C A001076 From _Enrique Navarrete_, Dec 16 2023: (Start)
%C A001076 a(n) is the number of compositions of n when there are P(k) sorts of parts k, with k,n > = 1, where P(k) = A006190(k) is the k-th 3-metallonacci number (see example below).
%C A001076 In general, the number of compositions with k-metallonacci number of parts is counted by the (k+1)-st metallonacci sequence (note k=1 and k=2 are the Fibonacci and the Pell numbers, respectively). (End).
%D A001076 A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 23.
%D A001076 S. Koshkin, Non-classical linear divisibility sequences ..., Fib. Q., 57 (No. 1, 2019), 68-80. See Table 1.
%D A001076 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A001076 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A001076 V. Thébault, Les Récréations Mathématiques. Gauthier-Villars, Paris, 1952, p. 282.
%H A001076 G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..200 from T. D. Noe)
%H A001076 Michael A. Allen and Kenneth Edwards, Fence tiling derived identities involving the metallonacci numbers squared or cubed, Fib. Q. 60:5 (2022) 5-17.
%H A001076 Paraskevas K. Alvanos and Konstantinos A. Draziotis, Integer Solutions of the Equation y^2 = Ax^4 + B, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.4.
%H A001076 Elwyn Berlekamp and Richard K. Guy, Fibonacci Plays Billiards (2003), arXiv:2002.03705 [math.HO], 2020. See p. 5.
%H A001076 D. Birmajer, J. B. Gil, and M. D. Weiner, On the Enumeration of Restricted Words over a Finite Alphabet, J. Int. Seq. 19 (2016) # 16.1.3, Example 8.
%H A001076 D. W. Boyd, Some integer sequences related to the Pisot sequences, Acta Arithmetica, 34 (1979), 295-305.
%H A001076 D. W. Boyd, Linear recurrence relations for some generalized Pisot sequences, Advances in Number Theory ( Kingston ON, 1991) 333-340, Oxford Sci. Publ., Oxford Univ. Press, New York, 1993.
%H A001076 Latham Boyle and Paul J. Steinhardt, Self-Similar One-Dimensional Quasilattices, arXiv preprint arXiv:1608.08220 [math-ph], 2016.
%H A001076 E. I. Emerson, Recurrent Sequences in the Equation DQ^2=R^2+N, Fib. Quart., 7 (1969), pp. 231-242, Thm. 1, p. 233.
%H A001076 M. C. Firengiz and A. Dil, Generalized Euler-Seidel method for second order recurrence relations, Notes on Number Theory and Discrete Mathematics, Vol. 20, 2014, No. 4, 21-32.
%H A001076 Yixing Fu, E. J. König, J. H. Wilson, Yang-Zhi Chou, and J. H. Pixley, Magic-angle semimetals, arXiv:1809.04604 [cond-mat.str-el], 2018.
%H A001076 Juan B. Gil and Aaron Worley, Generalized metallic means, arXiv:1901.02619 [math.NT], 2019.
%H A001076 Brian Hopkins and Stéphane Ouvry, Combinatorics of Multicompositions, arXiv:2008.04937 [math.CO], 2020.
%H A001076 INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 398
%H A001076 Milan Janjic, On Linear Recurrence Equations Arising from Compositions of Positive Integers, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.7.
%H A001076 Tanya Khovanova, Recursive Sequences
%H A001076 C. Pita, On s-Fibonomials, J. Int. Seq. 14 (2011) # 11.3.7.
%H A001076 Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H A001076 Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
%H A001076 Kai Wang, On k-Fibonacci Sequences And Infinite Series List of Results and Examples, 2020.
%H A001076 Shaoxiong (Steven) Yuan, Generalized Identities of Certain Continued Fractions, arXiv:1907.12459 [math.NT], 2019.
%H A001076 Index entries for sequences related to Chebyshev polynomials.
%H A001076 Index entries for linear recurrences with constant coefficients, signature (4,1).
%F A001076 a(n) = 4*a(n-1) + a(n-2), n > 1. a(0)=0, a(1)=1.
%F A001076 G.f.: x/(1 - 4*x - x^2).
%F A001076 a(n) = ((2+sqrt(5))^n - (2-sqrt(5))^n)/(2*sqrt(5)).
%F A001076 a(n) = A014445(n)/2 = F(3n)/2.
%F A001076 a(n) = ((-i)^(n-1))*S(n-1, 4*i), with i^2 = -1 and S(n, x) := U(n, x/2) Chebyshev's polynomials of the second kind. See A049310. S(-1, x) = 0.
%F A001076 a(n) = Sum_{i=0..n} Sum_{j=0..n} Fibonacci(i+j)*n!/(i!j!(n-i-j)!)/2. - _Paul Barry_, Feb 06 2004
%F A001076 E.g.f.: exp(2*x)*sinh(sqrt(5)*x)/sqrt(5). - _Vladeta Jovovic_, Sep 01 2004
%F A001076 a(n) = F(1) + F(4) + F(7) + ... + F(3n-2), for n > 0.
%F A001076 Conjecture: 2a(n+1) = a(n+2) - A001077(n+1). - _Creighton Dement_, Nov 28 2004
%F A001076 a(n) = Sum_{k=0..n} Sum_{j=0..n} C(n, j)*C(j, k)*F(j)/2. - _Paul Barry_, Feb 14 2005
%F A001076 a(n) = A048876(n) - A048875(n). - _Creighton Dement_, Mar 19 2005
%F A001076 Let M = {{0, 1}, {1, 4}}, v[1] = {0, 1}, v[n] = M.v[n - 1]; then a(n) = v[n][[1]]. - _Roger L. Bagula_, May 29 2005
%F A001076 a(n) = F(n, 4), the n-th Fibonacci polynomial evaluated at x=4. - _T. D. Noe_, Jan 19 2006
%F A001076 [A015448(n), a(n)] = [1,4; 1,3]^n * [1,0]. - _Gary W. Adamson_, Mar 21 2008
%F A001076 a(n) = (Sum_{k=0..n} Fibonacci(3*k-2)) + 1. - _Gary Detlefs_, Dec 26 2010
%F A001076 a(n) = (3*(-1)^n*F(n) + 5*F(n)^3)/2, n >= 0. See the general D. Jennings formula given in a comment on triangle A111125, where also the reference is given. Here the second (k=1) row [3,1] applies. - _Wolfdieter Lang_, Sep 01 2012
%F A001076 Sum_{k>=1} (-1)^(k-1)/(a(k)*a(k+1)) = (Sum_{k>=1} (-1)^(k-1)/(F_k*F_(k+1)))^3 = phi^(-3), where F_n is the n-th Fibonacci numbers (A000045) and phi is golden ratio (A001622). - _Vladimir Shevelev_, Feb 23 2013
%F A001076 G.f.: Q(0)*x/(2-4*x), where Q(k) = 1 + 1/(1 - x*(5*k-4)/(x*(5*k+1) - 2/Q(k+1) )); (continued fraction). - _Sergei N. Gladkovskii_, Oct 11 2013
%F A001076 a(-n) = -(-1)^n * a(n). - _Michael Somos_, Feb 23 2014
%F A001076 The o.g.f. A(x) = x/(1 - 4*x - x^2) satisfies A(x) + A(-x) + 8*A(x)*A(-x) = 0 or equivalently (1 + 8*A(x))*(1 + 8*A(-x)) = 1. The o.g.f. for A049660 equals -A(sqrt(x))*A(-sqrt(x)). - _Peter Bala_, Apr 02 2015
%F A001076 From _Rogério Serôdio_, Mar 30 2018: (Start)
%F A001076 Some properties:
%F A001076 (1) a(n)*a(n+1) = 4*Sum_{k=1..n} a(k)^2;
%F A001076 (2) a(n)^2 + a(n+1)^2 = a(2*n+1);
%F A001076 (3) a(n)^2 - a(n-2)^2 = 4*a(n-1)*(a(n) + a(n-2));
%F A001076 (4) a(m*(p+1)) = a(m*p)*a(m+1) + a(m*p-1)*a(m);
%F A001076 (5) a(n-k)*a(n+k) = a(n)^2 + (-1)^(n+k+1)*a(k)^2;
%F A001076 (6) a(n-1)*a(n+1) = a(n)^2 + (-1)^n (particular case of (5)!);
%F A001076 (7) a(2*n) = 2*a(n)*(2*a(n) + a(n-1));
%F A001076 (8) 3*Sum_{k=2..n+1} a(k)*a(k-1) is equal to a(n+1)^2 if n odd, and is equal to a(n+1)^2 - 1 if n is even;
%F A001076 (9) a(n) - a(n-2*k+1) = alpha(k)*a(n-2*k+1) + a(n-4*k+2), where alpha(k) = (2+sqrt(5))^(2*k-1) + (2-sqrt(5))^(2*k-1);
%F A001076 (10) 31|Sum_{k=n..n+9} a(k), for all positive n. (End)
%F A001076 O.g.f.: x*exp(Sum_{n >= 1} Lucas(3*n)*x^n/n) = x + 4*x^2 + 17*x^3 + .... - _Peter Bala_, Oct 11 2019
%F A001076 a(n) = Sum_{k=0..floor(n/2)} 4^(n-2*k-1)*C(n-k-1,n-2*k-1). - _Vladimir Kruchinin_, Oct 02 2022
%F A001076 a(n) = i^(n-1)*S(n-1, -4*i), with i = sqrt(-1), and the Chebyshev S-polynomials (see A049310) with S(n, -1) = 0. - _Gary Detlefs_ and _Wolfdieter Lang_, Mar 06 2023
%F A001076 G.f.: x/(1 - 4*x - x^2) = Sum_{n >= 0} x^(n+1) * ( Product_{k = 1..n} (m*k + 4 - m + x)/(1 + m*k*x) ) for arbitrary m (a telescoping series). - _Peter Bala_, May 08 2024
%e A001076 1 2 9 38 161 (A001077)
%e A001076 -,-,-,--,---, ...
%e A001076 0 1 4 17 72 (A001076)
%e A001076 G.f. = x + 4*x^2 + 17*x^3 + 72*x^4 + 305*x^5 + 1292*x^6 + 5473*x^7 + 23184*x^8 + ...
%e A001076 From _Enrique Navarrete_, Dec 16 2023: (Start)
%e A001076 From the comment on compositions with 3-metallonacci sorts of parts, A006190(k), there are A006190(1)=1 type of 1, A006190(2)=3 types of 2, A006190(3)=10 types of 3, A006190(4)=33 types of 4, A006190(5)=109 types of 5 and A006190(6)=360 types of 6. The following table gives the number of compositions of n=6:
%e A001076 Composition, number of such compositions, number of compositions of this type:
%e A001076 6, 1, 360;
%e A001076 5+1, 2, 218;
%e A001076 4+2, 2, 198;
%e A001076 3+3, 1, 100;
%e A001076 4+1+1, 3, 99;
%e A001076 3+2+1, 6, 180;
%e A001076 2+2+2, 1, 27;
%e A001076 3+1+1+1, 4, 40;
%e A001076 2+2+1+1, 6, 54;
%e A001076 2+1+1+1+1, 5, 15;
%e A001076 1+1+1+1+1+1, 1, 1;
%e A001076 for a total of a(6)=1292 compositions of n=6. (End)
%p A001076 A001076:=-1/(-1+4*z+z**2); # conjectured by _Simon Plouffe_ in his 1992 dissertation
%t A001076 Join[{0}, Denominator[Convergents[Sqrt[5], 30]]] (* _Harvey P. Dale_, Dec 10 2011 *)
%t A001076 a[ n_] := Fibonacci[3*n] / 2; (* _Michael Somos_, Feb 23 2014 *)
%t A001076 a[ n_] := ((2 + Sqrt[5])^n - (2 - Sqrt[5])^n) /(2 Sqrt[5]) // Simplify; (* _Michael Somos_, Feb 23 2014 *)
%t A001076 LinearRecurrence[{4, 1}, {0, 1}, 26] (* _Jean-François Alcover_, Sep 23 2017 *)
%t A001076 a[ n_] := Fibonacci[n, 4]; (* _Michael Somos_, Nov 02 2021 *)
%o A001076 (MuPAD) numlib::fibonacci(3*n)/2 $ n = 0..30; // _Zerinvary Lajos_, May 09 2008
%o A001076 (Sage) [lucas_number1(n,4,-1) for n in range(23)] # _Zerinvary Lajos_, Apr 23 2009
%o A001076 (Sage) [fibonacci(3*n)/2 for n in range(23)] # _Zerinvary Lajos_, May 15 2009
%o A001076 (PARI) {a(n) = fibonacci(3*n) / 2}; /* _Michael Somos_, Aug 11 2009 */
%o A001076 (PARI) {a(n) = imag( (2 + quadgen(20))^n )}; /* _Michael Somos_, Feb 23 2014 */
%o A001076 (PARI) {a(n) = polchebyshev(n-1, 2, 2*I)/I^(n-1)}; /* _Michael Somos_, Nov 02 2021 */
%o A001076 (Magma) I:=[0,1]; [n le 2 select I[n] else 4*Self(n-1) + Self(n-2): n in [1..30]]; // _G. C. Greubel_, Jan 24 2018
%o A001076 (GAP) a:=[0,1];; for n in [3..30] do a[n]:=4*a[n-1]+a[n-2]; od; a; # _Muniru A Asiru_, Mar 31 2018
%o A001076 (Maxima) a(n):=sum(4^(n-1-2*k)*binomial(n-k-1,n-2*k-1),k,0,floor((n)/2));/* _Vladimir Kruchinin_, Oct 02 2022 */
%Y A001076 Row n=4 of A073133, A172236 and A352361.
%Y A001076 Cf. A000045, A001077, A015448, A175183 (Pisano periods).
%Y A001076 Cf. A049660, A007805.
%Y A001076 Partial sums of A033887. First differences of A049652. Bisection of A059973.
%Y A001076 Third column of array A028412.
%K A001076 nonn,easy,cofr,frac,nice
%O A001076 0,3
%A A001076 _N. J. A. Sloane_
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