A032908 One of four 3rd-order recurring sequences for which the first derived sequence and the Galois transformed sequence coincide.

2, 2, 3, 6, 14, 35, 90, 234, 611, 1598, 4182, 10947, 28658, 75026, 196419, 514230, 1346270, 3524579, 9227466, 24157818, 63245987, 165580142, 433494438, 1134903171, 2971215074, 7778742050, 20365011075, 53316291174, 139583862446, 365435296163, 956722026042

Offset: 0

Author

Michele Elia (elia(AT)polito.it)

Keywords

• eigen
• nonn
• easy

Comments

a(n) is also a sequence with the property that the difference between the sum and product of two consecutive terms is equal to the square of the difference between those terms, i.e., a(n)*a(n+1) - (a(n)+ a(n+1)) = (a(n) - a(n + 1))^2. The difference between those two terms, a(n + 1) - a(n) = F(2n -2), the (2n - 2)th Fibonacci number. - John Baker, May 18 2010

Conjecture: consecutive terms of this sequence and consecutive terms of A101265 provide all the positive integer solutions of (a+b)*(a+b+1) == 0 (mod (a*b)). - Robert Israel, Aug 26 2015

Conjecture is true: see Mathematics Stack Exchange link. - Robert Israel, Sep 06 2015

Consecutive terms of this sequence and consecutive terms of A101879 provide all the positive integer pairs for which K = (a+1)/b + (b+1)/a is an integer. For this sequence, K = 3. - Andrey Vyshnevyy, Sep 18 2015

References

• L. E. Dickson, History of the Theory of Numbers, Dover, New York, 1971.

Links

• Robert Israel, Table of n, a(n) for n = 0..2154
• Michele Elia, A Note on derived linear recurring sequences, pp. 83-92 of Proceedings Seventh Int. Conference on Fibonacci Numbers and their Applications (Austria, 1996), Applications of Fibonacci Numbers, Volume 7.
• Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]
• Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 919.
• Robert Israel, Will Jagy et al., Diophantine equation (x+y)(x+y+1)-kxy=0, Mathematics Stack Exchange, Sep 1 2015.
• Hieu D. Nguyen and Douglas Taggart, Mining the OEIS: Ten Experimental Conjectures, 2013. Mentions this sequence. - From _N. J. A. Sloane_, Mar 16 2014
• Index entries for sequences related to Chebyshev polynomials.
• Index entries for linear recurrences with constant coefficients, signature (4,-4,1).

Crossrefs

Cf. A001519, A001622, A049310, A101265, A104457.

Programs

Magma

[2] cat [n le 1 select 2 else Floor((3*Self(n-1) + Sqrt(5*Self(n-1)^2 - 10*Self(n-1) + 1) - 1)/2): n in [1..30]]; // Vincenzo Librandi, Aug 27 2015

Maple

f:= proc(n) option remember; local x;
  x:= procname(n-1); (3*x + sqrt(5*x^2 - 10*x + 1) - 1)/2 end proc:
f(0):= 2: f(1):= 2:
map(f, [$0..30]); # Robert Israel, Aug 26 2015

Mathematica

Table[Fibonacci[2 n - 1] + 1, {n, 0, 27}] (* Michael De Vlieger, Aug 26 2015 *)
LinearRecurrence[{4,-4,1},{2,2,3},40] (* Harvey P. Dale, Apr 11 2018 *)

PARI

Vec((2-6*x+3*x^2)/(1-4*x+4*x^2-x^3)+O(x^66)) \\ Joerg Arndt, Jul 02 2013

Formula

G.f.: (2 - 6*x + 3*x^2)/((1 - x)*(1 - 3*x + x^2)).

a(n) = 4*a(n-1) - 4*a(n-2) + a(n-3).

a(n) = Fibonacci(2*n-1) + 1 = A001519(n) + 1. - Vladeta Jovovic, Mar 19 2003

a(n) = 3*a(n - 1) - a(n - 2) - 1. - N. Sato, Jan 21 2010

From Wolfdieter Lang, Aug 27 2014: (Start)

a(n) = 1 + S(n-1, 3) - S(n-2, 3) = 1 + A001519(n), with Chebyshev S-polynomials (see A049310). For n < 0, we have S(-1, x) = 0 and S(-2, x) = -1.

This follows from the partial fraction decomposition of the g.f., 1/(1 - x) + (1 - 2*x)/ (1 - 3*x + x^2), using the recurrence for S, or from A001519. (End)

From Robert Israel, Aug 26 2015: (Start)

(a(n) + a(n+1))*(a(n) + a(n+1) + 1) = 5*a(n)*a(n+1).

a(n+1) = (3*a(n) + sqrt(5*a(n)^2 - 10*a(n) + 1) - 1)/2 for n >= 1. (End)

a(n) = 1 + (2^(-1-n) * ((3 - sqrt(5))^n * (1 + sqrt(5)) + (-1 + sqrt(5)) * (3 + sqrt(5))^n)) / sqrt(5). - Colin Barker, Nov 02 2016

Sum_{n>=0} 1/a(n) = phi (A001622). - Amiram Eldar, Oct 05 2020

Product_{n>=1} (1 + 1/a(n)) = (3+sqrt(5))/2 = phi^2 (A104457). - Amiram Eldar, Nov 28 2024

Extensions

More terms from Ralf Stephan, Mar 10 2003

Index for Chebyshev polynomials and cross reference added by Wolfdieter Lang, Aug 27 2014