A084158 a(n) = A000129(n)*A000129(n+1)/2.

0, 1, 5, 30, 174, 1015, 5915, 34476, 200940, 1171165, 6826049, 39785130, 231884730, 1351523251, 7877254775, 45912005400, 267594777624, 1559656660345, 9090345184445, 52982414446326, 308804141493510, 1799842434514735

Offset

0,3

Comments

May be called Pell triangles.

References

S. Falcon, On the Sequences of Products of Two k-Fibonacci Numbers, American Review of Mathematics and Statistics, March 2014, Vol. 2, No. 1, pp. 111-120.

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..500

P. Barry, Symmetric Third-Order Recurring Sequences, Chebyshev Polynomials, and Riordan Arrays, JIS 12 (2009) 09.8.6

Index entries for linear recurrences with constant coefficients, signature (5,5,-1).

Formula

a(n) = ((sqrt(2)+1)^(2*n+1) - (sqrt(2)-1)^(2*n+1) - 2*(-1)^n)/16.

a(n) = 5*a(n-1) + 5*a(n-2) - a(n-3). - Mohamed Bouhamida, Sep 02 2006; corrected by Antonio Alberto Olivares, Mar 29 2008

a(n) = (-1/8)*(-1)^n + (( sqrt(2)+1)/16)*(3+2*sqrt(2))^n + ((-sqrt(2)+1)/16)*(3-2*sqrt(2))^n. - Antonio Alberto Olivares, Mar 30 2008

sqrt(a(n) - a(n-1)) = A000129(n). - Antonio Alberto Olivares, Mar 30 2008

O.g.f.: x/((1+x)(1-6*x+x^2)). - R. J. Mathar, May 18 2008

a(n) = A041011(n)*A041011(n+1). - R. K. Guy, May 18 2008

From Mohamed Bouhamida, Aug 30 2008: (Start)

a(n) = 6*a(n-1) - a(n-2) - (-1)^n.

a(n) = 7*(a(n-1) - a(n-2)) + a(n-3) - 2*(-1)^n. (End)

In general, for n>k+1, a(n+k) = A003499(k+1)*a(n-1) - a(n-k-2) - (-1)^n A000129(k+1)^2. - Charlie Marion, Jan 04 2012

For n>0, a(2n-1)*a(2n+1) = oblong(a(2n)); a(2n)*a(2n+2) = oblong(a(2n+1)-1). - Charlie Marion, Jan 09 2012

a(n) = A046729(n)/4. - Wolfdieter Lang, Mar 07 2012

a(n) = sum of squares of first n Pell numbers A000129 (A079291). - N. J. A. Sloane, Jun 18 2012

Maple

with(combinat): a:=n->fibonacci(n, 2)*fibonacci(n-1, 2)/2: seq(a(n), n=1..22); # Zerinvary Lajos, Apr 04 2008

Mathematica

LinearRecurrence[{5, 5, -1}, {0, 1, 5}, 30] (* Harvey P. Dale, Sep 07 2011 *)

Prog

(Magma) [Floor(((Sqrt(2)+1)^(2*n+1)-(Sqrt(2)-1)^(2*n+1)-2*(-1)^n)/16): n in [0..35]]; // Vincenzo Librandi, Jul 05 2011
(PARI) Pell(n)=([2, 1; 1, 0]^n)[2, 1];
a(n)=Pell(n)*Pell(n+1)/2 \\ Charles R Greathouse IV, Mar 21 2016
(PARI) a(n)=([0, 1, 0; 0, 0, 1; -1, 5, 5]^n*[0; 1; 5])[1, 1] \\ Charles R Greathouse IV, Mar 21 2016
(SageMath) [(lucas_number2(2*n+1, 2, -1) -2*(-1)^n)/16 for n in (0..30)] # G. C. Greubel, Aug 18 2022

Crossrefs

Cf. A000129, A001652, A001654, A002203, A003499.

Cf. A041011, A046729, A079291, A084159, A084175.

Sequence in context: A055838 A318591 A094972 * A111469 A241588 A229246

Adjacent sequences: A084155 A084156 A084157 * A084159 A084160 A084161

Keyword

easy,nonn

Author

Paul Barry, May 18 2003

Status

approved