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, 10490250465594899, 61141660359054660, 356359711688733060
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..500
- Paul Barry, Symmetric Third-Order Recurring Sequences, Chebyshev Polynomials, and Riordan Arrays, JIS 12 (2009) 09.8.6.
- Sergio 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.
- Ronald Orozco López, Generating Functions of Generalized Simplicial Polytopic Numbers and (s,t)-Derivatives of Partial Theta Function, arXiv:2408.08943 [math.CO], 2024. See p. 11.
- Ronald Orozco López, Simplicial d-Polytopic Numbers Defined on Generalized Fibonacci Polynomials, arXiv:2501.11490 [math.CO], 2025. See p. 6.
- Index entries for linear recurrences with constant coefficients, signature (5,5,-1).
Crossrefs
Programs
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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
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Maple
with(combinat): a:=n->fibonacci(n,2)*fibonacci(n-1,2)/2: seq(a(n), n=1..22); # Zerinvary Lajos, Apr 04 2008
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Mathematica
LinearRecurrence[{5,5,-1},{0,1,5},30] (* Harvey P. Dale, Sep 07 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
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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
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SageMath
[(lucas_number2(2*n+1,2,-1) -2*(-1)^n)/16 for n in (0..30)] # G. C. Greubel, Aug 18 2022
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
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) = (A002315(n) - (-1)^n)/8. - Adam Mohamed, Sep 05 2024
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*(sqrt(2)-1) (A163960). - Amiram Eldar, Dec 02 2024
G.f.: x * exp( Sum_{k>=1} Pell(3*k)/Pell(k) * x^k/k ). - Seiichi Manyama, May 07 2025
Comments