A099930 a(n) = Pell(n) * Pell(n-1) * Pell(n-2) / 10.
1, 12, 174, 2436, 34307, 482664, 6791772, 95567064, 1344731653, 18921807828, 266250046986, 3746422451772, 52716164405255, 741772724044560, 10437534301224120, 146867252940711408, 2066579075472320521, 29078974309550454492, 409172219409185308518, 5757490046038128779316
Offset: 3
Links
- Michael A. Allen and Kenneth Edwards, Fence tiling derived identities involving the metallonacci numbers squared or cubed, Fib. Q. 60:5 (2022) 5-17.
- N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, arXiv:math/0509316 [math.NT], 2005-2006.
- N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.
- 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. 13.
- Ronald Orozco López, Simplicial d-Polytopic Numbers Defined on Generalized Fibonacci Polynomials, arXiv:2501.11490 [math.CO], 2025. See p. 10.
- Index entries for linear recurrences with constant coefficients, signature (12,30,-12,-1).
Programs
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Mathematica
Drop[CoefficientList[Series[x^3/((1+2x-x^2)(1-14x-x^2)),{x,0,20}],x],3] (* or *) LinearRecurrence[{12,30,-12,-1},{1,12,174,2436},20] (* Harvey P. Dale, Feb 26 2012 *)
Formula
G.f.: x^3 / ((1+2*x-x^2)*(1-14*x-x^2)).
a(n) = 12*a(n-1) + 30*a(n-2) - 12*a(n-3) - a(n-4); a(3)=1, a(4)=12, a(5)=174, a(6)=2436. - Harvey P. Dale, Feb 26 2012
From Peter Bala, Mar 30 2015: (Start)
The following remarks assume an offset of 0.
The o.g.f. A(x) = 1/( (1 + 2*x - x^2)*(1 - 14*x - x^2) ). Hence A(x) (mod 4) = 1/(1 + 2*x - x^2)^2 (mod 4). It follows by Theorem 1 of Heninger et al. that sqrt(A(x)) = 1 + 6*x + 69*x^2 + 804*x^3 + ... has integral coefficients.
a(n+1) = (1/2)*Sum_{k=1..n-1} ( A014445(k)*A110272(n-k) ) for n > 1. - Michael A. Allen, Jan 25 2023