A164595 a(n) = 10*a(n-1) - 17*a(n-2) for n > 1; a(0) = 5, a(1) = 37.
5, 37, 285, 2221, 17365, 135893, 1063725, 8327069, 65187365, 510313477, 3994949565, 31274166541, 244827522805, 1916614396853, 15004076080845, 117458316061949, 919513867245125, 7198347299398117, 56351737250814045, 441145468418372461, 3453475150919885845
Offset: 0
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..144 from Vincenzo Librandi)
- Index entries for linear recurrences with constant coefficients, signature (10,-17).
Programs
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Magma
Z
:=PolynomialRing(Integers()); N :=NumberField(x^2-2); S:=[ ((5+3*r)*(5+2*r)^n+(5-3*r)*(5-2*r)^n)/2: n in [0..18] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 24 2009 -
Maple
seq(coeff(series( (5-13*x)/(1-10*x+17*x^2) , x, n+1), x, n), n = 0..25); # G. C. Greubel, Apr 21 2020
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Mathematica
CoefficientList[Series[(5 -13z)/(1 -10z +17z^2), {z,0,25}], z] (* Vladimir Joseph Stephan Orlovsky, Jun 12 2011 *) LinearRecurrence[{10,-17}, {5,37}, 25] (* G. C. Greubel, Aug 11 2017 *) -
PARI
my(x='x+O('x^25)); Vec((5-13*x)/(1-10*x+17*x^2)) \\ G. C. Greubel, Aug 11 2017 -
Sage
def A164595_list(prec): P.
= PowerSeriesRing(ZZ, prec) return P( (5-13*x)/(1-10*x+17*x^2) ).list() A164595_list(25) # G. C. Greubel, Apr 21 2020
Formula
a(n) = ((5 + sqrt(18))*(5 + sqrt(8))^n + (5 - sqrt(18))*(5 - sqrt(8))^n)/2.
G.f.: (5-13*x)/(1-10*x+17*x^2).
E.g.f.: exp(5*x)*(5*cosh(2*sqrt(2)*x) + 3*sqrt(2)*sinh(2*sqrt(2)*x)). - G. C. Greubel, Aug 11 2017
a(n) = (1/2)*Sum_{k=0..n} binomial(n,k)*2^(n-k)*(3*Q(2*k+1) + 2*Q(2*k)), where Q(n) are the Pell-Lucas numbers (A002203). - G. C. Greubel, Apr 21 2020
Extensions
Extended by Klaus Brockhaus and R. J. Mathar, Aug 24 2009
Comments