A017897 Expansion of 1/((1-3*x)*(1-5*x)*(1-9*x)).
1, 17, 202, 2090, 20251, 189707, 1745332, 15900020, 144066901, 1301455397, 11737424062, 105758621150, 952437144751, 8574983669087, 77190104636392, 694787214149480, 6253466332501801, 56283104147438777, 506557473488982322, 4559064943373269010
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..200
- Christian Brouder, William J. Keith, and Ângela Mestre, Closed forms for a multigraph enumeration, arXiv preprint arXiv:1301.0874 [math.CO], 2013-2015.
- Index entries for linear recurrences with constant coefficients, signature (17,-87,135).
Programs
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Magma
m:=20; R
:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-3*x)*(1-5*x)*(1-9*x)))); // Vincenzo Librandi, Jul 01 2013 -
Magma
I:=[1, 17, 202]; [n le 3 select I[n] else 17*Self(n-1)-87*Self(n-2)+135*Self(n-3): n in [1..20]]; // Vincenzo Librandi, Jul 01 2013
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Maple
a:= n -> (Matrix(3, (i,j)-> if (i=j-1) then 1 elif j=1 then [17, -87, 135][i] else 0 fi)^n)[1,1]: seq (a(n), n=0..25); # Alois P. Heinz, Aug 04 2008
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Mathematica
CoefficientList[Series[1 / ((1 - 3 x) (1 - 5 x) (1 - 9 x)), {x, 0, 30}], x] (* Vincenzo Librandi, Jul 01 2013 *) LinearRecurrence[{17,-87,135},{1,17,202},30] (* Harvey P. Dale, Sep 26 2014 *) a[n_]:=(9^(n+2) - 3*5^(n+2) + 2*3^(n+2))/24; Array[a, 30, 0] (* Stefano Spezia, Oct 04 2018 *) -
PARI
a(n) = (9^(n+2) - 3*5^(n+2) + 2*3^(n+2))/24; \\ Joerg Arndt, Aug 13 2013
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SageMath
def A017897(n): return (9^(n+2) -3*5^(n+2) +2*3^(n+2))//24 [A017897(n) for n in range(41)] # G. C. Greubel, Nov 09 2024
Formula
a(n) = term (1,1) in the 3 X 3 matrix [17,1,0; -87,0,1; 135,0,0]^n. - Alois P. Heinz, Aug 04 2008
From Vincenzo Librandi, Jul 01 2013: (Start)
a(n) = 17*a(n-1) - 87*a(n-2) + 135*a(n-3); a(0)=1, a(1)=17, a(2)=202.
a(n) = 14*a(n-1) - 45*a(n-2) + 3^n. (End)
a(n) = (9^(n+2) - 3*5^(n+2) + 2*3^(n+2))/24. - Yahia Kahloune, Aug 13 2013
E.g.f.: exp(3*x)*(6 - 25*exp(2*x) + 27*exp(6*x))/8. - Stefano Spezia, Nov 09 2024