A016316 Expansion of 1/((1-2x)*(1-8x)*(1-9x)).
1, 19, 255, 2975, 32231, 333759, 3353335, 32976175, 319155111, 3051352799, 28893830615, 271497720975, 2535105456391, 23548956856639, 217804673719095, 2007154559579375, 18439691005140071, 168959618797797279, 1544655767192730775, 14094055488835543375
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..200
- Index entries for linear recurrences with constant coefficients, signature (19,-106,144).
Programs
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Magma
I:=[1,19,255]; [n le 3 select I[n] else 19*Self(n-1)-106*Self(n-2)+144*Self(n-3): n in [1..20]]; // Vincenzo Librandi, Jun 26 2013
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Magma
m:=20; R
:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-2*x)*(1-8*x)*(1-9*x)))); // Vincenzo Librandi, Jun 26 2013 -
Mathematica
CoefficientList[Series[1 / ((1 - 2 x) (1 - 8 x) (1 - 9 x)), {x, 0, 20}], x] (* Vincenzo Librandi, Jun 26 2013 *) LinearRecurrence[{19,-106,144},{1,19,255},30] (* Harvey P. Dale, Dec 29 2021 *) -
PARI
a(n)=(9^n-2^n)/7-(8^n-2^n)/6 \\ Charles R Greathouse IV, Sep 24 2012
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Sage
[(9^n - 2^n)/7-(8^n - 2^n)/6 for n in range(2,20)] # Zerinvary Lajos, Jun 05 2009
Formula
a(n) = 2^(n+1)/21 - 4*8^(n+1)/3 + 9^(n+2)/7; a(n) = A016133(n+1) - A016131(n+1). - Zerinvary Lajos, Jun 05 2009 [corrected by R. J. Mathar, Mar 14 2011]
From Vincenzo Librandi, Jun 26 2013: (Start)
a(n) = 19*a(n-1) - 106*a(n-2) + 144*a(n-3).
a(n) = 17*a(n-1) - 72*a(n-2) + 2^n. (End)
E.g.f.: exp(2*x)*(2 - 224*exp(6*x) + 243*exp(7*x))/21. - Stefano Spezia, Jul 30 2022