A137232 a(n) = -a(n-1) + 7*a(n-2) + 3*a(n-3) with a(0) = a(1) = 0, a(2) = 1.
0, 0, 1, -1, 8, -12, 65, -125, 544, -1224, 4657, -11593, 40520, -107700, 356561, -988901, 3161728, -9014352, 28179745, -81795025, 252010184, -740036124, 2258722337, -6682944653, 20273892640, -60278338200, 182146752721, -543273442201, 1637465696648, -4893939533892, 14726379083825
Offset: 0
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, signature (-1,7,3).
Programs
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Magma
I:=[0,0,1]; [n le 3 select I[n] else -Self(n-1) +7*Self(n-2) +3*Self(n-3): n in [1..36]]; // G. C. Greubel, Apr 19 2021
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Mathematica
Table[((-3)^n + 5*Fibonacci[n,2] -Fibonacci[n+1,2])/14, {n,0,40}] (* G. C. Greubel, Apr 19 2021 *) LinearRecurrence[{-1,7,3},{0,0,1},40] (* Harvey P. Dale, Apr 26 2022 *) -
Sage
[((-3)^n +5*lucas_number1(n,2,-1) -lucas_number1(n+1,2,-1))/14 for n in (0..40)] # G. C. Greubel, Apr 19 2021
Formula
From R. J. Mathar, Mar 17 2008: (Start)
O.g.f.: x^2/((1+3*x)*(1-2*x-x^2)).
a(n) = ( (-3)^n + A135532(n) )/14. (End)
a(n) = (1/14)*( (-3)^n + 5*Pell(n) - Pell(n+1) ), where Pell(n) = A000129(n). - G. C. Greubel, Apr 19 2021