A141416 First differences of A133730.
-1, -1, 2, 0, 0, -2, 4, -2, 4, -6, 12, -10, 20, -22, 44, -42, 84, -86, 172, -170, 340, -342, 684, -682, 1364, -1366, 2732, -2730, 5460, -5462, 10924, -10922, 21844, -21846, 43692, -43690, 87380, -87382, 174764, -174762, 349524, -349526, 699052, -699050, 1398100, -1398102
Offset: 0
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, signature (0,1,0,2).
Programs
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Magma
I:=[-1,-1,2,0]; [n le 4 select I[n] else Self(n-2) +2*Self(n-4): n in [1..51]]; // G. C. Greubel, Mar 30 2021
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Mathematica
LinearRecurrence[{0,1,0,2}, {-1,-1,2,0}, 50] (* G. C. Greubel, Mar 30 2021 *) Differences[LinearRecurrence[{0,1,0,2},{1,0,-1,1},70]] (* Harvey P. Dale, Sep 04 2024 *) -
Sage
[((4*i^(n+1) - 2^((n+1)/2))*(1-(-1)^n) - 2*(4*i^n - 2^(n/2))*(1+(-1)^n))/12 for n in (0..50)] # G. C. Greubel, Mar 30 2021
Formula
a(2n) = (-1)^(n+1)*A084247(n).
a(2n+1) = -A078008(n).
a(2n) = -2*a(2n-1), n>0.
a(2n) + a(2n+1) = 2*(-1)^(n+1).
G.f.: (-1 -x +3*x^2 +x^3)/( (1-2*x^2)*(1+x^2) ). - R. J. Mathar, Jul 02 2011
a(n) = ((4*i^(n+1) - 2^((n+1)/2))*(1-(-1)^n) - 2*(4*i^n - 2^(n/2))*(1+(-1)^n))/12. - G. C. Greubel, Mar 30 2021