A137221 a(n) = 5*a(n-1) - 9*a(n-2) + 8*a(n-3) - 4*a(n-4), with a(0)=0, a(1)=0, a(2)=0, a(3)=1.

0, 0, 0, 1, 5, 16, 43, 107, 256, 597, 1365, 3072, 6827, 15019, 32768, 70997, 152917, 327680, 699051, 1485483, 3145728, 6640981, 13981013, 29360128, 61516459, 128625323, 268435456, 559240533, 1163220309, 2415919104, 5010795179

Offset: 0

Author

Paul Curtz, Mar 07 2008

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (5,-9,8,-4).

Crossrefs

Same recurrence as in A100335 (essentially first differences of this sequence).

Cf. A001787, A002264, A010892, A024495.

Programs

Magma

[n le 4 select Floor((n-1)/3) else 5*Self(n-1) -9*Self(n-2) +8*Self(n-3) -4*Self(n-4): n in [1..30]]; // G. C. Greubel, Jan 05 2022

Mathematica

Table[(1/3)*(2^(n-1)*(n-2) + ChebyshevU[n, 1/2]), {n, 0, 40}] (* G. C. Greubel, Jan 05 2022 *)
LinearRecurrence[{5,-9,8,-4},{0,0,0,1},40] (* Harvey P. Dale, Apr 30 2023 *)

Sage

[(1/3)*(2^(n-1)*(n-2) + chebyshev_U(n, 1/2)) for n in (0..40)] # G. C. Greubel, Jan 05 2022

Formula

Binomial transform of A002264; a(n+1) - 2*a(n) = A024495.

From R. J. Mathar, Mar 17 2008: (Start)

O.g.f.: x^3/((1-x+x^2)(1-2*x)^2).

a(n) = ( -3*2^n + A001787(n+1) + 2*A010892(n) )/6. (End)

a(n) = (1/3)*(2^(n-1)*(n-2) + ChebyshevU(n, 1/2)). - G. C. Greubel, Jan 05 2022

Extensions

More terms from R. J. Mathar, Mar 17 2008