A099855 a(n) = n*2^n - 2^(n/2)*sin(Pi*n/4).

0, 1, 6, 22, 64, 164, 392, 904, 2048, 4592, 10208, 22496, 49152, 106560, 229504, 491648, 1048576, 2227968, 4718080, 9960960, 20971520, 44041216, 92276736, 192940032, 402653184, 838856704, 1744822272, 3623870464, 7516192768

Offset: 0

Author

Paul Barry, Oct 28 2004

Keywords

• easy
• nonn

Comments

Related to binomial transform of A002265. Sequence is identical to its fourth differences (cf. A139756, A137221). See also A097064, A135035, A038504, A135356. - Paul Curtz, Jun 18 2008

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (6,-14,16,-8).

Crossrefs

Cf. A001787, A002265, A009545, A038504, A097064, A135035, A135356.

Cf. A137221, A139756.

Binomial transform of A047538.

Programs

Magma

I:=[0,1,6,22]; [n le 4 select I[n] else 6*Self(n-1) -14*Self(n-2) +16*Self(n-3) -8*Self(n-4): n in [1..51]]; // G. C. Greubel, Apr 20 2023

Mathematica

LinearRecurrence[{6,-14,16,-8},{0,1,6,22},30] (* Harvey P. Dale, Mar 22 2018 *)

SageMath

@CachedFunction
def a(n): # a = A099855
    if (n<5): return (0,1,6,22,64)[n]
    else: return 6*a(n-1) - 14*a(n-2) + 16*a(n-3) - 8*a(n-4)
[a(n) for n in range(51)] # G. C. Greubel, Apr 20 2023

Formula

G.f.: x/((1-2*x+2*x^2)*(1-4*x+4*x^2)).

a(n) = Sum_{k=0..n} 2^(k/2)*sin(Pi*k/4)*2^(n-k)*(n-k+1).

a(n) = 6*a(n-1) - 14*a(n-2) + 16*a(n-3) - 8*a(n-4).

a(n) = 2*A001787(n) - A009545(n).