A099462 Expansion of x/(1 - 4*x^2 - 4*x^3).

0, 1, 0, 4, 4, 16, 32, 80, 192, 448, 1088, 2560, 6144, 14592, 34816, 82944, 197632, 471040, 1122304, 2674688, 6373376, 15187968, 36192256, 86245376, 205520896, 489750528, 1167065088, 2781085696, 6627262464, 15792603136, 37633392640

Offset: 0

Author

Paul Barry, Oct 16 2004

Keywords

• easy
• nonn

Comments

Binomial transform is A099463.

Links

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

Crossrefs

Cf. A099463.

Programs

Magma

[n le 3 select (1+(-1)^n)/2 else 4*(Self(n-2) +Self(n-3)): n in [1..41]]; // G. C. Greubel, Nov 18 2021

Mathematica

LinearRecurrence[{0,4,4}, {0,1,0}, 40] (* G. C. Greubel, Nov 18 2021 *)

Sage

def a(n): return sum( 4^k*binomial(k, n-2*k-1) for k in (0..(n-1)//2) )
[a(n) for n in (0..40)] # G. C. Greubel, Nov 18 2021

Formula

a(n) = 4*a(n-2) + 4*a(n-3).

a(n) = Sum_{k=0..floor((n-1)/2)} binomial(k, n-2*k-1)*4^k.

a(n+1) = Sum_{k=0..floor(n/2)} C((n-k)/2, k)*(1+(-1)^(n-k))*2^(n-k). - Paul Barry, Sep 09 2005