A282153 Expansion of x*(1 - 2*x + 3*x^2)/((1 - x)*(1 - 2*x)*(1 - x + x^2)).
0, 1, 2, 5, 13, 30, 63, 127, 254, 509, 1021, 2046, 4095, 8191, 16382, 32765, 65533, 131070, 262143, 524287, 1048574, 2097149, 4194301, 8388606, 16777215, 33554431, 67108862, 134217725, 268435453, 536870910, 1073741823, 2147483647, 4294967294, 8589934589
Offset: 0
Links
- Colin Barker, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, signature (4,-6,5,-2).
Programs
-
Mathematica
LinearRecurrence[{4, -6, 5, -2}, {0, 1, 2, 5}, 34] (* Robert P. P. McKone, Feb 07 2021 *) -
PARI
concat(0, Vec(x*(1 - 2*x + 3*x^2) / ((1 - x)*(1 - 2*x)*(1 - x + x^2)) + O(x^50))) \\ Colin Barker, Feb 10 2017
Formula
From Colin Barker, Feb 10 2017: (Start)
G.f.: x*(1 - 2*x + 3*x^2)/((1 - x)*(1 - 2*x)*(1 - x + x^2)).
a(n) = 4*a(n-1) - 6*a(n-2) + 5*a(n-3) - 2*a(n-4) for n>3. (End)
From Bruno Berselli, Feb 10 2017: (Start)
a(n) = 2^n + ((-1)^floor(n/3) + (-1)^floor((n+1)/3))/2 - 2. Therefore:
a(3*k) = 8^k + (-1)^k - 2,
a(3*k+1) = 2*8^k + (-1)^k - 2,
a(3*k+2) = 4*8^k - 2. (End)
a(n+6*h) = a(n) + 2^n*(64^h - 1) with h>=0. For h=1, a(n+6) = a(n) + 63*2^n.
Extensions
More terms from Colin Barker, Feb 10 2017
Comments