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

Author

Paul Curtz, Feb 07 2017

Keywords

• nonn
• easy

Comments

After 0, partial sums of A281166.

Table of the first differences:

0, 1, 2, 5, 13, 30, 63, 127, 254, 509, 1021, 2046, ...

1, 1, 3, 8, 17, 33, 64, 127, 255, 512, 1025, 2049, ... A281166

0, 2, 5, 9, 16, 31, 63, 128, 257, 513, 1024, 2047, ...

2, 3, 4, 7, 15, 32, 65, 129, 256, 511, 1023, 2048, ...

repeat A281166.

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).

Crossrefs

Cf. A000079, A153237, A281166.

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.

a(n) - (a(n) mod 9) = A153237(n) = 9*A153234(n).

Extensions

More terms from Colin Barker, Feb 10 2017