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

0,3

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).

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).

Mathematica

LinearRecurrence[{4, -6, 5, -2}, {0, 1, 2, 5}, 34] (* Robert P. P. McKone, Feb 07 2021 *)

Prog

(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

Crossrefs

Cf. A000079, A153237, A281166.

Sequence in context: A057873 A116699 A290198 * A054127 A184052 A295057

Adjacent sequences: A282150 A282151 A282152 * A282154 A282155 A282156

Keyword

nonn,easy

Author

Paul Curtz, Feb 07 2017

Extensions

More terms from Colin Barker, Feb 10 2017

Status

approved