A280173 a(0) = 1, a(n+1) = 2*a(n) + periodic sequence of length 2: repeat [5, -4].

1, 7, 10, 25, 46, 97, 190, 385, 766, 1537, 3070, 6145, 12286, 24577, 49150, 98305, 196606, 393217, 786430, 1572865, 3145726, 6291457, 12582910, 25165825, 50331646, 100663297, 201326590, 402653185, 805306366, 1610612737, 3221225470, 6442450945, 12884901886

Offset: 0

Author

Paul Curtz, Dec 28 2016

Keywords

• nonn
• easy

Comments

a(n) mod 9 = period 2: repeat [1, 7].

The last digit from 7 is of period 4: repeat [7, 0, 5, 6].

The bisection A096045 = 1, 10, 46, ... is based on Bernoulli numbers.

a(n) is a companion to A051049(n).

With an initial 0, A051049(n) is an autosequence of the first kind.

With an initial 2, this sequence is an autosequence of the second kind.

See the reference.

Difference table:

1, 7, 10, 25, 46, 97, ... = this sequence.

6, 3, 15, 21, 51, 93, ... = 3*A014551(n)

-3, 12, 6, 30, 42, 102, ... = -3 followed by 6*A014551(n).

The main diagonal of the difference table gives A003945: 1, 3, 6, 12, 24, ...

Examples

			a(0) = 1, a(1) = 2*1 + 5 = 7, a(2) = 2*7 - 4 = 10, a(3) = 2*10 + 5 = 25.
		

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (2,1,-2).

Crossrefs

Cf. A003945, A005010, A010688, A010710, A014551, A051049, A096045, A199116.

Programs

Maple

seq(3*2^n-(-1)^n*(1+irem(n+1,2)),n=0..32); # Peter Luschny, Dec 29 2016

Mathematica

LinearRecurrence[{2,1,-2},{1,7,10},50] (* Paolo Xausa, Nov 13 2023 *)

PARI

Vec((1 + 5*x - 5*x^2) / ((1 - x)*(1 + x)*(1 - 2*x)) + O(x^40)) \\ Colin Barker, Dec 28 2016

Formula

a(2n) = 3*4^n - 2, a(2n+1) = 6*4^n + 1.

a(n+2) = a(n) + 9*2^n, a(0) = 1, a(1) = 7.

a(n) = 2*A051049(n+1) - A051049(n).

From Colin Barker, Dec 28 2016: (Start)

a(n) = 3*2^n - 2 for n even.

a(n) = 3*2^n + 1 for n odd.

a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) for n>2.

G.f.: (1 + 5*x - 5*x^2) / ((1 - x)*(1 + x)*(1 - 2*x)).

(End)