A176965 a(n) = 2^(n-1) - (2^n*(-1)^n + 2)/3.

1, 0, 6, 2, 26, 10, 106, 42, 426, 170, 1706, 682, 6826, 2730, 27306, 10922, 109226, 43690, 436906, 174762, 1747626, 699050, 6990506, 2796202, 27962026, 11184810, 111848106, 44739242, 447392426, 178956970, 1789569706, 715827882, 7158278826

Offset: 1

Author

Roger L. Bagula, Apr 29 2010

Keywords

• nonn
• easy

Comments

The ratio a(n+1)/a(n) approaches 10 for even n and 2/5 for odd n as n->infinity.

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (1,4,-4).

Crossrefs

Merger of A020988 (even n) and A020989 (odd n).

Programs

GAP

List([1..30], n-> (3*2^(n-1) -(-2)^n -2)/3); # G. C. Greubel, Dec 28 2019

Magma

[(3*2^(n-1) -(-2)^n -2)/3: n in [1..30]]; // G. C. Greubel, Dec 28 2019

Maple

seq( (3*2^(n-1) -(-2)^n -2)/3, n=1..30); # G. C. Greubel, Dec 28 2019

Mathematica

a[n_]:= a[n]= 2^(n-1)*If[n==1, 1, a[n-1]/2 +(-1)^(n-1)*Sqrt[(5 +4*(-1)^(n-1) )]/2]; Table[a[n], {n,30}]
LinearRecurrence[{1,4,-4}, {1,0,6}, 30] (* G. C. Greubel, Dec 28 2019 *)

PARI

vector(30, n, (3*2^(n-1) -(-2)^n -2)/3 ) \\ G. C. Greubel, Dec 28 2019

Sage

[(3*2^(n-1) -(-2)^n -2)/3 for n in (1..30)] # G. C. Greubel, Dec 28 2019

Formula

From R. J. Mathar, Apr 30 2010: (Start)

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

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

a(n) = A087231(n), n > 2. - R. J. Mathar, May 03 2010

a(2n-1) = A061547(2n), a(2n) = A061547(2n-1), n > 0. - Yosu Yurramendi, Dec 23 2016

a(n+1) = 2*A096773(n), n > 0. - Yosu Yurramendi, Dec 30 2016

a(2n-1) = A020989(n-1), a(2n) = A020988(n-1), n > 0. - Yosu Yurramendi, Jan 03 2017

a(2n-1) = (A083597(n-1) + A000302(n-1))/2, a(2n) = (A083597(n-1) - A000302(n-1))/2, n > 0. - Yosu Yurramendi, Mar 04 2017

a(n+2) = 4*a(n) + 2, a(1) = 1, a(2) = 0, n > 0. - Yosu Yurramendi, Mar 07 2017

a(n) = (-16 + (9 - (-1)^n) * 2^(n - (-1)^n))/24. - Loren M. Pearson, Dec 28 2019

E.g.f.: (3*exp(2*x) - 4*exp(x) + 3 - 2*exp(-2*x))/6. - G. C. Greubel, Dec 28 2019

a(n) = (2^n*5^(n mod 2) - 4)/6. - Heinz Ebert, Jun 29 2021