A094039 -id:A094039 - cp's OEIS frontend

A087211 a(n) = floor((1+2^n+3^n)/3).

1, 2, 4, 12, 32, 92, 264, 772, 2272, 6732, 20024, 59732, 178512, 534172, 1599784, 4793892, 14370752, 43090412, 129227544, 387595252, 1162610992, 3487483452, 10461751304, 31383855812, 94148771232, 282440721292, 847310979064

Offset: 0

Paul Barry, Aug 26 2003

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-1,-6).

Cf. A011782, A094039.

Table[Floor[(1+2^n+3^n)/3],{n,0,30}] (* or *) LinearRecurrence[{4,-1,-6},{1,2,4,12},30] (* Harvey P. Dale, May 21 2018 *)

G.f.: (1-2*x-3*x^2+4*x^3)/((1+x)*(1-2*x)*(1-3*x));
E.g.f.: (exp(3*x)+exp(2*x)+2*exp(0)-exp(-x))/3;
a(n) = (3^n+2^n+2*0^n-(-1)^n)/3.
a(n) = 2*A094039(n), n>0. - R. J. Mathar, Feb 13 2015
a(n) = 4*a(n-1) - a(n-2) - 6*a(n-3). - Wesley Ivan Hurt, Apr 25 2023

A094038 Binomial transform of (Pell(-n)+Pell(n))/2.

Original entry on oeis.org

0, 1, 2, 8, 24, 84, 280, 960, 3264, 11152, 38048, 129920, 443520, 1514304, 5170048, 17651712, 60266496, 205762816, 702517760, 2398545920, 8189147136, 27959497728, 95459694592, 325919784960, 1112759746560, 3799199420416

Offset: 0

Paul Barry, Apr 23 2004

Also binomial transform of Pell(n)(1-(-1)^n)/2.

a(n+2) = A088013(n+1) + A007070(n) - Creighton Dement, Oct 23 2004

Index entries for linear recurrences with constant coefficients, signature (4,0,-8,4).

Cf. A000129, A094039, A051450.

LinearRecurrence[{4,0,-8,4},{0,1,2,8},30] (* Harvey P. Dale, Jan 01 2016 *)

G.f.: x(1-2x)/((1-2x^2)(1-4x+2x^2));

a(n)=(sqrt(2)/8)((2+sqrt(2))^n-(2-sqrt(2))^n+(sqrt(2))^n-(-sqrt(2))^n).

cp's OEIS Frontend

A087211 a(n) = floor((1+2^n+3^n)/3).

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