A242563 a(n) = 2*a(n-1) - a(n-3) + 2*a(n-4), a(0)=a(1)=0, a(2)=2, a(3)=3.

0, 0, 2, 3, 6, 10, 21, 42, 86, 171, 342, 682, 1365, 2730, 5462, 10923, 21846, 43690, 87381, 174762, 349526, 699051, 1398102, 2796202, 5592405, 11184810, 22369622, 44739243, 89478486, 178956970, 357913941, 715827882, 1431655766, 2863311531, 5726623062, 11453246122

Offset

0,3

Comments

Generally, a(n) is an autosequence if its inverse binomial transform is (-1)^n*a(n). It is of the first kind if the main diagonal is 0's and the first two upper diagonals (just above the main one) are the same. It is of the second kind if the main diagonal is equal to the first upper diagonal multiplied by 2. If the first upper diagonal is an autosequence, the sequence is a super autosequence. Example: A113405. The first upper diagonal is A001045(n). Another super autosequence: 0, 0, 0 followed by A059633(n). The first upper diagonal is A000045(n).

Difference table of a(n):

0, 0, 2, 3, 6, 10, 21, 42, ...

0, 2, 1, 3, 4, 11, 21, 44, ...

2, -1, 2, 1, 7, 10, 23, 41, ...

-3, 3, -1, 6, 3, 13, 18, 45, ... .

This is an autosequence of the second kind. The main diagonal is 2*A001045(n) = A078008(n). More precisely it is a super autosequence, companion of A113405(n).

a(n+1) mod 10 = period 12: repeat 0, 2, 3, 6, 0, 1, 2, 6, 1, 2, 2, 5.

It is shifted A081374(n+1) mod 10 =

period 12: repeat 1, 2, 2, 5, 0, 2, 3, 6, 0, 1, 2, 6.

a(n) mod 9 = period 18:

repeat 0, 0, 2, 3, 6, 1, 3, 6, 5, 0, 0, 7, 6, 3, 8, 6, 3, 4 = c(n).

c(n) + c(n+9) = 0, 0, 9, 9, 9, 9, 9, 9, 9.

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (2,0,-1,2).

Formula

a(n+3) = 3*2^n - a(n), a(0)=a(1)=0, a(2)=2.

a(n) = 2*A113405(n+1) - A113405(n).

a(n+1) = 2*a(n) + period 6: repeat 0, 2, -1, 0, -2, 1. a(0)=0.

a(n) = 2^n - A081374(n+1).

a(n+3) = a(n+1) + A130755(n).

G.f.: x^2*(x-2) / ((x+1)*(2*x-1)*(x^2-x+1)). - Colin Barker, May 18 2014

a(n) = A024495(n) + A131531(n).

a(n+6) = a(n) + 21*2^n, a(0)=a(1)=0, a(2)=2, a(3)=3, a(4)=6, a(5)=10.

a(n) = A001045(n) - A092220(n).

a(n+12) = a(n) + 1365*2^n. First 12 values in the Data. (A024495(n+12) = A024495(n) + 1365*2^n).

a(3n) = A132805(n) = 3*A015565(n).

a(3n+1) = A132804(n) = 6*A015565(n).

a(3n+2) = A132397(n) = 2*A082311(n).

a(n) = 1/3*((-1)^n - 2*cos((n*Pi)/3) + 2^n). - Alexander R. Povolotsky, Jun 02 2014

Example

G.f. = 2*x^2 + 3*x^3 + 6*x^4 + 10*x^5 + 21*x^6 + 42*x^7 + 86*x^8 + ...

Mathematica

a[n_] := (m = Mod[n, 6]; 1/3*(2^n + (-1)^n + 1/120*(m-6)*(m+1)*(m^3-29*m+40))); Table[a[n], {n, 0, 35}] (* Jean-François Alcover, May 19 2014, a non-recursive formula, after Mathematica's RSolve *)
LinearRecurrence[{2, 0, -1, 2}, {0, 0, 2, 3}, 50] (* G. C. Greubel, Feb 21 2017 *)

Prog

(PARI) concat([0, 0], Vec(x^2*(x-2)/((x+1)*(2*x-1)*(x^2-x+1)) + O(x^100))) \\ Colin Barker, May 18 2014

Crossrefs

Cf. A000032, 1/(n+1), A164555/A027642 (all autosequences of 2nd kind). A007283, A175805.

Sequence in context: A138347 A211180 A265582 * A240513 A036650 A049889

Adjacent sequences: A242560 A242561 A242562 * A242564 A242565 A242566

Keyword

nonn,easy

Author

Paul Curtz, May 17 2014

Extensions

More terms from Colin Barker, May 18 2014

Status

approved