A191597 - OEIS

A191597

Expansion of x*(1+3*x)/ ( (1-4*x)*(1+x+x^2)).

0, 1, 6, 21, 85, 342, 1365, 5461, 21846, 87381, 349525, 1398102, 5592405, 22369621, 89478486, 357913941, 1431655765, 5726623062, 22906492245, 91625968981, 366503875926, 1466015503701, 5864062014805, 23456248059222, 93824992236885, 375299968947541

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

a(n) and successive differences define a square array T(0,k) = a(k), T(n,k) = T(n-1,k+1) - T(n-1,k):

0, 1, 6, 21, 85, 342,...

1, 5, 15, 64, 257, 1023,...

4, 10, 49, 193, 766, 3073,...

As with any sequence which obeys a homogeneous linear recurrence (we say it once, only once and we shall not repeat it), the recurrence is also valid for the rows of such arrays of higher order differences.

LINKS

Table of n, a(n) for n=0..25.

Index entries for linear recurrences with constant coefficients, signature (3,3,4).

FORMULA

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

a(n) = A024495(2*n).

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

a(n+1) - 4*a(n) = A132677(n).

a(n+3) - a(n) = 21*4^n.

a(n) = A178872(n) + 3*A178872(n-1) = (4^n-A061347(n+1))/3. - R. J. Mathar, Jun 08 2011

MAPLE

A061347 := proc(n) op(1+(n mod 3), [-2, 1, 1]) ; end proc: