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A191597 Expansion of x*(1+3*x)/ ( (1-4*x)*(1+x+x^2)). 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

a(n) = 1/3*4^n - 1/6*((-1/2-(1/2*i)*sqrt(3))^n + (-1/2 +(1/2*i)*sqrt(3))^n) + (1/6*i)*sqrt(3)*((-1/2+(1/2*I)*sqrt(3))^n - (-1/2-(1/2*i)*sqrt(3))^n), with n >= 0 and i = sqrt(-1). - Paolo P. Lava, Jul 01 2011

MAPLE

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

A191597 := proc(n) (4^n-A061347(n+1))/3 ; end proc:

seq(A191597(n), n=0..30) ; # R. J. Mathar, Jun 08 2011

PROG

(PARI) a(n)=([0, 1, 0; 0, 0, 1; 4, 3, 3]^n*[0; 1; 6])[1, 1] \\ Charles R Greathouse IV, Jul 06 2017

CROSSREFS

Sequence in context: A320649 A219596 A182251 * A088556 A316105 A137966

Adjacent sequences:  A191594 A191595 A191596 * A191598 A191599 A191600

KEYWORD

nonn,easy

AUTHOR

Paul Curtz, Jun 08 2011

STATUS

approved

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Last modified March 20 15:43 EDT 2019. Contains 321345 sequences. (Running on oeis4.)