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A011377 Expansion of 1/((1-x)*(1-2*x)*(1-x^2)). 12
1, 3, 8, 18, 39, 81, 166, 336, 677, 1359, 2724, 5454, 10915, 21837, 43682, 87372, 174753, 349515, 699040, 1398090, 2796191, 5592393, 11184798, 22369608, 44739229, 89478471, 178956956, 357913926 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Kenneth Edwards and Michael A. Allen, New combinatorial interpretations of the Fibonacci numbers squared, golden rectangle numbers, and Jacobsthal numbers using two types of tile, J. Int. Seq. 24 (2021) Article 21.3.8.

Index entries for linear recurrences with constant coefficients, signature (3,-1,-3,2).

FORMULA

From Paul Barry, Jul 29 2004: (Start)

a(n) = Sum_{k=0..n+2} floor((n-k+2)/2) * 2^k;

a(n) = Sum_{k=0..n+2} floor(k/2) * 2^(n-k+2). (End)

a(n) = Sum_{k=0..floor((n+2)/2)} binomial(n-k+2, k+2)*2^k. - Paul Barry, Oct 25 2004

a(n) = floor((2^(n+4) - 3*n - 6)/6). - David W. Wilson, Feb 26 2006

a(n) = (2^(n+5) - 6*n - 21 + (-1)^n)/12. - Hieronymus Fischer, Dec 02 2006

Row sums of triangle A135086. - Gary W. Adamson, Nov 18 2007

a(n) = 3*a(n-1) - a(n-2) - 3*a(n-3) + 2*a(n-4). - Paul Curtz, Jul 29 2008

G.f.: Q(0)/(3*x*(1-x)^2), where Q(k) = 1 - 1/(4^k - 2*x*16^k/(2*x*4^k - 1/(1 + 1/(2*4^k - 8*x*16^k/(4*x*4^k + 1/Q(k+1)))))); (continued fraction). - Sergei N. Gladkovskii, May 21 2013

From Michael A. Allen, Jan 11 2022: (Start)

a(n) = J(n+3) - ceiling((n+3)/2), where Jacobsthal number J(n) = A001045(n).

a(n) = Sum_{j=1..n+1} j*J(n+2-j). (End)

MATHEMATICA

Table[(2^(n+5) -6*n-21+(-1)^n)/12, {n, 0, 30}] (* Vladimir Joseph Stephan Orlovsky, Jan 29 2011, modified by G. C. Greubel, Jun 02 2019 *)

CoefficientList[Series[1/((1-x)(1-2x)(1-x^2)), {x, 0, 30}], x] (* or *) LinearRecurrence[{3, -1, -3, 2}, {1, 3, 8, 18}, 30] (* Harvey P. Dale, Apr 17 2017 *)

PROG

(Magma) [Floor((2^(n+4)-3*n-6)/6): n in [0..30]]; // Vincenzo Librandi, Aug 14 2011

(PARI) my(x='x+O('x^30)); Vec(1/((1-x)*(1-2*x)*(1-x^2))) \\ G. C. Greubel, Sep 26 2017

(Sage) [(2^(n+5) -6*n-21+(-1)^n)/12 for n in (0..30)] # G. C. Greubel, Jun 02 2019

(GAP) List([0..30], n-> (2^(n+5) -6*n -21 +(-1)^n)/12) # G. C. Greubel, Jun 02 2019

CROSSREFS

Partial sums of A000975.

Second partial sums of A001045.

Cf. A135086.

Sequence in context: A117713 A128552 A238361 * A178420 A036385 A196534

Adjacent sequences: A011374 A011375 A011376 * A011378 A011379 A011380

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified January 31 21:46 EST 2023. Contains 359981 sequences. (Running on oeis4.)