A084175 Jacobsthal oblong numbers.

0, 1, 3, 15, 55, 231, 903, 3655, 14535, 58311, 232903, 932295, 3727815, 14913991, 59650503, 238612935, 954429895, 3817763271, 15270965703, 61084037575, 244335800775, 977343902151, 3909374210503, 15637499638215, 62549992960455

Offset

0,3

Comments

Inverse binomial transform is A001019 doubled up.

Binomial transform is A084177.

Partial sums of A003683.

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..500

N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, arXiv:math/0509316 [math.NT], 2005-2006.

N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.

Index entries for linear recurrences with constant coefficients, signature (3,6,-8).

Formula

a(n) = A001045(n)*A001045(n+1).

a(n) = (2*4^n - (-2)^n - 1)/9;

a(n) = 3*a(n-1) + 6*a(n-2) - 8*a(n-3), a(0)=0, a(1)=1, a(2)=3.

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

E.g.f.: (2*exp(4*x) - exp(x) - exp(-2*x))/9.

a(n+1) - 4*a(n) = 1, -1, 3, -5, 11, ... = A001045(n+1) signed. - Paul Curtz, May 19 2008

a(n) = round(2^n/3) * round(2^(n+1)/3). - Gary Detlefs, Feb 10 2010

From Peter Bala, Mar 30 2015: (Start)

The shifted o.g.f. A(x) := 1/( (1 + 2*x)*(1 - x)*(1 - 4*x) ) = 1/(1 - 3*x - 6*x^2 + 8*x^3). Hence A(x) == 1/(1 - 3*x + 3*x^2 - x^3) (mod 9) == 1/(1 - x)^3 (mod 9). It follows by Theorem 1 of Heninger et al. that (A(x))^(1/3) = 1 + x + 4*x^2 + 10*x^3 + ... has integral coefficients.

Sum_{n >= 0} a(n+1)*x^n = exp( Sum_{n >= 1} J(3*n)/J(n)*x^n/n ), where J(n) = A001045(n) are the Jacobsthal numbers. Cf. A001656, A099930. (End)

Maple

for n from 1 to 25 do print(round(2^n/3)*round(2^(n+1)/3)) od; # Gary Detlefs, Feb 10 2010

Mathematica

Table[(2*4^n -(-2)^n -1)/9, {n, 0, 30}] (* Vladimir Joseph Stephan Orlovsky, Feb 05 2011, modified by G. C. Greubel, Sep 21 2019 *)
LinearRecurrence[{3, 6, -8}, {0, 1, 3}, 25] (* Jean-François Alcover, Sep 21 2017 *)

Prog

(Sage) [gaussian_binomial(n, 2, -2) for n in range(1, 26)] # Zerinvary Lajos, May 28 2009
(Magma) [(2*4^n-(-2)^n-1)/9: n in [0..30]]; // Vincenzo Librandi, Jun 04 2011
(PARI) a(n)=(2*4^n-(-2)^n-1)/9 \\ Charles R Greathouse IV, Sep 24 2015
(GAP) List([0..30], n-> (2^(2*n+1) -(-2)^n -1)/9); # G. C. Greubel, Sep 21 2019

Crossrefs

Except for initial terms, same as A015249 and A084152.

Cf. A001654, A001656, A084158, A084159, A099930.

Sequence in context: A261737 A015249 A084152 * A081951 A367520 A346542

Adjacent sequences: A084172 A084173 A084174 * A084176 A084177 A084178

Keyword

easy,nonn

Author

Paul Barry, May 18 2003

Status

approved