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Jacobsthal oblong numbers.
16

%I #51 Aug 22 2024 17:55:29

%S 0,1,3,15,55,231,903,3655,14535,58311,232903,932295,3727815,14913991,

%T 59650503,238612935,954429895,3817763271,15270965703,61084037575,

%U 244335800775,977343902151,3909374210503,15637499638215,62549992960455

%N Jacobsthal oblong numbers.

%C Inverse binomial transform is A001019 doubled up.

%C Binomial transform is A084177.

%C Partial sums of A003683.

%H Vincenzo Librandi, <a href="/A084175/b084175.txt">Table of n, a(n) for n = 0..500</a>

%H N. Heninger, E. M. Rains and N. J. A. Sloane, <a href="https://arxiv.org/abs/math/0509316">On the Integrality of n-th Roots of Generating Functions</a>, arXiv:math/0509316 [math.NT], 2005-2006.

%H N. Heninger, E. M. Rains and N. J. A. Sloane, <a href="http://dx.doi.org/10.1016/j.jcta.2006.03.018">On the Integrality of n-th Roots of Generating Functions</a>, J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.

%H Ronald Orozco López, <a href="https://arxiv.org/abs/2408.08943">Generating Functions of Generalized Simplicial Polytopic Numbers and (s,t)-Derivatives of Partial Theta Function</a>, arXiv:2408.08943 [math.CO], 2024. See p. 11.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,6,-8).

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

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

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

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

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

%F a(n+1) - 4*a(n) = 1, -1, 3, -5, 11, ... = A001045(n+1) signed. - _Paul Curtz_, May 19 2008

%F a(n) = round(2^n/3) * round(2^(n+1)/3). - _Gary Detlefs_, Feb 10 2010

%F From _Peter Bala_, Mar 30 2015: (Start)

%F 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.

%F 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)

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

%t 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 *)

%t LinearRecurrence[{3,6,-8}, {0,1,3}, 25] (* _Jean-François Alcover_, Sep 21 2017 *)

%o (Sage) [gaussian_binomial(n, 2, -2) for n in range(1, 26)] # _Zerinvary Lajos_, May 28 2009

%o (Magma) [(2*4^n-(-2)^n-1)/9: n in [0..30]]; // _Vincenzo Librandi_, Jun 04 2011

%o (PARI) a(n)=(2*4^n-(-2)^n-1)/9 \\ _Charles R Greathouse IV_, Sep 24 2015

%o (GAP) List([0..30], n-> (2^(2*n+1) -(-2)^n -1)/9); # _G. C. Greubel_, Sep 21 2019

%Y Except for initial terms, same as A015249 and A084152.

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

%K easy,nonn

%O 0,3

%A _Paul Barry_, May 18 2003