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A084175 Jacobsthal oblong numbers.

%I A084175 #59 Apr 25 2025 09:49:38
%S A084175 0,1,3,15,55,231,903,3655,14535,58311,232903,932295,3727815,14913991,
%T A084175 59650503,238612935,954429895,3817763271,15270965703,61084037575,
%U A084175 244335800775,977343902151,3909374210503,15637499638215,62549992960455
%N A084175 Jacobsthal oblong numbers.
%C A084175 Inverse binomial transform is A001019 doubled up.
%C A084175 Binomial transform is A084177.
%C A084175 Partial sums of A003683.
%H A084175 Vincenzo Librandi, <a href="/A084175/b084175.txt">Table of n, a(n) for n = 0..500</a>
%H A084175 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 A084175 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 A084175 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 A084175 Ronald Orozco López, <a href="https://arxiv.org/abs/2501.11490">Simplicial d-Polytopic Numbers Defined on Generalized Fibonacci Polynomials</a>, arXiv:2501.11490 [math.CO], 2025. See p. 6.
%H A084175 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,6,-8).
%F A084175 a(n) = A001045(n)*A001045(n+1).
%F A084175 a(n) = (2*4^n - (-2)^n - 1)/9;
%F A084175 a(n) = 3*a(n-1) + 6*a(n-2) - 8*a(n-3), a(0)=0, a(1)=1, a(2)=3.
%F A084175 G.f.: x/((1+2*x)*(1-x)*(1-4*x)).
%F A084175 E.g.f.: (2*exp(4*x) - exp(x) - exp(-2*x))/9.
%F A084175 a(n+1) - 4*a(n) = 1, -1, 3, -5, 11, ... = A001045(n+1) signed. - _Paul Curtz_, May 19 2008
%F A084175 a(n) = round(2^n/3) * round(2^(n+1)/3). - _Gary Detlefs_, Feb 10 2010
%F A084175 From _Peter Bala_, Mar 30 2015: (Start)
%F A084175 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 A084175 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 A084175 for n from 1 to 25 do print(round(2^n/3)*round(2^(n+1)/3)) od; # _Gary Detlefs_, Feb 10 2010
%t A084175 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 A084175 LinearRecurrence[{3,6,-8}, {0,1,3}, 25] (* _Jean-François Alcover_, Sep 21 2017 *)
%o A084175 (Sage) [gaussian_binomial(n, 2, -2) for n in range(1, 26)] # _Zerinvary Lajos_, May 28 2009
%o A084175 (Magma) [(2*4^n-(-2)^n-1)/9: n in [0..30]]; // _Vincenzo Librandi_, Jun 04 2011
%o A084175 (PARI) a(n)=(2*4^n-(-2)^n-1)/9 \\ _Charles R Greathouse IV_, Sep 24 2015
%o A084175 (GAP) List([0..30], n-> (2^(2*n+1) -(-2)^n -1)/9); # _G. C. Greubel_, Sep 21 2019
%o A084175 (Python)
%o A084175 def A084175(n): return ((m:=1<<n)|1)//3*((m<<1|1)//3) # _Chai Wah Wu_, Apr 25 2025
%Y A084175 Except for initial terms, same as A015249 and A084152.
%Y A084175 Cf. A001654, A001656, A084158, A084159, A099930.
%K A084175 easy,nonn
%O A084175 0,3
%A A084175 _Paul Barry_, May 18 2003