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%I #38 Mar 08 2024 12:10:52
%S 1,5,23,103,457,2021,8927,39415,174001,768101,3390599,14966887,
%T 66067129,291634565,1287337487,5682582967,25084135393,110726731589,
%U 488771441783,2157541529575,9523849084969,42040303802789
%N G.f. (1-x)/(7*x^2-6*x+1).
%C A floretion-generated sequence relating to the second binomial transform of Pell numbers A000129.
%C Floretion Algebra Multiplication Program, FAMP Code: (a(n)) = jesforseq[ + .5'i + .5i' + 2'jj' + .5'ij' + .5'ji' ]; A000004 = vesforseq.
%H Vincenzo Librandi, <a href="/A102285/b102285.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6,-7).
%F a(n) = A086351(n+1) - 3*A086351(n) (FAMP result); Inversion gives A027649 (SuperSeeker result); Inverse binomial transform of A007070 (SuperSeeker result);
%F From Al Hakanson (hawkuu(AT)gmail.com), Jul 25 2009: (Start)
%F a(n) = ((1+sqrt(2))*(3+sqrt(2))^n + (1-sqrt(2))*(3-sqrt(2))^n)/2 offset 0.
%F Third binomial transform of 1,2,2,4,4. (End)
%F a(n) = 6*a(n-1) - 7*a(n-2) for n > 1; a(0)=1, a(1)=5. - _Philippe Deléham_, Sep 19 2009
%F a(n) = A081179(n) + A086351(n). - _Joseph M. Shunia_, Sep 09 2019
%F a(n) = A081179(n+1)-A081179(n). - _R. J. Mathar_, Sep 11 2019
%t CoefficientList[Series[(1-x)/(7x^2-6x+1),{x,0,30}],x] (* or *) LinearRecurrence[{6,-7},{1,5},30] (* _Harvey P. Dale_, Dec 10 2017 *)
%o (Magma) [Floor(((1+Sqrt(2))*(3+Sqrt(2))^n+(1-Sqrt(2))*(3-Sqrt(2))^n)/2): n in [0..30]]; // _Vincenzo Librandi_, Oct 12 2011
%Y Cf. A086351, A027649, A007070 (inv. Binom. Trans.), A081179, A163350 (Binomial Trans.).
%K nonn,easy
%O 0,2
%A _Creighton Dement_, Feb 19 2005
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