This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A081179 #56 May 06 2025 11:50:58
%S A081179 0,1,6,29,132,589,2610,11537,50952,224953,993054,4383653,19350540,
%T A081179 85417669,377052234,1664389721,7346972688,32431108081,143157839670,
%U A081179 631929281453,2789470811028,12313319895997,54353623698786
%N A081179 3rd binomial transform of (0,1,0,2,0,4,0,8,0,16,...).
%C A081179 Binomial transform of 0, 1, 4, 14, 48, ... (A007070 with offset 1) and second binomial transform of A000129. - _R. J. Mathar_, Dec 10 2011
%H A081179 Vincenzo Librandi, <a href="/A081179/b081179.txt">Table of n, a(n) for n = 0..300</a>
%H A081179 Sergio Falcon, <a href="http://dx.doi.org/10.9734/BJMCS/2014/11783">Iterated Binomial Transforms of the k-Fibonacci Sequence</a>, British Journal of Mathematics & Computer Science, 4 (22): 2014.
%H A081179 Yassine Otmani, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL28/Otmani/otmani10.html">The 2-Pascal Triangle and a Related Riordan Array</a>, J. Int. Seq. (2025) Vol. 28, Issue 3, Art. No. 25.3.5. See p. 12.
%H A081179 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6,-7).
%F A081179 a(n) = 6*a(n-1) - 7*a(n-2), a(0)=0, a(1)=1.
%F A081179 G.f.: x/(1-6*x+7*x^2).
%F A081179 a(n) = ((3+sqrt(2))^n - (3-sqrt(2))^n)/(2*sqrt(2)). [Corrected by Al Hakanson (hawkuu(AT)gmail.com), Dec 27 2008]
%F A081179 a(n) = 3^(n-1) Sum_{i>=0} binomial(n, 2i+1) * (2/9)^i. - _Sergio Falcon_, Mar 15 2016
%F A081179 a(n) = 2^(-1/2)*7^(n/2)*sinh(n*arcsinh(sqrt(2/7))). - _Robert Israel_, Mar 15 2016
%F A081179 E.g.f.: exp(3*x)*sinh(sqrt(2)*x)/sqrt(2). - _Ilya Gutkovskiy_, Aug 12 2017
%F A081179 a(n) = 7^((n-1)/2)*ChebyshevU(n-1, 3/sqrt(7)). - _G. C. Greubel_, Jan 14 2024
%p A081179 f:= gfun:-rectoproc({a(n) = 6*a(n-1)-7*a(n-2), a(0)=0, a(1)=1},a(n),remember):
%p A081179 map(f, [$0..50]); # _Robert Israel_, Mar 15 2016
%t A081179 CoefficientList[Series[x/(1-6 x +7 x^2), {x,0,30}], x] (* _Vincenzo Librandi_, Aug 06 2013 *)
%t A081179 LinearRecurrence[{6,-7}, {0,1}, 41] (* _G. C. Greubel_, Jan 14 2024 *)
%o A081179 (Sage) [lucas_number1(n,6,7) for n in range(0, 23)] # _Zerinvary Lajos_, Apr 22 2009
%o A081179 (Magma) I:=[0, 1]; [n le 2 select I[n] else 6*Self(n-1)-7*Self(n-2): n in [1..30]]; // _Vincenzo Librandi_, Aug 06 2013
%Y A081179 Cf. A081180, A081182, A081183, A081184, A081185, A153593.
%K A081179 easy,nonn
%O A081179 0,3
%A A081179 _Paul Barry_, Mar 11 2003