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 A099459 #49 Jul 27 2025 11:15:55
%S A099459 1,7,40,217,1159,6160,32689,173383,919480,4875913,25856071,137109280,
%T A099459 727060321,3855438727,20444528200,108412748857,574888488199,
%U A099459 3048504677680,16165536349969,85722212350663,454565659304920
%N A099459 Expansion of 1/(1 - 7*x + 9*x^2).
%C A099459 Associated to the knot 9_48 by the modified Chebyshev transform A(x) -> (1/(1+x^2)^2)*A(x/(1+x^2)). See A099460 and A099461.
%H A099459 G. C. Greubel, <a href="/A099459/b099459.txt">Table of n, a(n) for n = 0..1000</a>
%H A099459 Dror Bar-Natan, <a href="http://katlas.org/wiki/9_48">9 48</a>, The Knot Atlas.
%H A099459 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 A099459 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (7,-9).
%F A099459 a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k)*(-9)^k*7^(n-2*k).
%F A099459 a(n) = Sum{k=0..n} binomial(2*n-k+1, k) * 3^k. - _Paul Barry_, Jan 17 2005
%F A099459 a(n) = 7*a(n-1) - 9*a(n-2), n >= 2. - _Vincenzo Librandi_, Mar 18 2011
%F A099459 a(n) = ((7 + sqrt(13))^(n+1) - (7 - sqrt(13))^(n+1))/(2^(n+1)*sqrt(13)). - _Rolf Pleisch_, May 19 2011
%F A099459 a(n) = 3^(n-1)*ChebyshevU(n-1, 7/6). - _G. C. Greubel_, Nov 18 2021
%F A099459 From _Peter Bala_, Jul 23 2025: (Start)
%F A099459 The following products telescope:
%F A099459 Product_{k >= 1} 1 + 3^k/a(k) = (1 + sqrt(13))/2.
%F A099459 Product_{k >= 1} 1 - 3^k/a(k) = (1 + sqrt(13))/14,
%F A099459 Product_{k >= 1} 1 + (-3)^k/a(k) = (13 + sqrt(13))/26.
%F A099459 Product_{k >= 1} 1 - (-3)^k/a(k) = (13 + sqrt(13))/14. (End)
%t A099459 LinearRecurrence[{7,-9},{1,7},30] (* _Harvey P. Dale_, Jan 06 2012 *)
%o A099459 (Sage) [lucas_number1(n,7,9) for n in range(1, 22)] # _Zerinvary Lajos_, Apr 23 2009
%o A099459 (Magma) [n le 2 select 7^(n-1) else 7*Self(n-1) -9*Self(n-2): n in [1..31]]; // _G. C. Greubel_, Nov 18 2021
%Y A099459 Cf. A002450, A004187, A016153, A099460, A099461, A102902.
%K A099459 easy,nonn
%O A099459 0,2
%A A099459 _Paul Barry_, Oct 16 2004