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 A120689 #29 Dec 27 2022 04:57:40
%S A120689 0,3,30,252,2040,16368,131040,1048512,8388480,67108608,536870400,
%T A120689 4294966272,34359736320,274877902848,2199023247360,17592186028032,
%U A120689 140737488322560,1125899906777088,9007199254609920,72057594037665792
%N A120689 a(n) = 10*a(n-1) - 16*a(n-2), with a(0) = 0 and a(1) = 3.
%C A120689 a(n) is a leg in a Pythagorean triangle along with A081342(n) (the hypotenuse) and 4^n. Example: a(4) = 2040, A081342(4) = 2056; then sqrt(2056^2 - 2040^2) = 256 = 4^4. Characteristic polynomial of M = x^2 -10*x + 16.
%C A120689 Order of modular group of degree 2^(n-1)+1. - _Artur Jasinski_, Aug 04 2007
%H A120689 G. C. Greubel, <a href="/A120689/b120689.txt">Table of n, a(n) for n = 0..1000</a>
%H A120689 E. Mathieu, <a href="http://portail.mathdoc.fr/JMPA/PDF/JMPA_1860_2_5_A2_0.pdf">Mémoire sur le nombre de valeurs que peut acquérir une fonction quand on y permute ses variables de toutes les manières possibles</a>, Journ. de math. pures et appliquées (2) 5 (1860), 9-42 (see p. 39).
%H A120689 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (10,-16).
%F A120689 a(n) = 8^n - A081342(n).
%F A120689 Given M = 2 X 2 matrix [5,3; 3,5]; M^n * [1,0] = [A081342(a), a(n)]. E.g. a(4) = 2040, right term in = M^4 * [1,0] = [2056, 2040] = [A081342(4), a(4)].
%F A120689 a(n) = 2^(n-1)*(4^n - 1). - _Artur Jasinski_, Aug 04 2007
%F A120689 From _R. J. Mathar_, Feb 16 2011: (Start)
%F A120689 a(n) = 3*A016131(n-1).
%F A120689 G.f.: 3*x / ( (1-2*x)*(1-8*x) ). (End)
%F A120689 E.g.f.: (1/2)*(exp(8*x) - exp(2*x)). - _G. C. Greubel_, Dec 27 2022
%p A120689 a[0]:=0: a[1]:=3; for n from 2 to 20 do a[n]:=10*a[n-1]-16*a[n-2] end do: seq(a[n], n = 0 .. 20); # _Emeric Deutsch_, Aug 16 2007
%p A120689 seq(binomial(2^n,2)*(2^n + 1),n=0..19); # _Zerinvary Lajos_, Jan 07 2008
%t A120689 Table[2^(n-1) (4^n-1), {n,0,20}] (* _Artur Jasinski_, Aug 04 2007 *)
%o A120689 (Magma) [2^(n-1)*(4^n-1): n in [0..30]]; // _G. C. Greubel_, Dec 27 2022
%o A120689 (SageMath)
%o A120689 A120689=BinaryRecurrenceSequence(10,-16,0,3)
%o A120689 [A120689(n) for n in range(31)] # _G. C. Greubel_, Dec 27 2022
%Y A120689 Cf. A016131, A081342.
%K A120689 nonn,easy
%O A120689 0,2
%A A120689 _Gary W. Adamson_, Jun 25 2006
%E A120689 Edited by _N. J. A. Sloane_ at the suggestion of _Andrew S. Plewe_, Jul 13 2007
%E A120689 More terms from _Emeric Deutsch_, Aug 16 2007