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 A083333 #15 Dec 27 2024 03:51:07
%S A083333 1,1,6,10,44,84,344,680,2736,5456,21856,43680,174784,349504,1398144,
%T A083333 2796160,11184896,22369536,89478656,178956800,715828224,1431655424,
%U A083333 5726623744,11453245440,45812985856,91625967616,366503878656
%N A083333 a(n) = 10*a(n-2) - 16*a(n-4) for n>=4, with a(0)=a(1)=1, a(2)=6, a(3)=10.
%H A083333 G. C. Greubel, <a href="/A083333/b083333.txt">Table of n, a(n) for n = 0..1000</a>
%H A083333 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,10,0,-16).
%F A083333 G.f.: (1+x-4*x^2)/(1-10*x^2+16*x^4).
%F A083333 Limit_{n -> oo} A083332(n)/a(n) = 3.
%F A083333 a(n) = A001045(n+1)*A016116(n). - _R. J. Mathar_, Jul 08 2009
%F A083333 From _G. C. Greubel_, Dec 27 2024: (Start)
%F A083333 a(n) = (1/3)*2^((n-3)/2)*( (1-(-1)^n)*(2^(n+1) - 1) + (1+(-1)^n)*sqrt(2)*(2^(n+1) + 1) ).
%F A083333 E.g.f.: (1/3)*(2*cosh(2*sqrt(2)*x) + cosh(sqrt(2)*x)) + (1/(3*sqrt(2)))*(2*sinh(2*sqrt(2)*x) - sinh(sqrt(2)*x)). (End)
%t A083333 CoefficientList[Series[(1+x-4x^2)/(1-10x^2+16x^4), {x, 0, 30}], x]
%t A083333 LinearRecurrence[{0,10,0,-16},{1,1,6,10},30] (* _Harvey P. Dale_, Aug 04 2024 *)
%o A083333 (Magma) I:=[1,1,6,10]; [n le 4 select I[n] else 10*Self(n-2) -16*Self(n-4): n in [1..41]]; // _G. C. Greubel_, Dec 27 2024
%o A083333 (SageMath)
%o A083333 def A083333(n): return 2^((n-1)/2)*( (n%2)*(2^(n+1) -1) + ((n+1)%2)*sqrt(2)*(2^(n+1) +1))/3
%o A083333 print([A083333(n) for n in range(41)]) # _G. C. Greubel_, Dec 27 2024
%Y A083333 Cf. A016131, A082412 (bisections).
%Y A083333 Cf. A001045, A016116, A083332.
%K A083333 easy,nonn
%O A083333 0,3
%A A083333 Mario Catalani (mario.catalani(AT)unito.it), Apr 24 2003