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 A085282 #46 Jul 20 2025 15:20:34
%S A085282 1,3,10,35,126,463,1730,6555,25126,97223,379050,1486675,5858126,
%T A085282 23166783,91869970,365088395,1453179126,5791193143,23100202490,
%U A085282 92207099715,368247268126,1471245680303,5879752544610,23503319648635,93966207005126,375723613252263,1502470808704330,6008612301903155,24030636408870126
%N A085282 Expansion of (1 - 5*x + 5*x^2)/((1-x)*(1-3*x)*(1-4*x)).
%C A085282 Binomial transform of A085281.
%C A085282 Number of walks of length 2n+1 between two adjacent vertices in the cycle graph C_12. - _Herbert Kociemba_, Jul 05 2004
%H A085282 Vincenzo Librandi, <a href="/A085282/b085282.txt">Table of n, a(n) for n = 0..500</a>
%H A085282 Mircea Merca, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL15/Merca1/merca6.html">A Note on Cosine Power Sums</a> J. Integer Sequences, Vol. 15 (2012), Article 12.5.3.
%H A085282 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (8,-19,12).
%F A085282 a(n) = 4^n/3 + 3^n/2 + 1/6.
%F A085282 a(n) = Sum_{k=-floor(n/6)..floor(n/6)} binomial(2*n, n+6*k)/2. - _Mircea Merca_, Jan 28 2012
%F A085282 a(n) = 8*a(n-1) - 19*a(n-2) + 12*a(n-3) for n>2. - _Colin Barker_, Feb 07 2020
%F A085282 E.x.p.: (1/6)*(exp(x) + 3*exp(3*x) + 2*exp(4*x)). - _G. C. Greubel_, Nov 11 2024
%t A085282 CoefficientList[Series[(1 - 5*x + 5*x^2)/((1-x)*(1-3*x)*(1-4*x)), {x, 0, 50}], x] (* _Stefano Spezia_, Sep 09 2018 *)
%t A085282 Table[(2^(2*n+1) +3^(n+1) +1)/6, {n,0,40}] (* _G. C. Greubel_, Nov 11 2024 *)
%t A085282 LinearRecurrence[{8,-19,12},{1,3,10},30] (* _Harvey P. Dale_, Jul 20 2025 *)
%o A085282 (Magma) [4^n/3+3^n/2+1/6: n in [0..35]]; // _Vincenzo Librandi_, May 29 2011
%o A085282 (PARI) apply( {A085282(n)=(4^n*2+3^(n+1))\/6}, [0..29]) \\ _M. F. Hasler_, Feb 07 2020
%o A085282 (SageMath)
%o A085282 def A085282(n): return (2^(2*n+1) +3^(n+1) +1)//6
%o A085282 [A085282(n) for n in range(41)] # _G. C. Greubel_, Nov 11 2024
%Y A085282 Cf. A085281.
%K A085282 easy,nonn
%O A085282 0,2
%A A085282 _Paul Barry_, Jun 25 2003