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 A082505 #55 Aug 27 2025 08:20:44
%S A082505 0,1,6,12,24,48,96,192,384,768,1536,3072,6144,12288,24576,49152,98304,
%T A082505 196608,393216,786432,1572864,3145728,6291456,12582912,25165824,
%U A082505 50331648,100663296,201326592,402653184,805306368,1610612736,3221225472
%N A082505 a(n) = sum of (n-1)-th row terms of triangle A134059.
%C A082505 a(n) is the least number x such that gcd(2^x, x-phi(x)) = 2^n. If cototient is replaced by totient, analogous values are different: A053576.
%H A082505 Vincenzo Librandi, <a href="/A082505/b082505.txt">Table of n, a(n) for n = 0..200</a>
%H A082505 <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (2).
%F A082505 a(n) = A007283(n-1) for n>1, with a(0) = 0 and a(1) = 1.
%F A082505 G.f.: x * (1 + 4*x) / (1 - 2*x) = x / (1 - 6*x / (1 + 4*x)). - _Michael Somos_, Jun 15 2012
%F A082505 Starting (1, 6, 12, 24, 48, ...) = binomial transform of [1, 5, 1, 5, 1, 5, ...]. - _Gary W. Adamson_, Nov 18 2007
%F A082505 a(n+1) = Sum_{k=0..n} A109466(n,k)*A144706(k). - _Philippe Deléham_, Oct 30 2008
%F A082505 a(n) = (-6*n + 16) * a(n-1) + 2 * Sum_{k=1..n-1} a(k) * a(n-k) if n>1. - _Michael Somos_, Jul 23 2011
%F A082505 E.g.f.: (-3 - 4*x + 3*exp(2*x))/2. - _Ilya Gutkovskiy_, Jul 04 2016
%F A082505 a(n) = 3*2^(n-1) - (3/2)*[n=0] - 2*[n=1]. - _G. C. Greubel_, Apr 27 2021
%e A082505 G.f. = x + 6*x^2 + 12*x^3 + 24*x^4 + 48*x^5 + 96*x^6 + 192*x^7 + 384*x^8 + ...
%p A082505 0,1,seq(3*2^(n-1), n=2..40); # _G. C. Greubel_, Apr 27 2021
%t A082505 {0}~Join~Map[Total, {{1}}~Join~Table[3 Binomial[n, k], {n, 30}, {k, 0, n}]] (* _Michael De Vlieger_, Jul 03 2016, after _Harvey P. Dale_ at A134059 *)
%t A082505 Table[3*2^(n-1) -(3/2)*Boole[n==0] -2*Boole[n==1], {n,0,40}] (* _G. C. Greubel_, Apr 27 2021 *)
%t A082505 Join[{0,1},NestList[2#&,6,30]] (* _Harvey P. Dale_, Jan 22 2024 *)
%o A082505 (Magma) [0, 1] cat [ &+[ 3*Binomial(n,k): k in [0..n] ]: n in [1..30] ]; // _Klaus Brockhaus_, Dec 02 2009
%o A082505 (PARI) {a(n) = local(A); if( n<1, 0, A = vector(n); A[1] = 1; for( k=2, n, A[k] = (-6*k + 16) * A[k-1] + 2 * sum( j=1, k-1, A[j] * A[k-j])); A[n])} /* _Michael Somos_, Jul 23 2011 */
%o A082505 (PARI) a(n)=if(n<2,n,3<<(n-1)) \\ _Charles R Greathouse IV_, Jun 16 2012
%o A082505 (Sage) [0,1]+[3*2^(n-1) for n in (2..40)] # _G. C. Greubel_, Apr 27 2021
%Y A082505 Cf. A000010, A007283, A009195, A050339, A134059.
%Y A082505 Essentially the same as A003945 (and perhaps also A058764).
%K A082505 nonn,easy,changed
%O A082505 0,3
%A A082505 _Labos Elemer_, Apr 28 2003
%E A082505 More terms from _Klaus Brockhaus_, Dec 02 2009