A082505 a(n) = sum of (n-1)-th row terms of triangle A134059.

0, 1, 6, 12, 24, 48, 96, 192, 384, 768, 1536, 3072, 6144, 12288, 24576, 49152, 98304, 196608, 393216, 786432, 1572864, 3145728, 6291456, 12582912, 25165824, 50331648, 100663296, 201326592, 402653184, 805306368, 1610612736, 3221225472

Offset: 0

Author

Labos Elemer, Apr 28 2003

Keywords

• nonn
• easy
• changed

Comments

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.

Examples

			G.f. = x + 6*x^2 + 12*x^3 + 24*x^4 + 48*x^5 + 96*x^6 + 192*x^7 + 384*x^8 + ...
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Index entries for linear recurrences with constant coefficients, signature (2).

Crossrefs

Cf. A000010, A007283, A009195, A050339, A134059.

Essentially the same as A003945 (and perhaps also A058764).

Programs

Magma

[0, 1] cat [ &+[ 3*Binomial(n,k): k in [0..n] ]: n in [1..30] ]; // Klaus Brockhaus, Dec 02 2009

Maple

0,1,seq(3*2^(n-1), n=2..40); # G. C. Greubel, Apr 27 2021

Mathematica

{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 *)
Table[3*2^(n-1) -(3/2)*Boole[n==0] -2*Boole[n==1], {n,0,40}] (* G. C. Greubel, Apr 27 2021 *)
Join[{0,1},NestList[2#&,6,30]] (* Harvey P. Dale, Jan 22 2024 *)

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 */

PARI

a(n)=if(n<2,n,3<<(n-1)) \\ Charles R Greathouse IV, Jun 16 2012

Sage

[0,1]+[3*2^(n-1) for n in (2..40)] # G. C. Greubel, Apr 27 2021

Formula

a(n) = A007283(n-1) for n>1, with a(0) = 0 and a(1) = 1.

G.f.: x * (1 + 4*x) / (1 - 2*x) = x / (1 - 6*x / (1 + 4*x)). - Michael Somos, Jun 15 2012

Starting (1, 6, 12, 24, 48, ...) = binomial transform of [1, 5, 1, 5, 1, 5, ...]. - Gary W. Adamson, Nov 18 2007

a(n+1) = Sum_{k=0..n} A109466(n,k)*A144706(k). - Philippe Deléham, Oct 30 2008

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

E.g.f.: (-3 - 4*x + 3*exp(2*x))/2. - Ilya Gutkovskiy, Jul 04 2016

a(n) = 3*2^(n-1) - (3/2)*[n=0] - 2*[n=1]. - G. C. Greubel, Apr 27 2021

Extensions

More terms from Klaus Brockhaus, Dec 02 2009