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

0,3

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.

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Index entries for linear recurrences with constant coefficients, signature (2).

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

Example

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

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 *)

Prog

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

Crossrefs

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

Sequence in context: A332043 A362487 A229926 * A091629 A089529 A300915

Adjacent sequences: A082502 A082503 A082504 * A082506 A082507 A082508

Keyword

nonn,easy

Author

Labos Elemer, Apr 28 2003

Extensions

More terms from Klaus Brockhaus, Dec 02 2009

Status

approved