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A082505 a(n) = sum of (n-1)-th row terms of triangle A134059. 13
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

If cototient is replaced by totient, analogous values are different: A053576.

Sequence is generated by the southwest direction of circles put around circle(s). See illustration. - Odimar Fabeny, Aug 09 2008

LINKS

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

Odimar Fabeny, Illustration for this sequence

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

FORMULA

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

EXAMPLE

Apart from a(0)=0, a(1)=1 it equals A007283(n) = 3*2^n.

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

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 *)

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

CROSSREFS

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

Sequence in context: A160728 A229926 * A091629 A089529 A300915 A001766

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

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Last modified October 23 13:02 EDT 2018. Contains 316528 sequences. (Running on oeis4.)