A140428 a(n) = A000045(n) + A113405(n).

0, 1, 1, 3, 5, 9, 15, 27, 49, 91, 169, 317, 599, 1143, 2197, 4251, 8269, 16161, 31711, 62435, 123273, 243963, 483745, 960725, 1910503, 3803295, 7577933, 15109499, 30143973, 60166553, 120136687, 239955563, 479396897, 957961755, 1914577241

Offset: 0

Author

Paul Curtz, Jun 19 2008

Keywords

• nonn
• easy

Comments

The inverse binomial transform yields the sequence (-1)^(n+1)*a(n). This property is inherited from the A000045 and A113405 sequences, which have the same property individually. The same sign flipping behavior under inverse binomial transform is found in A001045 and for the sequence with two zeros followed by A000975.

This is often, but not here, related to the recurrences a(n)=2a(n-1)+a(n-2)-2a(n-3) associated with denominators 1-2x-x^2+2x^3=(x-1)(2x-1)(x+1) in the o.g.f., which transform into the similar -(x-1)(2x+1)/(1+x)^4 under the inverse binomial transform, see A137241.

Examples

			a(n) and the repeated differences in the followup rows are:
    0,   1,   1,   3,   5,   9,  15, ...
    1,   0,   2,   2,   4,   6,  12, ...
   -1,   2,   0,   2,   2,   6,  10, ...
    3,  -2,   2,   0,   4,   4,  10, ...
   -5,   4,  -2,   4,   0,   6,   6, ...
    9,  -6,   6,  -4,   6,   0,  12, ...
  -15,  12, -10,  10,  -6, -12,   0, ...
The main diagonal consists of zeros.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (3,-1,-3,3,-1,-2).

Programs

Magma

I:=[0,1,1,3,5,9]; [n le 6 select I[n] else 3*Self(n-1)-Self(n-2) -3*Self(n-3)+3*Self(n-4)-Self(n-5)-2*Self(n-6): n in [1..30]]; // G. C. Greubel, Jan 15 2018

Mathematica

CoefficientList[Series[-x (1 - 2 x - 3 x^4 + x^2)/((1 - x - x^2) (2 x - 1) (1 + x) (x^2 - x + 1)), {x, 0, 50}], x] (* G. C. Greubel, Oct 11 2017 *)
LinearRecurrence[{3,-1,-3,3,-1,-2}, {0,1,1,3,5,9}, 30] (* G. C. Greubel, Jan 15 2018 *)

PARI

a(n)=([0,1,0,0,0,0; 0,0,1,0,0,0; 0,0,0,1,0,0; 0,0,0,0,1,0; 0,0,0,0,0,1; -2,-1,3,-3,-1,3]^n*[0;1;1;3;5;9])[1,1] \\ Charles R Greathouse IV, Oct 03 2016

Formula

O.g.f.: -x*(1-2*x-3*x^4+x^2)/((1-x-x^2)*(2*x-1)*(1+x)*(x^2-x+1)). - R. J. Mathar, Jul 10 2008

a(n)= -A128834(n)/3 + 2^n/9 + A000045(n) - (-1)^n/9. - R. J. Mathar, Jul 10 2008

Extensions

Edited and extended by R. J. Mathar, Jul 10 2008