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A140428
a(n) = A000045(n) + A113405(n).
2
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
0,4
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.
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
EXAMPLE
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.
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 *)
PROG
(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
(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
CROSSREFS
Sequence in context: A018298 A017913 A307677 * A340849 A027154 A217350
KEYWORD
nonn,easy
AUTHOR
Paul Curtz, Jun 19 2008
EXTENSIONS
Edited and extended by R. J. Mathar, Jul 10 2008
STATUS
approved