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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 (list; graph; refs; listen; history; text; internal format)
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.

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).

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

Adjacent sequences:  A140425 A140426 A140427 * A140429 A140430 A140431

KEYWORD

nonn,easy

AUTHOR

Paul Curtz, Jun 19 2008

EXTENSIONS

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

STATUS

approved

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Last modified October 26 04:54 EDT 2021. Contains 348256 sequences. (Running on oeis4.)