A105384 -id:A105384 - cp's OEIS frontend

A008616 Expansion of 1/((1-x^2)(1-x^5)).

1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 2, 2, 2, 3, 2, 3, 2, 3, 3, 3, 3, 3, 3, 4, 3, 4, 3, 4, 4, 4, 4, 4, 4, 5, 4, 5, 4, 5, 5, 5, 5, 5, 5, 6, 5, 6, 5, 6, 6, 6, 6, 6, 6, 7, 6, 7, 6, 7, 7, 7, 7, 7, 7, 8, 7, 8, 7, 8, 8, 8, 8, 8, 8, 9, 8, 9, 8, 9, 9, 9, 9, 9, 9, 10, 9, 10, 9, 10, 10, 10, 10, 10, 10

Offset: 0

N. J. A. Sloane

Number of partitions of n into parts of size two and five.

It appears that, for n >= 2, a(n-2) is also (1) the number of partitions of 3n that are 6-term arithmetic progressions and (2) floor(n/2) - floor(2n/5). - John W. Layman, Jun 29 2009

G. E. Andrews, K. Eriksson, Integer Partitions, Cambridge Univ. Press, 2004. page 30, Exercise 48.
D. J. Benson, Polynomial Invariants of Finite Groups, Cambridge, 1993, p. 100.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 213
M. Janjic, On Linear Recurrence Equations Arising from Compositions of Positive Integers, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.7.
Index entries for two-way infinite sequences
Index entries for linear recurrences with constant coefficients, signature (0, 1, 0, 0, 1, 0, -1).

Cf. A000217, A025810.

Cf. A008615. - John W. Layman, Jun 29 2009

m:=100; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-x^2)*(1-x^5)))); // Wesley Ivan Hurt, Dec 27 2021

CoefficientList[Series[1 / ((1 - x^2) (1 - x^5)),{x, 0, 100}], x] (* Vincenzo Librandi, Jun 21 2013 *)

{a(n) = if( n<-6, -a(-7 - n), polcoeff( 1 / (1 - x^2) / (1 - x^5) + x * O(x^n), n))} /* Michael Somos, Jan 25 2005 */

a(n) = floor(n/10+(3+(-1)^n)/4) \\ Charles R Greathouse IV, Jun 19 2013

G.f.: 1/((1-x^2)(1-x^5)) = 1/((x-1)^2*(1+x)*(1+x+x^2+x^3+x^4)).
Euler transform of finite sequence [0, 1, 0, 0, 1].
a(n) = -a(-7 - n) = a(n - 10) + 1 = a(n - 2) + a(n - 5) - a(n - 7). - Michael Somos, Jan 25 2005
A000217(a(n)) = A025810(n). - Michael Somos, Dec 15 2002
a(n) = 7/20 + n/10 + (-1)^n/4 + (A105384(n) + 2*(A010891(n) + A105384(n+4)))/5. - R. J. Mathar, Jun 28 2009
a(n) = floor(n/10 + (3 + (-1)^n)/4). - Tani Akinari, Jun 20 2013

A138112 a(n)=3a(n-1)-4a(n-2)+2a(n-3)-a(n-4), a(0)=a(1)=a(2)=0, a(3)=1, a(4)=3.

Original entry on oeis.org

0, 0, 0, 1, 3, 5, 5, 0, -13, -34, -55, -55, 0, 144, 377, 610, 610, 0, -1597, -4181, -6765, -6765, 0, 17711, 46368, 75025, 75025, 0, -196418, -514229, -832040, -832040, 0, 2178309, 5702887, 9227465, 9227465, 0, -24157817, -63245986, -102334155, -102334155

Offset: 0

Paul Curtz, May 04 2008

sign

Obeys also the recurrence a(n)=5a(n-1)-10a(n-2)+10a(n-3)-5a(n-4)+2a(n-5), so the sequence is identical to its fifth differences (cf. A135356). a(n) = A138110(0,n): if A138110 is interpreted as an array with five rows, this is the top row.
The first differences are represented by A100334(n-1).
The 2nd differences are represented by A103311(n).
The 3rd differences are essentially represented by -A138003(n-2).
The 4th differences are represented by -A105371(n).
A102312 contains the absolute values of the terms which occur in pairs, for example a(5)=a(6)=5=A102312(1), a(10)=a(11)= -55 = -A102312(2).
Inverse BINOMIAL transform yields two zeros followed by A105384. - R. J. Mathar, Jul 04 2008

Index entries for linear recurrences with constant coefficients, signature (3, -4, 2, -1).

Cf. A138003, A103311, A105371.

CoefficientList[Series[x^3/(1-3x+4x^2-2x^3+x^4),{x,0,45}],x] (* or *) LinearRecurrence[{3,-4,2,-1},{0,0,0,1},45] (* Harvey P. Dale, Jun 22 2011 *)

O.g.f.: x^3/(1-3x+4x^2-2x^3+x^4). - R. J. Mathar, Jul 04 2008

Edited and extended by R. J. Mathar, Jul 04 2008

A105385 Expansion of (1-x^2)/(1-x^5).

Original entry on oeis.org

1, 0, -1, 0, 0, 1, 0, -1, 0, 0, 1, 0, -1, 0, 0, 1, 0, -1, 0, 0, 1, 0, -1, 0, 0, 1, 0, -1, 0, 0, 1, 0, -1, 0, 0, 1, 0, -1, 0, 0, 1, 0, -1, 0, 0, 1, 0, -1, 0, 0, 1, 0, -1, 0, 0, 1, 0, -1, 0, 0, 1, 0, -1, 0, 0, 1, 0, -1, 0, 0, 1, 0, -1, 0, 0, 1, 0, -1, 0, 0, 1, 0, -1, 0, 0, 1, 0, -1, 0, 0, 1

Offset: 0

Paul Barry, Apr 02 2005

Periodic {1,0,-1,0,0}.

Binomial transform is A103311(n+1). Consecutive pair sums of A105384.

Index entries for linear recurrences with constant coefficients, signature (-1,-1,-1,-1).

Cf. A092202 (essentially the same).
Cf. A198517 (absolute values).
Cf. A103311, A105384

CoefficientList[Series[(1-x^2)/(1-x^5),{x,0,100}],x] (* or *) PadRight[{},100,{1,0,-1,0,0}] (* or *) LinearRecurrence[{-1,-1,-1,-1},{1,0,-1,0},100] (* Harvey P. Dale, Mar 10 2013 *)

G.f.: (1+x)/(1 + x + x^2 + x^3 + x^4);
a(n) = sqrt(1/5 - 2*sqrt(5)/25)*cos(4*Pi*n/5 + Pi/10) + sqrt(5)*sin(4*Pi*n/5 + Pi/10)/5 + sqrt(2*sqrt(5)/25 + 1/5)*cos(2*Pi*n/5 + 3*Pi/10) + sqrt(5)*sin(2*Pi*n/5 + 3*Pi/10)/5.
a(n) = A092202(n+1). - R. J. Mathar, Aug 28 2008
a(n) = a(n-1) - a(n-2) - a(n-3) - a(n-4); a(0)=1, a(1)=0, a(2)=-1, a(3)=0. - Harvey P. Dale, Mar 10 2013

cp's OEIS Frontend

A008616 Expansion of 1/((1-x^2)(1-x^5)).

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A138112 a(n)=3a(n-1)-4a(n-2)+2a(n-3)-a(n-4), a(0)=a(1)=a(2)=0, a(3)=1, a(4)=3.

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A105385 Expansion of (1-x^2)/(1-x^5).

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