A026922 Number of partitions of n into an odd number of parts, the greatest being 2; also, a(n+3) = number of partitions of n+1 into an even number of parts, each <=2.

0, 1, 0, 1, 1, 2, 1, 2, 2, 3, 2, 3, 3, 4, 3, 4, 4, 5, 4, 5, 5, 6, 5, 6, 6, 7, 6, 7, 7, 8, 7, 8, 8, 9, 8, 9, 9, 10, 9, 10, 10, 11, 10, 11, 11, 12, 11, 12, 12, 13, 12, 13, 13, 14, 13, 14, 14, 15, 14, 15, 15, 16, 15, 16, 16, 17, 16, 17, 17, 18, 17, 18, 18, 19, 18, 19, 19, 20, 19, 20, 20, 21

Offset

1,6

Comments

a(n) is also the number of partitions of n into two parts, the larger being odd (the conjugate of the defining partition). Example: a(10) = 3 because we have 55, 73 and 91. - Emeric Deutsch, Nov 12 2008

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Formula

a(2*n + 1) = a(2*n - 2) = A004526(n).

a(n) = floor((n+2)/4) - [n == 3 mod 4] = floor((1/8)*{2*n - 1 + 3*(-1)^n + 2*(-1)^[(n-1)/2]}). - Ralf Stephan, Jun 09 2005

a(n) = A008624(n-2). - R. J. Mathar, Oct 23 2008

From Emeric Deutsch, Nov 12 2008: (Start)

G.f. = sum(sum(x^(2*i-1+j), j=1..2*i-1), i=1..infinity).

G.f. = x^2*(1-x+x^2)/[(1+x)*(1-x)^2*(1+x^2)]. (End)

From Michael Somos, Oct 14 2008: (Start)

Euler transform of length 6 sequence [ 0, 1, 1, 1, 0, -1].

a(n) = a(n-1) + a(n-4) - a(n-5). a(1 - n) = -a(n).

G.f.: x^2 * (1 - x + x^2) / (1 - x - x^4 + x^5). (End)

Example

a(10)=3 because we have 22222, 2221111 and 211111111. - Emeric Deutsch, Nov 12 2008

Maple

G:=x^2*(x^2-x+1)/((x+1)*(1-x)^2*(x^2+1)): Gser:= series(G, x=0, 105): seq(coeff(Gser, x, n), n=1..82); # Emeric Deutsch, Nov 12 2008
a := proc(n): if (n mod 4 = 3) then floor((n+2)/4) - 1 else floor((n+2)/4) fi: end: seq(a(n), n=1..82); # Johannes W. Meijer, Oct 10 2013

Mathematica

CoefficientList[Series[x (1 - x + x^2) / (1 - x - x^4 + x^5), {x, 0, 100}], x] (* Vincenzo Librandi, Aug 15 2013 *)

Prog

(PARI) {a(n) = n \ 2 - ((n + 1) \ 4)} /* Michael Somos, Oct 14 2008 */

Crossrefs

Column 2 of A026920.

Sequence in context: A033922 A008624 A059169 * A178696 A161090 A349219

Adjacent sequences: A026919 A026920 A026921 * A026923 A026924 A026925

Keyword

nonn,easy

Author

Clark Kimberling

Extensions

More terms from Emeric Deutsch, Nov 12 2008

Status

approved