cp's OEIS Frontend

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.

%I A026922 #25 Aug 02 2025 13:58:08
%S A026922 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,
%T A026922 8,9,9,10,9,10,10,11,10,11,11,12,11,12,12,13,12,13,13,14,13,14,14,15,
%U A026922 14,15,15,16,15,16,16,17,16,17,17,18,17,18,18,19,18,19,19,20,19,20,20,21
%N 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.
%C A026922 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
%H A026922 Vincenzo Librandi, <a href="/A026922/b026922.txt">Table of n, a(n) for n = 1..1000</a>
%H A026922 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,1,-1).
%F A026922 a(2*n + 1) = a(2*n - 2) = A004526(n).
%F A026922 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
%F A026922 a(n) = A008624(n-2).  - _R. J. Mathar_, Oct 23 2008
%F A026922 From _Emeric Deutsch_, Nov 12 2008: (Start)
%F A026922 G.f. = sum(sum(x^(2*i-1+j), j=1..2*i-1), i=1..infinity).
%F A026922 G.f. = x^2*(1-x+x^2)/[(1+x)*(1-x)^2*(1+x^2)]. (End)
%F A026922 From _Michael Somos_, Oct 14 2008: (Start)
%F A026922 Euler transform of length 6 sequence [ 0, 1, 1, 1, 0, -1].
%F A026922 a(n) = a(n-1) + a(n-4) - a(n-5). a(1 - n) = -a(n).
%F A026922 G.f.: x^2 * (1 - x + x^2) / (1 - x - x^4 + x^5). (End)
%e A026922 a(10)=3 because we have 22222, 2221111 and 211111111. - _Emeric Deutsch_, Nov 12 2008
%p A026922 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
%p A026922 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
%t A026922 CoefficientList[Series[x (1 - x + x^2) / (1 - x - x^4 + x^5), {x, 0, 100}], x] (* _Vincenzo Librandi_, Aug 15 2013 *)
%o A026922 (PARI) {a(n) = n \ 2 - ((n + 1) \ 4)} /* _Michael Somos_, Oct 14 2008 */
%Y A026922 Column 2 of A026920. Essentially the same as A008624.
%K A026922 nonn,easy
%O A026922 1,6
%A A026922 _Clark Kimberling_
%E A026922 More terms from _Emeric Deutsch_, Nov 12 2008