cp's OEIS Frontend

A163985 Sum of all isolated parts of all partitions of n.

%I A163985 #22 Apr 21 2025 08:38:29
%S A163985 0,1,2,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41,43,45,47,
%T A163985 49,51,53,55,57,59,61,63,65,67,69,71,73,75,77,79,81,83,85,87,89,91,93,
%U A163985 95,97,99,101,103,105,107,109,111,113
%N A163985 Sum of all isolated parts of all partitions of n.
%C A163985 Note that for n >= 3 the isolated parts of all partitions of n are n and n-1.
%H A163985 G. C. Greubel, <a href="/A163985/b163985.txt">Table of n, a(n) for n = 0..5000</a>
%H A163985 Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polpa2dt.jpg">Illustration of the shell model of partitions (2D view)</a>.
%H A163985 Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polpa3dt.jpg">Illustration of the shell model of partitions (3D view)</a>.
%H A163985 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1).
%F A163985 a(n) = n for n<3, a(n) = 2*n-1 for n>=3.
%F A163985 a(n) = A140139(n), n>=1.
%F A163985 a(n) = A130773(n-1), n >=2. - _R. J. Mathar_, Jan 25 2023
%F A163985 From _Stefano Spezia_, Apr 21 2025: (Start)
%F A163985 G.f.: x*(1 + 2*x^2 - x^3)/(1 - x)^2.
%F A163985 E.g.f.: 1 - x^2/2 - exp(x)*(1 - 2*x). (End)
%e A163985 For n=4, the five partitions of 4 are {(4);(2,2);(3,1);(2,1,1);(1,1,1,1)}. Since 1 and 2 are repeated parts and 3 and 4 are not repeated parts (or isolated parts) then a(4) = 3 + 4 = 7.
%t A163985 Join[{0, 1, 2}, Table[2 n - 1, {n, 3, 60}]] (* _Vincenzo Librandi_, Dec 23 2015 *)
%o A163985 (Magma) [0,1,2] cat [2*n-1: n in [3..60]]; // _Vincenzo Librandi_, Dec 23 2015
%o A163985 (PARI) a(n) = if (n<3, n, 2*n-1); \\ _Michel Marcus_, Dec 23 2015
%Y A163985 Cf. A000041, A005408, A140139, A163986.
%K A163985 easy,nonn,less
%O A163985 0,3
%A A163985 _Omar E. Pol_, Aug 14 2009