cp's OEIS Frontend

A194805 Number of parts that are visible in one of the three views of the section model of partitions version "tree" with n sections.

%I A194805 #37 May 12 2020 22:54:22
%S A194805 0,1,2,4,7,11,17,25,36,51,71,97,132,177,235,310,406,527,681,874,1116,
%T A194805 1418,1793,2256,2829,3532,4393,5445,6727,8282,10168,12445,15190,18491,
%U A194805 22452,27192,32859,39613,47651,57199,68522,81920,97756,116434,138435
%N A194805 Number of parts that are visible in one of the three views of the section model of partitions version "tree" with n sections.
%C A194805 The mentioned view of the section model looks like a tree (see example). Note that every column contains the same parts. For more information about the section model of partitions see A135010 and A194803.
%C A194805 Number of partitions of 2n-1 such that n-1 or n is a part, for n >=1. - _Clark Kimberling_, Mar 01 2014
%H A194805 Robert Price, <a href="/A194805/b194805.txt">Table of n, a(n) for n = 0..5000</a>
%F A194805 a(n) = A084376(n) - 1.
%F A194805 a(n) = A000041(n) + A000041(n-1) - 1, if n >= 1.
%F A194805 a(n) = A000041(n) + A000065(n-1), if n >= 1.
%e A194805 Illustration of one of the three views with seven sections:
%e A194805 .
%e A194805 .                   1
%e A194805 .                 2 1
%e A194805 .                   1 3
%e A194805 .                 2 1
%e A194805 .               4   1
%e A194805 .                   1 3
%e A194805 .                   1   5
%e A194805 .                 2 1
%e A194805 .               4   1
%e A194805 .             3     1
%e A194805 .           6       1
%e A194805 .                     3
%e A194805 .                       5
%e A194805 .                         4
%e A194805 .                           7
%e A194805 .
%e A194805 There are 25 parts that are visible, so a(7) = 25.
%e A194805 Using the formula we have a(7) = p(7) + p(7-1) - 1 = 15 + 11 - 1 = 25, where p(n) is the number of partitions of n.
%t A194805 Table[Count[IntegerPartitions[2 n - 1],  p_ /; Or[MemberQ[p, n - 1], MemberQ[p, n]]], {n, 50}]  (* _Clark Kimberling_, Mar 01 2014 *)
%t A194805 Table[PartitionsP[n] + PartitionsP[n-1] - 1, {n, 0, 44}] (* _Robert Price_, May 12 2020 *)
%Y A194805 Cf. A000041, A000065, A084376, A135010, A138121, A141285, A194550, A194803, A194804.
%K A194805 nonn
%O A194805 0,3
%A A194805 _Omar E. Pol_, Jan 27 2012