A140106 -id:A140106 - cp's OEIS frontend

A186235 Total Wiener index of double-star trees with n nodes.

10, 18, 57, 82, 169, 220, 374, 460, 700, 830, 1175, 1358, 1827, 2072, 2684, 3000, 3774, 4170, 5125, 5610, 6765, 7348, 8722, 9412, 11024, 11830, 13699, 14630, 16775, 17840, 20280, 21488, 24242, 25602, 28689, 30210, 33649, 35340, 39150, 41020

Offset: 4

Washington Bomfim, Feb 15 2011

For the trees of a given order, it appears that the Wiener indexes are very close. For n=8, the indexes are 54, 57, and 58.

The second Bomfim link refers to formulas of the total Wiener index, and the average Wiener index of those trees.

			The first Bomfim link shows a way to find a(8).

Vincenzo Librandi, Table of n, a(n) for n = 4..10000
Rundan Xing, Bo Zho, Ordering trees having small reverse Wiener indices
W. Bomfim, Example
W. Bomfim, Formulas
Index entries for linear recurrences with constant coefficients, signature (1,3,-3,-3,3,1,-1).

Cf. A122681, A140106, A168559.

[ IsEven(n) select (n-2)*(2*n-3)*(7*n-4)/24 else (n-3)*(n-1)*(7*n-8)/12: n in [4..43] ]; // Bruno Berselli, Feb 17 2011

a[n_]:= a[n] = -a[n-7] + a[n-6] + 3a[n-5] - 3a[n-4] - 3a[n-3] + 3a[n-2] + a[n-1]; a[0]=-1; a[1]=0; a[2]=0; a[3]=0; a[4]=10; a[5]=18; a[6]=57; a /@ Range[4, 43] (* Jean-François Alcover, Jun 01 2011, after recurrence signature *)
LinearRecurrence[{1,3,-3,-3,3,1,-1},{10,18,57,82,169,220,374},40] (* Harvey P. Dale, Mar 25 2013 *)

for(n=4,43,if(n%2,print1((1/12)*(7*n^3+53*n)-3*n^2-2,", "), print1((1/24)*(14*n^3-57*n^2+70*n)-1,", ")))

G.f.: x^4*(10+8*x+9*x^2+x^3)/((1+x)^3*(1-x)^4). Also a(n)=(n*(28*n^2-129*n+176)+3*(5*n^2-12*n+8)*(-1)^n-72)/48. - Bruno Berselli, Feb 15 2011
For even n, a(n)=(14*n^3-57*n^2+70*n)/24-1, otherwise a(n)=(7*n^3+53*n)/12-3*n^2-2.
With d=floor((n-2)/2), a(n)=d((n-2)*(n-1)+n*(d+3)/2-d^2/3-3*d/2-13/6).

A337698 Number of compositions of n that are not strictly increasing.

Original entry on oeis.org

0, 0, 1, 2, 6, 13, 28, 59, 122, 248, 502, 1012, 2033, 4078, 8170, 16357, 32736, 65498, 131026, 262090, 524224, 1048500, 2097063, 4194200, 8388486, 16777074, 33554267, 67108672, 134217506, 268435200, 536870616, 1073741484, 2147483258, 4294966848, 8589934080

Offset: 0

Gus Wiseman, Oct 06 2020

nonn

			The a(2) = 1 through a(5) = 13 compositions:
  (11)  (21)   (22)    (32)
        (111)  (31)    (41)
               (112)   (113)
               (121)   (122)
               (211)   (131)
               (1111)  (212)
                       (221)
                       (311)
                       (1112)
                       (1121)
                       (1211)
                       (2111)
                       (11111)

A000009 counts the complement.
A047967 is the unordered version.
A056823 is the weak version.
A140106 counts the unordered case of length 3.
A242771 counts the case of length 3.
A333255 is the complement of a ranking sequence (using standard compositions A066099) for these compositions.
A337481 counts these compositions that are not strictly decreasing.
A337482 counts these compositions that are not weakly decreasing.
A001523 counts unimodal compositions, with complement A115981.
A007318 and A097805 count compositions by length.
A218004 counts strictly increasing or weakly decreasing compositions.
Cf. A000212, A128422, A332745, A332834, A332835, A337484.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!Less@@#&]],{n,0,15}]

a(n) = 2^(n-1) - A000009(n) for n > 0.

cp's OEIS Frontend

A186235 Total Wiener index of double-star trees with n nodes.

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