A152060 -id:A152060 - cp's OEIS frontend

A348447 Irregular triangle read by rows: T(n,k) (n >= 0) is the number of ways of selecting k objects from n objects arranged in a circle with no two selected objects having unit separation (i.e. having exactly one object between them).

1, 1, 1, 1, 2, 1, 1, 3, 1, 4, 4, 1, 5, 5, 1, 6, 9, 1, 7, 14, 7, 1, 8, 20, 16, 4, 1, 9, 27, 30, 9, 1, 10, 35, 50, 25, 1, 11, 44, 77, 55, 11, 1, 12, 54, 112, 105, 36, 4, 1, 13, 65, 156, 182, 91, 13, 1, 14, 77, 210, 294, 196, 49, 1, 15, 90, 275, 450, 378, 140, 15

Offset: 0

N. J. A. Sloane, Oct 23 2021

For odd n, T(n,k) is the number of independent vertex sets of size k in an n-cycle. For even n, T(n,k) is the number of independent vertex sets of size k in two disjoint (n/2)-cycles. Here, a cycle of length one or two is taken to be a path. - Andrew Howroyd, Jan 01 2024

			Triangle begins:
  1;
  1,   1;
  1,   2,   1;
  1,   3;
  1,   4,   4;
  1,   5,   5;
  1,   6,   9;
  1,   7,  14,   7;
  1,   8,  20,  16,   4;
  1,   9,  27,  30,   9;
  1,  10,  35,  50,  25;
  1,  11,  44,  77,  55,  11;
  1,  12,  54, 112, 105,  36,   4;
  1,  13,  65, 156, 182,  91,  13;
  ...
If n = 2 and we call the objects 0 and 1, the permitted sets of objects are {}, {0}, {1}, and {0,1}. If n = 3 and we call the objects 0, 1, and 2, then the permitted sets of objects are {}, {0}, {1}, and {2}; {0,1} is not a permitted set in this case since the objects lie in a circle and 2 lies between 0 and 1 in one direction. - _Michael A. Allen_, Apr 25 2022

Andrew Howroyd, Table of n, a(n) for n = 0..2578 (rows 0..100)
John Konvalina, On the number of combinations without unit separation., Journal of Combinatorial Theory, Series A 31.2 (1981): 101-107. See Table II (which erroneously lacks the n=k=2 element).

See A348445 for the case when the n objects are on a line.
The triangles A034807, A061896, A152060 are very similar.
The k=2 column is (essentially) A347553.

T(n) = {[Vecrev(p) | p <- Vec(-3 + y*x + y*(2 + y)*x^2 + (4 - 3*x - y*x^3)/((1 + y*x^2)*(1 - x - y*x^2)) + O(x*x^n))]}
{ my(A=T(10)); for(i=1, #A, print(A[i])) } \\ Andrew Howroyd, Jan 01 2024

G.f.: -3 + y*x + y*(2 + y)*x^2 + (4 - 3*x - y*x^3)/((1 + y*x^2)*(1 - x - y*x^2)). - Andrew Howroyd, Jan 01 2024

Corrected by Michael A. Allen, Apr 25 2022

Terms a(56) and beyond from Andrew Howroyd, Jan 01 2024

A126570 Triangle read by rows: row n gives coefficients (ignoring the alternating signs) of the characteristic polynomial of the n X n matrix with 2's in the main diagonal, 1's in the super- and subdiagonals, and 1 in the upper-right corner, with other elements zeros.

Original entry on oeis.org

1, 1, 2, 1, 4, 3, 1, 6, 10, 5, 1, 8, 21, 20, 4, 1, 10, 36, 56, 35, 7, 1, 12, 55, 120, 126, 56, 6, 1, 14, 78, 220, 330, 252, 84, 9, 1, 16, 105, 364, 715, 792, 462, 120, 8, 1, 18, 136, 560, 1365, 2002, 1716, 792, 165, 11, 1, 20, 171, 816, 2380, 4368, 5005, 3432, 1287, 220, 10

Offset: 0

Gary W. Adamson, Dec 28 2006

			First few rows of the triangle are:
  1;
  1, 2;
  1, 4, 3;
  1, 6, 10, 5;
  1, 8, 21, 20, 4;
  1, 10, 36, 56, 35, 7;
  1, 12, 55, 120, 126, 56, 6;
  ...
Charpoly of the 4 X 4 matrix [2,1,0,1; 1,2,1,0; 0,1,2,1; 0,0,1,2] = x^4 - 8*x^3 + 21*x^2 - 20*x + 4; with a root sqrt(3)+2.

William G. Harter, Physics Department, University of Arkansas; personal communication.

Cf. A152060, A053123.

M[i_,j_,n_]:=If[i==j,2,If[Abs[i-j]==1,1,If[j==n&&i==1,1,0]]]; row[0]=1; row[n_]:=Reverse[Abs[CoefficientList[CharacteristicPolynomial[Table[M[i,j,n],{i,n},{j,n}],x],x]]]; Array[row,11,0]//Flatten (* Stefano Spezia, Jun 30 2025 *)

It seems that for n > 2, T(n, k) = binomial(2n+1-k, k) - [k=n] * (-1)^n. - Andrei Zabolotskii, Jun 29 2025

Edited by N. J. A. Sloane, Aug 10 2019

Edited and extended by Andrei Zabolotskii, Jun 29 2025

cp's OEIS Frontend

A348447 Irregular triangle read by rows: T(n,k) (n >= 0) is the number of ways of selecting k objects from n objects arranged in a circle with no two selected objects having unit separation (i.e. having exactly one object between them).

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