A309937 Irregular triangle read by rows: T(n,k) is the number of compositions of n with 2k parts and circular differences all equal to 1 or -1, (n >= 3, 1 <= k <= n/3).

2, 0, 2, 0, 2, 2, 0, 0, 4, 2, 0, 2, 0, 2, 0, 2, 0, 6, 0, 4, 0, 2, 2, 0, 6, 0, 0, 2, 0, 8, 2, 0, 8, 0, 2, 0, 4, 0, 12, 0, 2, 0, 6, 0, 10, 0, 2, 0, 16, 0, 2, 2, 0, 6, 0, 20, 0, 0, 4, 0, 18, 0, 12, 2, 0, 8, 0, 30, 0, 2, 0, 2, 0, 16, 0, 30, 0, 2, 0, 6, 0, 40, 0, 14, 0, 4, 0, 20, 0, 52, 0, 2

Offset: 3

Author

Andrew Howroyd, Aug 23 2019

Keywords

• nonn
• tabf

Comments

All values are even since the parts must alternate between even and odd and therefore a composition is never equal to its reversal.

The longest compositions will consist of alternating 1's and 2's. The number of parts cannot then exceed n / 3.

Examples

			Triangle begins:
  2;
  0;
  2;
  0, 2;
  2, 0;
  0, 4;
  2, 0, 2;
  0, 2, 0;
  2, 0, 6;
  0, 4, 0,  2;
  2, 0, 6,  0;
  0, 2, 0,  8;
  2, 0, 8,  0,  2;
  0, 4, 0, 12,  0;
  2, 0, 6,  0, 10;
  0, 2, 0, 16,  0,   2;
  2, 0, 6,  0, 20,   0;
  0, 4, 0, 18,  0,  12;
  2, 0, 8,  0, 30,   0,   2;
  0, 2, 0, 16,  0,  30,   0;
  2, 0, 6,  0, 40,   0,  14;
  0, 4, 0, 20,  0,  52,   0,   2;
  2, 0, 6,  0, 42,   0,  42,   0;
  0, 2, 0, 16,  0,  78,   0,  16;
  2, 0, 8,  0, 50,   0,  84,   0,  2;
  0, 4, 0, 18,  0,  96,   0,  56,  0;
  2, 0, 6,  0, 50,   0, 140,   0, 18;
  0, 2, 0, 16,  0, 116,   0, 128,  0, 2;
  ...
For n = 11 there are a total of 8 compositions:
  k = 1: (56), (65)
  k = 3: (121232), (123212), (212123), (212321), (232121), (321212)
		

Crossrefs

Row sums are A325589.

Cf. A309931, A309938, A309939, A325558, A325590.

Programs

PARI

step(R,n)={matrix(n,n,i,j, if(i>j, if(j>1, R[i-j,j-1]) + if(j+1<=n, R[i-j,j+1])))}
T(n)={my(v=vector(n\3)); for(k=1, n, my(R=matrix(n,n,i,j,i==j&&abs(i-k)==1), m=0); while(R, m++; if(m%2==0, v[m/2]+=R[n,k]); R=step(R,n))); v}
for(n=3, 24, print(T(n)))