A256107 Irregular triangle read by rows, T(n,k) is the number of pentagrams on the k layers at n iterations of a pentagram fractal (see comment).

1, 2, 4, 1, 7, 2, 2, 12, 4, 5, 2, 1, 20, 7, 10, 4, 2, 2, 33, 12, 18, 8, 4, 4, 2, 1, 54, 20, 31, 14, 7, 8, 4, 2, 2, 88, 33, 52, 24, 12, 14, 8, 4, 4, 2, 1, 143, 54, 86, 40, 20, 24, 14, 7, 8, 4, 2, 2, 232, 88, 141, 66, 33, 40, 24, 12, 14, 8, 4, 4, 2, 1, 376, 143, 230, 108, 54, 66

Offset: 0

Author

Kival Ngaokrajang, Mar 14 2015

Keywords

• nonn
• tabf

Comments

Refer to A255870, the number of pentagrams on one side of the outer layer (including pentagrams on two vertices) at n iterations would be T(n,0), the next layers k >= 1 T(n,k) are the number of pentagrams toward the center. For k >= 2, the row length is A032766. The first differences of A255870 = 5*(rows sum - 1). T(n,k) = A000071 with a shift for k = 0 or k mod 3 = 1. T(n,2) = A006327 with a shift. For k >= 3, T(n,k) = 2*A000071 with a shift for k mod 3 = 0 or 2. See illustration in the links.

Examples

			Irreuglar triangle begins:
n/k  0  1  2  3  4  5  6  7  8 ...
0   1
1   2
2   4  1
3   7  2  2
4  12  4  5  2  1
5  20  7 10  4  2  2
6  33 12 18  8  4  4  2  1
7  54 20 31 14  7  8  4  2  2
8  88 33 52 24 12 14  8  4  4
...
		

Links

• Kival Ngaokrajang, Illustration of initial terms, T(n,k) for n = 0..10, k = 0..13

Crossrefs

Cf. A255870, A032766, A000071, A006327.

Programs

PARI

{for(n=0, 20, if(n<2, lk=0, lk=floor(3*(n-2)/2)+1); for (k=0, lk, if(k<>0, if(k<>2, if(Mod(k,3)==1, t=fibonacci(n+1-2*(k-1)/3)-1, t=2*(fibonacci(n+2-ceil((2*k+1)/3))-1)), t=fibonacci(n+2)-3), t=fibonacci(n+3-2*k/3)-1); print1(t, ", ")))}