A367076 Irregular triangle read by rows: T(n,k) (0 <= n, 0 <= k < 2^n). T(n,k) = -Sum_{i=0..k} A365968(n,i).

0, 1, 0, 3, 4, 3, 0, 6, 10, 12, 12, 12, 10, 6, 0, 10, 18, 24, 28, 32, 34, 34, 32, 34, 34, 32, 28, 24, 18, 10, 0, 15, 28, 39, 48, 57, 64, 69, 72, 79, 84, 87, 88, 89, 88, 85, 80, 85, 88, 89, 88, 87, 84, 79, 72, 69, 64, 57, 48, 39, 28, 15, 0, 21, 40, 57, 72, 87

Offset: 0

Author

John Tyler Rascoe, Nov 05 2023

Keywords

• nonn
• easy
• look
• tabf

Examples

			Triangle begins:
    k=0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15
n=0:  0;
n=1:  1,  0;
n=2:  3,  4,  3,  0;
n=3:  6, 10, 12, 12, 12, 10,  6,  0;
n=4; 10, 18, 24, 28, 32, 34, 34, 32, 34, 34, 32, 28, 24, 18, 10,  0;
		

Links

• John Tyler Rascoe, Rows n = 0..12, flattened
• Wikipedia, Blancmange curve.
• Index entries for sequences related to binary expansion of n

Crossrefs

Cf. A000217 (column k=0), A028552 (column k=1), A192021 (row sums).

Cf. A004074, A249071, A274575, A277914, A296062, A365968.

Programs

Mathematica

nmax=10; row[n_]:=Join[CoefficientList[Series[1/(1-x)*Sum[ i/(1+x^2^(i-1))*Product[1+x^2^j,{j,0,i-2}],{i,n}],{x,0,2^n-1}],x],{0}]; Array[row,6,0] (* Stefano Spezia, Dec 23 2023 *)

Python

def row_gen(n):
    x = 0
    for k in range(2**n):
        b = bin(k)[2:].zfill(n)
        x += sum((-1)**(int(b[n-i])+1)*i for i in range(1,n+1))
        yield(-x)
def A367076_row_n(n): return(list(row_gen(n)))

Formula

T(n,k) = Sum_{i=0..n} abs(k + 1 - (2^i) * round((k+1)/2^i)) * i.

G.f. for n-th row: 1/(1-x) * Sum_{i=1..n} (i/(1+x^2^(i-1)) * Product_{j=0..i-2} 1 + x^2^j).