A369580 a(n) := f(n, n), where f(0,0) = 1/3, f(0,k) = 0 and f(k,0) = 3^(k-1) if k > 0, and f(n, m) = f(n, m-1) + f(n-1, m) + 3*f(n-1, m-1) otherwise.
2, 16, 138, 1216, 10802, 96336, 861114, 7708416, 69072354, 619380496, 5557080938, 49879087296, 447852531986, 4022246329936, 36132550233498, 324645166734336, 2917340834679234, 26219438520320016, 235672871308226634, 2118552629658530496, 19046140604787232242, 171241206828437556816
Offset: 1
Keywords
Links
- Tadayoshi Kamegai, Table of n, a(n) for n = 1..100
- Bartosz Sobolewski and Lukas Spiegelhofer, Decomposing the sum-of-digits correlation measure, arXiv:2411.07779 [math.NT], 2024. See p. 16.
Programs
-
Python
def lis(n): table = [[0]*(n+1) for _ in range(n+1)] table[1][1] = 2 for i in range(1, n+1) : table[i][0] = 3**(i-1) for i in range(1, n+1) : for j in range(1, n+1) : if (i == 1 and j == 1) : continue table[i][j] = table[i][j-1] + table[i-1][j] + 3*table[i-1][j-1] return [int(table[i][i]) for i in range(1, n+1)]
Formula
Limit_{n->oo} a(n)/3^(2n-1) = 1/2.
a(n) = Sum_{i>=n} Sum_{j=0..n-1} binomial(i-1,n-1)*binomial(i-1,j)*3^(2n-1)/2^(2i-1).
9*a(n) - a(n+1) = 2*A162326(n) (conjectured).
a(n) = 3^(2n-1)*A(n, n) where A(0, k) = 0 for k > 0, A(k, 0) = 1 for k >= 0 and A(n, m) = (A(n-1, m) + A(n, m-1) + A(n-1, m-1))/3.
Comments