0,3

Row sums = A024787, the numbers of 3's in all partitions of n, where A024787 starts with offset 1: (0, 0, 1, 1, 2, 4, 6, 9, 15,...). Triangle A173239 row sums start with the first "1" of A024787.

Let the triangle = M as an infinite lower triangular matrix. Then Lim_{n->inf} = A173241, the Euler transform of A051064 (the ruler function for 3).

Let P(x) = polcoeff A000041 = (1 + x + 2x^2 + 3x^3 + 5x^4 + 7x^5 + ...), then P(x) = A(x) / A(x^3), where A(x) = polcoeff A173241: (1 + x + 2x^2 + 4x^3 + 6x^4 + ...)

Refer to A173238 comments for three conjectures relating A000041 to the infinite set of generalized ruler function sequences.

Table of n, a(n) for n=0..69.

T(n,k) = A000041(n-3*k) for k=0..floor(n/3).

First few rows of the triangle =

1;

1;

2;

3, 1;

5, 1;

7, 2;

11, 3, 1;

15, 5, 1;

22, 7, 2;

30, 11, 3, 1;

42, 15, 5, 1;

56, 22, 7, 2;

77, 30, 11, 3, 1;

101, 42, 15, 5, 1;

135, 56, 22, 7, 2;

176, 77, 30, 11, 3, 1;

231, 101, 42, 15, 5, 1;

297, 135, 56, 22, 7, 2;

385, 176, 77, 30, 11, 3, 1;

490, 231, 101, 42, 15, 5, 1;

627, 297, 135, 56, 22, 7, 2;

792, 385, 176, 77, 30, 11, 3, 1;

1002,490, 231, 101, 42, 15, 5, 1;

1255, 627, 297, 135, 56, 22, 7, 2;

1575, 792, 385, 176, 77, 30, 11, 3, 1;

...

Cf. A000041, A173238, A173241, A051064, A024787.

Sequence in context: A326584 A166333 A322966 * A214055 A066909 A095195

Adjacent sequences:  A173236 A173237 A173238 * A173240 A173241 A173242

nonn,tabf,easy

Gary W. Adamson, Feb 13 2010

approved