0,4

The number of terms in each row equal the Padovan spiral numbers (A134816, with offset).

G.f. for row n: Sum_{k=0..A000931(n+5)-1} (x^{T(n-1,k)+T(n-2,k)} - 1)/(x-1) = Sum_{k=0..A000931(n+6)-1} T(n,k)*x^k for n>1 with T(0,0)=T(1,0)=1, where A000931 is the Padovan sequence.

Triangle begins:

[1],

[1],

[1,1],

[2,1],

[2,2,1],

[3,2,2,1],

[4,3,3,2,1],

[5,4,4,3,3,1,1],

[7,5,5,5,4,3,3,1,1],

[9,7,7,7,5,5,5,4,3,1,1,1],

[12,9,9,9,8,7,7,7,5,4,4,4,1,1,1,1],

[16,12,12,12,12,9,9,9,8,8,8,7,5,4,4,4,1,1,1,1,1],

[21,16,16,16,16,13,12,12,12,12,12,11,9,8,8,8,5,5,5,5,4,1,1,1,1,1,1,1],

[28,21,21,21,21,20,16,16,16,16,16,16,13,13,12,12,12,12,11,11,8,6,5,5,5,5,5,5,1,1,1,1,1,1,1,1,1],

[37,28,28,28,28,28,22,21,21,21,21,21,20,20,20,20,18,16,16,16,14,13,12,12,12,12,12,11,6,6,6,6,6,5,5,5,5,1,1,1,1,1,1,1,1,1,1,1,1],

[49,37,37,37,37,37,33,28,28,28,28,28,28,28,28,28,27,22,22,21,21,21,20,20,20,20,20,18,17,17,16,16,14,12,12,12,12,7,6,6,6,6,6,6,6,6,6,6,5,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1],

...

ILLUSTRATION OF RECURRENCE.

Start out with row 0 and row 1 consisting of a single '1'.

To obtain any given row of this irregular triangle, first

sum the prior two rows term-by-term; for rows 7 and 8 we get:

[5,4,4,3,3,1,1] + [7,5,5,5,4,3,3,1,1] = [12,9,9,8,7,4,4,1,1].

Place markers in an array so that the number of contiguous markers

in each row correspond to the term-by-term sums like so:

--------------------------

12:o o o o o o o o o o o o

9: o o o o o o o o o - - -

9: o o o o o o o o o - - -

8: o o o o o o o o - - - -

7: o o o o o o o - - - - -

4: o o o o - - - - - - - -

4: o o o o - - - - - - - -

1: o - - - - - - - - - - -

1: o - - - - - - - - - - -

--------------------------

Then count the markers by columns to obtain the desired row;

here, the number of markers in each column yields row 9:

[9,7,7,7,5,5,5,4,3,1,1,1].

Continuing in this way generates all the rows of this triangle.

(PARI) {T(n, k)=local(G000931=(1-x^2)/(1-x^2-x^3+x*O(x^(n+6)))); if(n<0, 0, if(n<2&k==0, 1, polcoeff(sum(j=0, polcoeff(G000931, n+5)-1, (x^(T(n-1, j)+T(n-2, j)) - 1)/(x-1)), k) ))};

/* To print, use Padovan g.f. to get the number of terms in row n: */

for(n=0, 10, for(k=0, polcoeff((1-x^2)/(1-x^2-x^3+x*O(x^(n+6))), n+6)-1, print1(T(n, k), ", ")); print(""))

Cf. A134816, A000045, A000931; A152546 (row squared sums).

Sequence in context: A095686 A105258 A160696 this_sequence A109967 A000119 A097368

Adjacent sequences: A152542 A152543 A152544 this_sequence A152546 A152547 A152548

nonn,tabf

Paul D. Hanna (pauldhanna(AT)juno.com), Dec 13 2008