A152545 Padovan-Fibonacci triangle, read by rows, where the first column equals the Padovan spiral numbers (A134816), while the row sums equal the Fibonacci numbers (A000045).

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

Offset

0,5

Comments

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

Links

Paul D. Hanna, Table of n, a(n) for n = 0..261

Formula

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.

Example

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.

Prog

(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(""))

Crossrefs

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

Sequence in context: A208244 A272171 A160696 * A359538 A256660 A109967

Adjacent sequences: A152542 A152543 A152544 * A152546 A152547 A152548

Keyword

nonn,tabf

Author

Paul D. Hanna, Dec 13 2008

Status

approved