

A172119


Sum the k preceding elements in the same column and add 1 every time.


9



1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 4, 4, 2, 1, 1, 5, 7, 4, 2, 1, 1, 6, 12, 8, 4, 2, 1, 1, 7, 20, 15, 8, 4, 2, 1, 1, 8, 33, 28, 16, 8, 4, 2, 1, 1, 9, 54, 52, 31, 16, 8, 4, 2, 1, 1, 10, 88, 96, 60, 32, 16, 8, 4, 2, 1, 1, 11, 143, 177, 116, 63, 32, 16, 8, 4, 2, 1, 1, 12, 232, 326, 224, 124, 64, 32, 16
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OFFSET

0,5


COMMENTS

Columns are related to Fibonacci nstep numbers. Are there closed forms for the sequences in the columns?
We denote by a(n,k) the number which is in the (n+1)th row and (k+1)thcolumn. With help of the definition, we have also the recurrence relation: a(n+k+1,k)=2*a(n+k,k)a(n,k). We see on the main diagonal the numbers 1,2,4, 8, ..., which is clear from the formula for the general term d(n)=2^n. [From Richard Choulet, Jan 31 2010]


LINKS

Table of n, a(n) for n=0..86.
Eric W. Weisstein, MathWorld: Fibonacci nStep Number
Wikipedia, Fibonacci number


FORMULA

The general term in the nth row and kth column is given by: a(n,k)=sum((1)^j binomial(nk*j,n(k+1)*j)*2^(n(k+1)*j),j=0..floor(n/(k+1))).For example: a(5,3)=binomial(5,5)*2^5binomial(2,1)*2^1= 28. The generating fonction of the (k+1)th column satisfies: psi(k)(z)=1/(12*z+z^(k+1)) (for k=0 we have the known result psi(0)(z)=1/(1z)). [From Richard Choulet, Jan 31 2010]


EXAMPLE

Triangle begins:
n\k....0....1....2....3....4....5....6....7....8....9...10

0......1
1......1....1
2......1....2....1
3......1....3....2....1
4......1....4....4....2....1
5......1....5....7....4....2....1
6......1....6...12....8....4....2....1
7......1....7...20...15....8....4....2....1
8......1....8...33...28...16....8....4....2....1
9......1....9...54...52...31...16....8....4....2....1
10.....1...10...88...96...60...32...16....8....4....2....1


MAPLE

for k from 0 to 20 do for n from 0 to 20 do b(n):=sum((1)^j*binomial(nk*j, n(k+1)*j)*2^(n(k+1)*j), j=0..floor(n/(k+1))):od: seq(b(n), n=0..20):od; [From Richard Choulet, Jan 31 2010]


CROSSREFS

Cf. k=0 A000012, k=1 A000027, k=2 A000071, k=3 A008937.
Cf. (1((1)^T(n, k)))/2 = A051731, see formula by Hieronymus Fischer in A022003.
Sequence in context: A055794 A092905 A052509 * A228125 A227588 A093628
Adjacent sequences: A172116 A172117 A172118 * A172120 A172121 A172122


KEYWORD

nonn,tabl


AUTHOR

Mats Granvik, Jan 26 2010


STATUS

approved



