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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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Columns are related to Fibonacci n-step 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)-th-column. 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. - Richard Choulet, Jan 31 2010

LINKS

Table of n, a(n) for n=0..86.

Eric W. Weisstein, MathWorld: Fibonacci n-Step Number

Wikipedia, Fibonacci number

FORMULA

The general term in the n-th row and k-th column is given by: a(n,k) = Sum_{j=0..floor(n/(k+1))} ((-1)^j binomial(n-k*j,n-(k+1)*j)*2^(n-(k+1)*j)). For example: a(5,3) = binomial(5,5)*2^5 - binomial(2,1)*2^1 = 28. The generating function of the (k+1)-th column satisfies: psi(k)(z)=1/(1-2*z+z^(k+1)) (for k=0 we have the known result psi(0)(z)=1/(1-z)). - 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(n-k*j, n-(k+1)*j)*2^(n-(k+1)*j), j=0..floor(n/(k+1))):od: seq(b(n), n=0..20):od; # 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

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Last modified December 9 21:41 EST 2016. Contains 278987 sequences.