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A172119 Sum the k preceding elements in the same column and add 1 every time. 10
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.

T. Langley, J. Liese, J. Remmel, Generating Functions for Wilf Equivalence Under Generalized Factor Order , J. Int. Seq. 14 (2011) # 11.4.2

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

Wikipedia, Fibonacci number

FORMULA

T(n,0)=1.

T(n,1)=n.

T(n,2)=A000071(n+1).

T(n,3)=A008937(n-2).

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

A172119 := proc(n, k)

    option remember;

    if k = 0 then

        1;

    elif k > n then

        0;

    else

        1+add(procname(n-k+i, k), i=0..k-1) ;

    end if;

end proc:

seq(seq(A172119(n, k), k=0..n), n=0..12) ; # R. J. Mathar, Sep 16 2017

CROSSREFS

Cf. A000071, 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,changed

AUTHOR

Mats Granvik, Jan 26 2010

STATUS

approved

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Last modified September 20 05:38 EDT 2017. Contains 292259 sequences.