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A172119
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Sum the k preceding elements in the same column and add 1 every time.
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9
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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
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0,5
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COMMENTS
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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. [From Richard Choulet, Jan 31 2010]
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LINKS
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Table of n, a(n) for n=0..86.
Eric W. Weisstein, MathWorld: Fibonacci n-Step Number
Wikipedia, Fibonacci number
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FORMULA
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The general term in the n-th row and k-th column is given by: a(n,k)=sum((-1)^j binomial(n-k*j,n-(k+1)*j)*2^(n-(k+1)*j),j=0..floor(n/(k+1))).For example: a(5,3)=binomial(5,5)*2^5-binomial(2,1)*2^1= 28. The generating fonction 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)). [From Richard Choulet, Jan 31 2010]
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EXAMPLE
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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
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MAPLE
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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; [From Richard Choulet, Jan 31 2010]
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CROSSREFS
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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 * A093628 A186807 A114282
Adjacent sequences: A172116 A172117 A172118 * A172120 A172121 A172122
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KEYWORD
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nonn,tabl
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AUTHOR
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Mats Granvik, Jan 26 2010
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STATUS
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approved
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