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Search: id:A098925
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| A098925 |
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Distribution of the number of ways for a child to climb a staircase having r steps (one step or two steps at a time). |
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+0
3
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| 1, 1, 1, 1, 2, 1, 1, 3, 1, 3, 4, 1, 1, 6, 5, 1, 4, 10, 6, 1, 1, 10, 15, 7, 1, 5, 20, 21, 8, 1, 1, 15, 35, 28, 9, 1, 6, 35, 56, 36, 10, 1, 1, 21, 70, 84, 45, 11, 1, 7, 56, 126, 120, 55, 12, 1, 1, 28, 126, 210, 165, 66, 13, 1, 8, 84, 252, 330, 220, 78, 14, 1, 1, 36, 210, 462, 495, 286, 91
(list; graph; listen)
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OFFSET
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0,5
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COMMENT
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Note that the row sums in the example yield the terms of Fibonacci's sequence(A000045). Were the child capable of taking three steps at a time, the row sums of the resulting table would add to the tribonacci sequence (A000073) etc.
Essentially the same as A030528 (without the 0's), where one can find additional information. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 29 2005
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REFERENCES
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Mario Livio, The Golden Ratio, (2002) page 100.
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FORMULA
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a(n) = abs(A092865)
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EXAMPLE
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There are 13 ways for the child to climb a staircase with six steps since the partitions of 6 into 1's and 2's are 222, 2211, 21111 and 111111; and these can be permuted in 1 + 6 + 5 + 1 = 13 ways.
The general cases can be readily shown by displacing Pascal's Triangle (A007318) as follows:
1
..1
..1..1
.....2..1
.....1..3..1
........3..4..1
........1..6..5..1
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MAPLE
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T:=(n, k)->sum((-1)^(n+i)*binomial(n, i)*binomial(i+k+1, 2*k+1), i=0..n): 1, 1, seq(seq(T(n, k), k=floor(n/2)..n), n=1..16); (Deutsch)
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CROSSREFS
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Cf. A000045, A000073, A000078, A007318, A092865.
Cf. A030528.
Adjacent sequences: A098922 A098923 A098924 this_sequence A098926 A098927 A098928
Sequence in context: A035667 A102426 A092865 this_sequence A052920 A089141 A003687
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KEYWORD
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easy,nonn
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AUTHOR
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Alford Arnold (Alford1940(AT)aol.com), Oct 19 2004
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EXTENSIONS
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More terms from Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 29 2005
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