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A186425
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Antidiagonal sums of A179748.
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1
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1, 1, 2, 2, 3, 4, 5, 7, 10, 14, 20, 30, 45, 68, 104, 161, 251, 393, 618, 976, 1547, 2459, 3917, 6251, 9993, 15999, 25647, 41157, 66108, 106272, 170961, 275202, 443250, 714265, 1151486, 1857057, 2995991, 4834907, 7804653, 12601553, 20351114, 32872743, 53107823, 85811996, 138674777, 224130364, 362286475, 585661676, 946848156, 1530906874, 2475418234, 4002917308, 6473364232, 10469027150, 16931802383, 27385369011, 44294592612, 71646979665
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OFFSET
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1,3
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COMMENTS
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a(n+1)/a(n) tends to the golden ratio. [Note added by Joerg Arndt, Mar 16 2013: this is only a conjecture so far!]
Grows slower than the Fibonacci sequence. More complicated than the Fibonacci sequence.
In terms of Dirichlet convolutions the divisibility related table A051731 can be described by the recurrence:
T(n,1)=1, k>1: T(n,k) = (sum from i = 1 to k-1 of T(n-i,k-1)) - 1*(sum from i = 1 to k-1 of T(n-i,k))
The silver means can be found as limiting ratios of the antidiagonal sums of the tables described by the following similar recurrences:
T(n,1)=1, k>1: T(n,k) = (sum from i = 1 to k-1 of T(n-i,k-1)) + 0*(sum from i = 1 to k-1 of T(n-i,k)). --> antidiagonal sums limiting ratio tends to the golden ratio, A001622.
T(n,1)=1, k>1: T(n,k) = (sum from i = 1 to k-1 of T(n-i,k-1)) + 1*(sum from i = 1 to k-1 of T(n-i,k)). --> antidiagonal sums limiting ratio tends to the silver ratio, A014176.
T(n,1)=1, k>1: T(n,k) = (sum from i = 1 to k-1 of T(n-i,k-1)) + 2*(sum from i = 1 to k-1 of T(n-i,k)). --> antidiagonal sums limiting ratio tends to the bronze ratio, A098316
The limiting ratio becomes apparent after the first 275 terms or so, of the antidiagonal sums.
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LINKS
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Table of n, a(n) for n=1..58.
Mats Granvik, Does this ratio converge to the Golden ratio?
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MATHEMATICA
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Clear[a, t]; nn = 58; t[n_, 1] = 1; t[n_, k_] := t[n, k] = If[n >= k, Sum[t[n - i, k - 1], {i, 1, k - 1}], 0]; a = Table[Total[Table[t[n - k + 1, k], {k, 1, nn}]], {n, 1, nn}]; a (* Mats Granvik, Apr 27 2013 *)
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CROSSREFS
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Cf. A001622, A179748, cumulative sums of A186426.
Sequence in context: A173674 A018128 A032189 * A034395 A032232 A175306
Adjacent sequences: A186422 A186423 A186424 * A186426 A186427 A186428
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KEYWORD
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nonn
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AUTHOR
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Mats Granvik, Feb 21 2011
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STATUS
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approved
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