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%I #32 Sep 24 2017 12:58:25
%S 1,1,2,2,3,4,5,7,10,14,20,30,45,68,104,161,251,393,618,976,1547,2459,
%T 3917,6251,9993,15999,25647,41157,66108,106272,170961,275202,443250,
%U 714265,1151486,1857057,2995991,4834907,7804653,12601553,20351114,32872743,53107823,85811996,138674777,224130364,362286475
%N Antidiagonal sums of A179748.
%C 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!]
%C Grows slower than the Fibonacci sequence. More complicated than the Fibonacci sequence.
%C The divisibility related table A051731 can be described by the recurrence:
%C T(n,1) = 1, k > 1: T(n,k) = (Sum_{i=1..k-1} T(n-i,k-1)) - 1*(Sum_{i=1..k-1} T(n-i,k)).
%C The silver means can be found as limiting ratios of the antidiagonal sums of the tables described by the following similar recurrences:
%C T(n,1) = 1, k > 1: T(n,k) = (Sum_{i=1..k-1} T(n-i,k-1)) + 0*(Sum_{i=1..k-1} T(n-i,k)). --> antidiagonal sums limiting ratio tends to the golden ratio, A001622.
%C T(n,1) = 1, k > 1: T(n,k) = (Sum_{i=1..k-1} T(n-i,k-1)) + 1*(Sum_{i=1..k-1} T(n-i,k)). --> antidiagonal sums limiting ratio tends to the silver ratio, A014176.
%C T(n,1) = 1, k > 1: T(n,k) = (Sum_{i=1..k-1} T(n-i,k-1)) + 2*(Sum_{i=1..k-1} T(n-i,k)). --> antidiagonal sums limiting ratio tends to the bronze ratio, A098316
%C The limiting ratio becomes apparent after the first 275 terms or so of the antidiagonal sums.
%C The empirical observation that the ratio a(n+1)/a(n) tends to the golden ratio 1.6180339887498... has been verified up to a(1500)/a(1499) which gives the first 65 digits of A001622. - _Mats Granvik_, Sep 16 2017
%H Mats Granvik, <a href="http://math.stackexchange.com/questions/164510/does-this-ratio-converge-to-the-golden-ratio">Does this ratio converge to the Golden ratio?</a>
%t 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 *)
%Y Cf. A001622, A179748, cumulative sums of A186426.
%K nonn
%O 1,3
%A _Mats Granvik_, Feb 21 2011
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