A186425 Antidiagonal sums of A179748.

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

Offset

1,3

Comments

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.

The divisibility related table A051731 can be described by the recurrence:

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)).

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_{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.

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.

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

The limiting ratio becomes apparent after the first 275 terms or so of the antidiagonal sums.

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

Links

Table of n, a(n) for n=1..47.

Mats Granvik, Does this ratio converge to the Golden ratio?

Mathematica

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 *)

Crossrefs

Cf. A001622, A179748, cumulative sums of A186426.

Sequence in context: A018128 A032189 A316077 * A327662 A034395 A032232

Adjacent sequences: A186422 A186423 A186424 * A186426 A186427 A186428

Keyword

nonn

Author

Mats Granvik, Feb 21 2011

Status

approved