A113533 Ascending descending base exponent transform of the infinite Fibonacci word (A003842).

1, 3, 6, 5, 7, 12, 10, 15, 14, 14, 23, 16, 20, 27, 21, 30, 27, 25, 40, 28, 37, 38, 32, 49, 36, 40, 53, 39, 54, 49, 43, 68, 45, 55, 66, 50, 71, 60, 56, 83, 57, 74, 75, 61, 92, 67, 73, 94, 68, 93, 84, 72, 113, 75, 94, 101, 79, 116, 89, 91, 122, 86, 115, 108, 90

Offset

1,2

Comments

The infinite Fibonacci word b(n) is the fixed point of the morphism 1->12, 2->1, starting from b(1) = 2. This transform a(n) of that sequence b(n) satisfies n <= a(n) <= 4*n, but that is not a tight bound.

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

Formula

a(n) = Sum_{k=1..n} A003842(k)^(A003842(n-k+1)). - G. C. Greubel, May 18 2017

Example

a(1) = A003842(1)^A003842(1) = 1^1 = 1.
a(2) = A003842(1)^A003842(2) + A003842(2)^A003842(1) = 1^2 + 2^1 = 3.
a(3) = 1^1 + 2^2 + 1^1 = 6.
a(4) = 1^1 + 2^1 + 1^2 + 1^1 = 5.
a(5) = 1^2 + 2^1 + 1^1 + 1^2 + 2^1 = 7.
a(6) = 1^1 + 2^2 + 1^1 + 1^1 + 2^2 + 1^1 = 12.
a(7) = 1^2 + 2^1 + 1^2 + 1^1 + 2^1 + 1^2 + 2^1 = 10.
a(8) = 1^1 + 2^2 + 1^1 + 1^2 + 2^1 + 1^1 + 2^2 + 1^1 = 15.
a(9) = 1^1 + 2^1 + 1^2 + 1^1 + 2^2 + 1^1 + 2^1 + 1^2 + 1^1 = 14.
a(10) = 1^2 + 2^1 + 1^1 + 1^2 + 2^1 + 1^2 + 2^1 + 1^1 + 1^2 + 2^1 = 14.

Mathematica

A003842[n_] := n + 1 - Floor[((1 + Sqrt[5])/2)*Floor[2*(n + 1)/(1 + Sqrt[5])]]; Table[Sum[A003842[k]^(A003842[n - k + 1]), {k, 1, n}], {n, 1, 50}] (* G. C. Greubel, May 18 2017 *)

Crossrefs

Cf. A003842, A005408, A087316, A113122, A113153, A113154, A113208, A113231, A113257, A113258, A113271, A113320, A113336, A113498.

Sequence in context: A322887 A175650 A272976 * A330525 A201418 A123688

Adjacent sequences: A113530 A113531 A113532 * A113534 A113535 A113536

Keyword

easy,nonn

Author

Jonathan Vos Post, Jan 13 2006

Extensions

Corrected and extended by Giovanni Resta, Jun 13 2016

Status

approved