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

%I A113533 #13 May 19 2017 02:37:44
%S A113533 1,3,6,5,7,12,10,15,14,14,23,16,20,27,21,30,27,25,40,28,37,38,32,49,
%T A113533 36,40,53,39,54,49,43,68,45,55,66,50,71,60,56,83,57,74,75,61,92,67,73,
%U A113533 94,68,93,84,72,113,75,94,101,79,116,89,91,122,86,115,108,90
%N A113533 Ascending descending base exponent transform of the infinite Fibonacci word (A003842).
%C A113533 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.
%H A113533 G. C. Greubel, <a href="/A113533/b113533.txt">Table of n, a(n) for n = 1..1000</a>
%F A113533 a(n) = Sum_{k=1..n} A003842(k)^(A003842(n-k+1)). - _G. C. Greubel_, May 18 2017
%e A113533 a(1) = A003842(1)^A003842(1) = 1^1 = 1.
%e A113533 a(2) = A003842(1)^A003842(2) + A003842(2)^A003842(1) = 1^2 + 2^1 = 3.
%e A113533 a(3) = 1^1 + 2^2 + 1^1 = 6.
%e A113533 a(4) = 1^1 + 2^1 + 1^2 + 1^1 = 5.
%e A113533 a(5) = 1^2 + 2^1 + 1^1 + 1^2 + 2^1 = 7.
%e A113533 a(6) = 1^1 + 2^2 + 1^1 + 1^1 + 2^2 + 1^1 = 12.
%e A113533 a(7) = 1^2 + 2^1 + 1^2 + 1^1 + 2^1 + 1^2 + 2^1 = 10.
%e A113533 a(8) = 1^1 + 2^2 + 1^1 + 1^2 + 2^1 + 1^1 + 2^2 + 1^1 = 15.
%e A113533 a(9) = 1^1 + 2^1 + 1^2 + 1^1 + 2^2 + 1^1 + 2^1 + 1^2 + 1^1 = 14.
%e A113533 a(10) = 1^2 + 2^1 + 1^1 + 1^2 + 2^1 + 1^2 + 2^1 + 1^1 + 1^2 + 2^1 = 14.
%t A113533 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 *)
%Y A113533 Cf. A003842, A005408, A087316, A113122, A113153, A113154, A113208, A113231, A113257, A113258, A113271, A113320, A113336, A113498.
%K A113533 easy,nonn
%O A113533 1,2
%A A113533 _Jonathan Vos Post_, Jan 13 2006
%E A113533 Corrected and extended by _Giovanni Resta_, Jun 13 2016