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A113534 Ascending descending base exponent transform of the flipped tribonacci substitution (A092782).

%I A113534 #15 May 19 2017 02:37:53
%S A113534 1,3,6,7,20,10,39,12,26,19,20,43,21,78,24,53,30,57,43,88,61,59,56,43,
%T A113534 90,42,155,46,109,53,122,75,105,114,73,122,62,197,63,172,71,136,96,
%U A113534 183,140,122,139,86,179,81,304,83,185,98,153,162,160,261,121,192,107,236,126
%N A113534 Ascending descending base exponent transform of the flipped tribonacci substitution (A092782).
%C A113534 The flipped tribonacci substitution (A092782) b(n) is the fixed point of the morphism 1 -> 12, 2 -> 13, 3 -> 1, starting from b(1) = 1. The transformed sequence a(n) satisfies n <= a(n) <= 27 n but the bound can be determined to be much tighter.
%H A113534 V. F. Sirvent, <a href="http://dx.doi.org/10.1016/S0893-9659(98)00121-9">Semigroups and the self-similar structure of the flipped tribonacci substitution</a>, Applied Math. Letters, 12 (1999), 25-29. [Contains many further references.]
%F A113534 a(n) = Sum_{k=1..n} A092782(k)^(A092782(n-k+1)). - _G. C. Greubel_, May 17 2017
%e A113534 a(1) = A092782(1)^A092782(1) = 1^1 = 1.
%e A113534 a(2) = A092782(1)^A092782(2) + A092782(2)^A092782(1) = 1^2 + 2^1 = 3.
%e A113534 a(3) = 1^1 + 2^2 + 1^1 = 6.
%e A113534 a(4) = 1^3 + 2^1 + 1^2 + 3^1 = 7.
%e A113534 a(5) = 1^1 + 2^3 + 1^1 + 3^2 + 1^1 = 20.
%e A113534 a(6) = 1^2 + 2^1 + 1^3 + 3^1 + 1^2 + 2^1 = 10.
%e A113534 a(7) = 1^1 + 2^2 + 1^1 + 3^3 + 1^1 + 2^2 + 1^1 = 39.
%e A113534 a(8) = 1^1 + 2^1 + 1^2 + 3^1 + 1^3 + 2^1 + 1^2 + 1^1 = 12.
%e A113534 a(9) = 1^2 + 2^1 + 1^1 + 3^2 + 1^1 + 2^3 + 1^1 + 1^2 + 2^1 = 26.
%e A113534 a(10) = 1^1 + 2^2 + 1^1 + 3^1 + 1^2 + 2^1 + 1^3 + 1^1 + 2^2 + 1^1 = 19.
%e A113534 a(11) = 1^3 + 2^1 + 1^2 + 3^1 + 1^1 + 2^2 + 1^1 + 1^3 + 2^1 + 1^2 + 3^1 = 20.
%e A113534 a(12) = 1^1 + 2^3 + 1^1 + 3^2 + 1^1 + 2^1 + 1^2 + 1^1 + 2^3 + 1^1 + 3^2 + 1^1 = 43.
%t A113534 A092782[n_] := Nest[Function[l, {Flatten[(l /. {1 -> {1, 2}, 2 -> {1, 3}, 3 -> {1}})]}], {1}, n][[1]]; Table[Sum[(A092782[k][[k]])^((A092782[n - k + 1][[n - k + 1]])), {k, 1, n}], {n, 1, 10}] (* _G. C. Greubel_, May 18 2017 *)
%Y A113534 Cf. A005408, A087316, A092782, A113122, A113153, A113154, A113208, A113231, A113257, A113258, A113271, A113320, A113336, A113498.
%K A113534 easy,nonn
%O A113534 1,2
%A A113534 _Jonathan Vos Post_, Jan 13 2006
%E A113534 a(3) corrected by _Giovanni Resta_, Jun 13 2016
%E A113534 a(13) onward from _G. C. Greubel_, May 18 2017