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A113497 Ascending descending base exponent transform of sequence A000034(n) = 1 + n mod 2.

%I A113497 #24 Mar 13 2017 04:30:57
%S A113497 1,3,6,6,11,9,16,12,21,15,26,18,31,21,36,24,41,27,46,30,51,33,56,36,
%T A113497 61,39,66,42,71,45,76,48,81,51,86,54,91,57,96,60,101,63,106,66,111,69,
%U A113497 116,72,121,75,126,78,131,81,136,84,141,87,146,90,151,93,156,96,161,99,166,102,171
%N A113497 Ascending descending base exponent transform of sequence A000034(n) = 1 + n mod 2.
%C A113497 A000034 = 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, ... = continued fraction for (sqrt(3)+1)/2 (cf. A040001) = base 3 digital root of n+1. In general, the ascending descending base exponent transform of any simple periodic sequence can be written as a periodic set of interleaved sequences.
%H A113497 G. C. Greubel, <a href="/A113497/b113497.txt">Table of n, a(n) for n = 1..1000</a>
%H A113497 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,2,0,-1).
%F A113497 a(n) = Sum_{i=1..n} A000034(i)^A000034(n-i+1).
%F A113497 a(2*n) = 3*n; a(2*n+1) = 5*n+1.
%F A113497 From _Colin Barker_, Jun 16 2012: (Start)
%F A113497 a(n) = (-3+3*(-1)^n+8*n-2*(-1)^n*n)/4.
%F A113497 a(n) = 2*a(n-2)-a(n-4).
%F A113497 G.f.: x*(1+3*x+4*x^2)/((1-x)^2*(1+x)^2). (End)
%F A113497 E.g.f.: (1/2)*(3*(x-1)*sinh(x) + 5*x*cosh(x)). - _G. C. Greubel_, Mar 12 2017
%e A113497 a(1) = 1^1 = 1.
%e A113497 a(2) = 1^2 + 2^1 = 3.
%e A113497 a(3) = 1^1 + 2^2 + 1^1 = 6.
%e A113497 a(4) = 1^2 + 2^1 + 1^2 + 2^1 = 6.
%e A113497 a(5) = 1^1 + 2^2 + 1^1 + 2^2 + 1^1 = 11.
%e A113497 a(6) = 1^2 + 2^1 + 1^2 + 2^1 + 1^2 + 2^1 = 9.
%t A113497 Table[(-3 + 3*(-1)^n + 8*n - 2*(-1)^n*n)/4, {n,1,50}] (* _G. C. Greubel_, Mar 12 2017 *)
%o A113497 (PARI) x='x +O('x^50); Vec(x*(1+3*x+4*x^2)/((1-x)^2*(1+x)^2)) \\ _G. C. Greubel_, Mar 12 2017
%Y A113497 Cf. A000034, A113320, A005408, A113122, A113153, A113154, A113336, A113271, A113258, A113257, A113231, A087316, A113208.
%K A113497 nonn,easy
%O A113497 1,2
%A A113497 _Jonathan Vos Post_, Jan 10 2006
%E A113497 Definition improved by _M. F. Hasler_, Jan 13 2012