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A309791 Expansion of (1 + 8*x - 6*x^2 + 12*x^3 - 18*x^4)/(1 - x - 9*x^4 + 9*x^5).

%I A309791 #20 Apr 19 2023 02:40:37
%S A309791 1,9,3,15,6,78,24,132,51,699,213,1185,456,6288,1914,10662,4101,56589,
%T A309791 17223,95955,36906,509298,155004,863592,332151,4583679,1395033,
%U A309791 7772325,2989356,41253108,12555294,69950922,26904201,371277969,112997643,629558295,242137806,3341501718,1016978784,5666024652
%N A309791 Expansion of (1 + 8*x - 6*x^2 + 12*x^3 - 18*x^4)/(1 - x - 9*x^4 + 9*x^5).
%C A309791 This sequence and its companion A309792 describe the additive constants which occur in an infinite series of maps from the row indices in the table defined by A307048 to the arithmetic progression contained in a specific column of that table. Only rows with indices of the form 6*j - 2 are concerned, and j is mapped to the unique term in that row (cf. example).
%C A309791 Conjecture: Any finite subset of these maps can build chains of finite length only.
%H A309791 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,9,-9).
%F A309791 a(n) = (1/48)*(-32+2^(n/2)*(42*(1+(-1)^n)-2*(-i)^n+105*sqrt(2)*(1-(-1)^n)+11*i*(-i)^n*sqrt(2)-i^(n+1)*(-2*i+11*sqrt(2)))), where i=sqrt(-1). - _Stefano Spezia_, Aug 19 2019
%e A309791 The maps for k >= 0 start with:
%e A309791    3*k +  1 ->  8*k +  2 ( 4->10,   7->18,  10->26, ...)
%e A309791    9*k +  9 ->  8*k +  8 ( 9-> 8,  18->16,  27->24, ...)
%e A309791    9*k +  3 -> 16*k +  5 ( 3-> 5,  12->21,  21->37, ...)
%e A309791   27*k + 15 -> 16*k +  9 (15-> 9,  42->25,  69->41, ...)
%e A309791   27*k +  6 -> 32*k +  7 ( 6-> 7,  33->39,  60->71, ...)
%e A309791   81*k + 78 -> 32*k + 31 (78->31, 159->63, 240->95, ...)
%e A309791           ^            ^
%e A309791           |            |
%e A309791           A309791      A309792
%e A309791 Chains:
%e A309791    33 ->  39 ->  69 ->  41
%e A309791   114 -> 135 -> 120 -> 213 -> 75 -> 133
%t A309791 LinearRecurrence[{1, 0, 0, 9, -9}, {1, 9, 3, 15, 6}, 32] (* or *) CoefficientList[Series[(1 + 8*x - 6*x^2 + 12*x^3 - 18*x^4)/(1 - x - 9*x^4 + 9*x^5), {x, 0, 40}], x]
%Y A309791 Cf. A307048, A309792.
%K A309791 nonn,easy
%O A309791 0,2
%A A309791 _Georg Fischer_, Aug 17 2019