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A309791
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Expansion of (1 + 8*x - 6*x^2 + 12*x^3 - 18*x^4)/(1 - x - 9*x^4 + 9*x^5).
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2
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1, 9, 3, 15, 6, 78, 24, 132, 51, 699, 213, 1185, 456, 6288, 1914, 10662, 4101, 56589, 17223, 95955, 36906, 509298, 155004, 863592, 332151, 4583679, 1395033, 7772325, 2989356, 41253108, 12555294, 69950922, 26904201, 371277969, 112997643, 629558295, 242137806, 3341501718, 1016978784, 5666024652
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OFFSET
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0,2
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COMMENTS
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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).
Conjecture: Any finite subset of these maps can build chains of finite length only.
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LINKS
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FORMULA
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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
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EXAMPLE
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The maps for k >= 0 start with:
3*k + 1 -> 8*k + 2 ( 4->10, 7->18, 10->26, ...)
9*k + 9 -> 8*k + 8 ( 9-> 8, 18->16, 27->24, ...)
9*k + 3 -> 16*k + 5 ( 3-> 5, 12->21, 21->37, ...)
27*k + 15 -> 16*k + 9 (15-> 9, 42->25, 69->41, ...)
27*k + 6 -> 32*k + 7 ( 6-> 7, 33->39, 60->71, ...)
81*k + 78 -> 32*k + 31 (78->31, 159->63, 240->95, ...)
^ ^
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Chains:
33 -> 39 -> 69 -> 41
114 -> 135 -> 120 -> 213 -> 75 -> 133
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MATHEMATICA
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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]
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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