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

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

Offset: 0

Author

Georg Fischer, Aug 17 2019

Keywords

• nonn
• easy

Comments

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.

Examples

			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, ...)
          ^            ^
          |            |
          A309791      A309792
Chains:
   33 ->  39 ->  69 ->  41
  114 -> 135 -> 120 -> 213 -> 75 -> 133
		

Links

• Index entries for linear recurrences with constant coefficients, signature (1,0,0,9,-9).

Crossrefs

Cf. A307048, A309792.

Programs

Mathematica

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]

Formula

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