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

0,2

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.

Links

Table of n, a(n) for n=0..39.

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

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

Example

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

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]

Crossrefs

Cf. A307048, A309792.

Sequence in context: A248312 A329653 A170874 * A103935 A040077 A195312

Adjacent sequences: A309788 A309789 A309790 * A309792 A309793 A309794

Keyword

nonn,easy

Author

Georg Fischer, Aug 17 2019

Status

approved