A340844 Numerators of coefficients in the expansion given in A340825 (see Comments).

1, -1, 13, -113, 1187, -877, 14569, -176017, 1745717, -88217, -147635381, 3238110769, 63045343657, -24855467017, 20362710600601, 13053665468881, -20331497188291717, 352228802894258, 25895914464827930459, -1879649533452408510071, -39932104233601587228899

Offset

1,3

Comments

Let x_0 = 1. The positive number x_1 such that x_1 + x_1^2 = x_0 is x_1 = 1/phi = (sqrt(5) - 1)/2 = 0.61803.... The positive number x_2 such that x_2 + x_2^2 = x_1 is x_2 = 0.43168.... In general, for k > 0, if x_k + x_k^2 = x_(k-1), then (using the positive root) x_k = (sqrt(4*x_(k-1) + 1) - 1)/2.

For large k, writing just "x" in place of "x(k)", k = 1/x - log(x) + c_0 + (1/2)*x - (1/3)*x^2 + (13/36)*x^3 - (113/240)*x^4 + ... + c_n*x^n + ... where c_0 = -1.32912232216454200165271262369745253672... (A340875) and the numerator of c_n is a(n).

Links

Table of n, a(n) for n=1..21.

Example

Numerators and denominators of coefficients c_1 through c_22 are as follows:
.
   j          numerator               denominator
  --  ------------------------   --------------------
   1                         1 / 2
   2                        -1 / 3
   3                        13 / 36
   4                      -113 / 240
   5                      1187 / 1800
   6                      -877 / 945
   7                     14569 / 11760
   8                   -176017 / 120960
   9                   1745717 / 1360800
  10                    -88217 / 259875
  11                -147635381 / 109771200
  12                3238110769 / 1556755200
  13               63045343657 / 23610787200
  14              -24855467017 / 1489863375
  15            20362710600601 / 817296480000
  16            13053665468881 / 380007936000
  17        -20331497188291717 / 94479473088000
  18           352228802894258 / 1856156927625
  19      25895914464827930459 / 20062993991040000
  20   -1879649533452408510071 / 443497761907200000
  21  -39932104233601587228899 / 10244798300056320000
  22    3188454294515019195931 / 60266321845104375

Crossrefs

Cf. A340745, A340824, A340825, A340845 (denominators), A340875.

Sequence in context: A095680 A126534 A361915 * A127827 A089569 A048383

Adjacent sequences: A340841 A340842 A340843 * A340845 A340846 A340847

Keyword

sign,frac

Author

Jon E. Schoenfield, Jan 24 2021

Status

approved