A340825 a(n) is the smallest k such that A340745(k) - k = n.

1, 2, 4, 10, 28, 76, 206, 561, 1524, 4143, 11261, 30611, 83210, 226188, 614843, 1671317, 4543110, 12349453, 33569293, 91250800, 248045393, 674257283, 1832821321, 4982124892, 13542819562, 36813200321, 100068653479, 272014802350, 739412894299, 2009932634301

Offset

-1,2

Comments

Let x_0 = 1. If x_1 + x_1^2 = x_0, then (using the positive root) x_1 = 1/phi = (sqrt(5) - 1)/2 = 0.61803....

If x_2 + x_2^2 = x_1, then (using the positive root) 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.

The real-valued sequence {x_0, x_1, x_2, ...} begins {1.0, 0.61803..., 0.43168..., ...}; reciprocals are {1.0, 1.61803..., 2.31651..., ...} (see Example section). Truncating each of those reciprocals to integer gives a sequence in which the terms that appear twice are {1, 3, 9, 27, 75, ...}; incrementing each of these by 1 gives this sequence (after a(-1)=1).

For large values of k, we have x_k = x_(k-1) - x_(k-1)^2 + 2*x_(k-1)^3 - 5*x_(k-1)^4 + 14*x_(k-1)^5 - 42*x_(k-1)^6 + ...; the coefficients are Catalan numbers, with alternating signs.

Also, for large k, writing just "x" in place of "x(k)", k = 1/x - log(x) + c0 + (1/2)*x - (1/3)*x^2 + (13/36)*x^3 - (113/240)*x^4 + (1187/1800)*x^5 - (877/945)*x^6 + (14569/11760)*x^7 - (176017/120960)*x^8 + (1745717/1360800)*x^9 - (88217/259875)*x^10 - (147635381/109771200)*x^11 + (3238110769/1556755200)*x^12 - ... where c0 = -1.32912232216454200165271262369745253672... (A340875).

Links

Table of n, a(n) for n=-1..28.

Example

The integers that appear as the integer part of 1/x_k for two values of k are 1, 3, 9, 27, 75, 205, 560, ...; adding 1 to each of these gives the terms of this sequence. The sequence of values of k such that floor(1/x_k) = floor(1/x_(k-1)) is 1, 4, 11, 30, 79, ... (A340824).
                   x_k =
       k  (sqrt(4*x_(k-1)+1)-1)/2             1/x_k
      --  -----------------------     ---------------------
       0  1.000000000000000000000      1.000000000000000000
  ==>  1  0.618033988749894848205 ==>  1.618033988749894848
       2  0.431683416590579253080      2.316512429173132330
       3  0.325641215414164782161      3.070864352131090453
  ==>  4  0.258710231520680616003 ==>  3.865328379639529750
       5  0.213239252649965007521      4.689568114560563292
       6  0.180616817783666735278      5.536582984192242122
       7  0.156214003038388944800      6.401474775307141564
       8  0.137349200233583838142      7.280712216011038260
       9  0.122373842825663615744      8.171680948392062100
      10  0.110224420050249722669      9.072399741764251779
  ==> 11  0.100186987571581497269 ==>  9.981336141936806057
      12  0.091765990549965195674     10.897283340013805067
      13  0.084607552594016010352     11.819275813335917533
       .             .                          .
       .             .                          .
       .             .                          .
      28  0.038387172830009749995     26.050368554837540742
      29  0.037016920431758602512     27.014672974849948323
  ==> 30  0.035739601328629245203 ==> 27.980166616994380100
      31  0.034546163889171732220     28.946774038591399743
      32  0.033428686788751296736     29.914426681472229087
       .             .                          .
       .             .                          .
       .             .                          .
      77  0.013510205832731656267     74.018117294502236762
      78  0.013332451567920705001     75.004960258479861760
  ==> 79  0.013159284791691852519 ==> 75.991971891310727604
      80  0.012990530898662741115     76.979147950215110113
      81  0.012826024006838737492     77.966484349850560106

Crossrefs

Cf. A340745, A340824, A340844, A340845, A340875.

Sequence in context: A244485 A128933 A173781 * A106362 A271896 A148109

Adjacent sequences: A340822 A340823 A340824 * A340826 A340827 A340828

Keyword

nonn

Author

Jon E. Schoenfield, Jan 22 2021

Status

approved