A340825 - OEIS

%I #20 Jan 26 2021 05:08:35

%S 1,2,4,10,28,76,206,561,1524,4143,11261,30611,83210,226188,614843,

%T 1671317,4543110,12349453,33569293,91250800,248045393,674257283,

%U 1832821321,4982124892,13542819562,36813200321,100068653479,272014802350,739412894299,2009932634301

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

%C 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....

%C If x_2 + x_2^2 = x_1, then (using the positive root) x_2 = 0.43168....

%C 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.

%C 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).

%C 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.

%C 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).

%e 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).

%e                    x_k =

%e        k  (sqrt(4*x_(k-1)+1)-1)/2             1/x_k

%e       --  -----------------------     ---------------------

%e        0  1.000000000000000000000      1.000000000000000000

%e   ==>  1  0.618033988749894848205 ==>  1.618033988749894848

%e        2  0.431683416590579253080      2.316512429173132330

%e        3  0.325641215414164782161      3.070864352131090453

%e   ==>  4  0.258710231520680616003 ==>  3.865328379639529750

%e        5  0.213239252649965007521      4.689568114560563292

%e        6  0.180616817783666735278      5.536582984192242122

%e        7  0.156214003038388944800      6.401474775307141564

%e        8  0.137349200233583838142      7.280712216011038260

%e        9  0.122373842825663615744      8.171680948392062100

%e       10  0.110224420050249722669      9.072399741764251779

%e   ==> 11  0.100186987571581497269 ==>  9.981336141936806057

%e       12  0.091765990549965195674     10.897283340013805067

%e       13  0.084607552594016010352     11.819275813335917533

%e        .             .                          .

%e        .             .                          .

%e        .             .                          .

%e       28  0.038387172830009749995     26.050368554837540742

%e       29  0.037016920431758602512     27.014672974849948323

%e   ==> 30  0.035739601328629245203 ==> 27.980166616994380100

%e       31  0.034546163889171732220     28.946774038591399743

%e       32  0.033428686788751296736     29.914426681472229087

%e        .             .                          .

%e        .             .                          .

%e        .             .                          .

%e       77  0.013510205832731656267     74.018117294502236762

%e       78  0.013332451567920705001     75.004960258479861760

%e   ==> 79  0.013159284791691852519 ==> 75.991971891310727604

%e       80  0.012990530898662741115     76.979147950215110113

%e       81  0.012826024006838737492     77.966484349850560106

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

%K nonn

%O -1,2

%A _Jon E. Schoenfield_, Jan 22 2021