A094766 Trajectory of 11 under repeated application of the map n -> n + 2*square excess of n (see A094765).

11, 15, 27, 31, 43, 57, 73, 91, 111, 133, 157, 183, 211, 241, 273, 307, 343, 381, 421, 463, 507, 553, 601, 651, 703, 757, 813, 871, 931, 993, 1057, 1123, 1191, 1261, 1333, 1407, 1483, 1561, 1641, 1723, 1807, 1893, 1981, 2071, 2163, 2257, 2353, 2451, 2551, 2653

Offset

0,1

Comments

The trajectory of 3 gives A002061 and 5 gives essentially the same trajectory as 3.

Links

Robert Israel, Table of n, a(n) for n = 0..10000

S. H. Weintraub, An interesting recursion, Amer. Math. Monthly, 111 (No. 6, 2004), 528-530.

Index entries for linear recurrences with constant coefficients, signature (3, -3, 1).

Formula

Numbers given satisfy a(n) = n^2 + 5n + 7, for n>2. - Ralf Stephan, Dec 04 2004

From Robert Israel, Oct 23 2015: (Start)

This is because for x = m^2 + 5*m + 7, (m+2)^2 < x < (m+3)^2 so A094765(x) = x + 2*(x-(m+2)^2) = m^2 + 7*m + 13 = (m+1)^2 + 5*(m+1) + 7.

Similarly, for any positive integer k, the trajectory of k^2 + k + 1 is n^2 + (2k+1) n + k^2 + k + 1 for n >= 0.

G.f.: 4 + 2*x + 6*x^2 + (7-8*x+3*x^2)/(1-x)^3. (End)

Maple

f:= n -> 3*n - 2*floor(sqrt(n))^2:
g:= proc(n) option remember; f(procname(n-1)) end proc:
g(0):= 11:
seq(g(n), n=0..100); # Robert Israel, Oct 23 2015

Mathematica

NestList[3*#-2*Floor[Sqrt[#]]^2&, 11, 50] (* Harvey P. Dale, Feb 26 2022 *)

Prog

(PARI) lista(nn) = {print1(n=11, ", "); for (k=2, nn, m = 3*n - 2*sqrtint(n)^2; print1(m, ", "); n = m; ); } \\ Michel Marcus, Oct 23 2015
(PARI) Vec(4+2*x+6*x^2+(7-8*x+3*x^2)/(1-x)^3 + O(x^100)) \\ Altug Alkan, Oct 23 2015

Crossrefs

Cf. A002061, A053186, A094765.

Sequence in context: A256498 A054280 A214863 * A009407 A009433 A275242

Adjacent sequences: A094763 A094764 A094765 * A094767 A094768 A094769

Keyword

nonn

Author

N. J. A. Sloane, Jun 10 2004

Status

approved