%I #8 Sep 30 2024 12:59:12

%S 18,24,26,32,68,74,76,82,118,124,126,132,168,174,176,182,218,224,226,

%T 232,268,274,276,282,318,324,326,332,368,374,376,382,418,424,426,432,

%U 468,474,476,482,518,524,526,532,568,574,576,582,618,624,626,632,668,674

%N Natural numbers whose iterated squaring modulo 100 eventually settles at the attractor 76.

%C The natural numbers decompose into six categories under the operation of repeated squaring modulo 100, four of which consist of numbers that eventually settle at the attractors 0 (cf. A008592), 1 (cf. A376506), 25 (cf. A017329), or 76 (this sequence), and two of which eventually enter one of the 4-cycles 16, 56, 36, 96 (cf. A376508) or 21, 41, 81, 61 (cf. A376509).

%C The first-order differences of the numbers in this sequence repeat with a fixed period of length four: 6, 2, 6, 36, ...

%D Alexander K. Dewdney, Computer-Kurzweil. Mit einem Computer-Mikroskop untersuchen wir ein Objekt von faszinierender Struktur in der Ebene der komplexen Zahlen. In: Spektrum der Wissenschaft, Oct 1985, p. 8-14, here p. 11-13 (Iterations on a finite set), 14 (Iteration diagram).

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,1,-1).

%F G.f.: 2*x*(9 + 3*x + x^2 + 3*x^3 + 9*x^4)/((1 - x)^2*(1 + x + x^2 + x^3)). - _Stefano Spezia_, Sep 26 2024

%e 18^2 = 24 -> 24^2 = 76 -> 76^2 = 76 -> ... (mod 100).

%Y Cf. A008592, A017329, A376506, A376508, A376509.

%K nonn,easy

%O 1,1

%A _Martin Renner_, Sep 25 2024