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%I #9 Sep 30 2024 12:58:58
%S 1,7,43,49,51,57,93,99,101,107,143,149,151,157,193,199,201,207,243,
%T 249,251,257,293,299,301,307,343,349,351,357,393,399,401,407,443,449,
%U 451,457,493,499,501,507,543,549,551,557,593,599,601,607,643,649,651,657
%N Natural numbers whose iterated squaring modulo 100 eventually settles at the attractor 1.
%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 (this sequence), 25 (cf. A017329), or 76 (cf. A376507), 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, 36, 6, 2, ...
%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.: x*(1 + 6*x + 36*x^2 + 6*x^3 + x^4)/((1 - x)^2*(1 + x + x^2 + x^3)). - _Stefano Spezia_, Sep 26 2024
%e 7^2 = 49 -> 49^2 = 1 -> 1^2 = 1 -> ... (mod 100).
%Y Cf. A008592, A017329, A376507, A376508, A376509.
%K nonn,easy
%O 1,2
%A _Martin Renner_, Sep 25 2024