OFFSET
1,1
COMMENTS
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 (cf. A376507), and two of which eventually enter one of the 4-cycles 16, 56, 36, 96 (cf. A376508) or 21, 41, 81, 61 (this sequence).
The first-order differences of the numbers in this sequence repeat with a fixed period of length sixteen: 6, 2, 2, 4, 2, 2, 2, 4, 2, 2, 2, 4, 2, 2, 6, 6, ...
REFERENCES
Alexander K. Dewdney, Computer-Kurzweil. Using a computer microscope, we examine an object of fascinating structure in the plane of complex numbers. In: Scientific American, October 1985, pp. 8-14, here pp. 11-13 (Iterations on a finite set), 14 (Iteration diagram).
LINKS
Index entries for linear recurrences with constant coefficients, signature (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1).
FORMULA
Gf: x*(3 + 6*x + 2*x^2 + 2*x^3 + 4*x^4 + 2*x^5 + 2*x^6 + 2*x^7 + 4*x^8 + 2*x^9 + 2*x^10 + 2*x^11 + 4*x^12 + 2*x^13 + 2*x^14 + 6*x^15 + 3*x^16)/(1 - x - x^16 + x^17). - Charles R Greathouse IV, May 16 2026
EXAMPLE
3^2 = 9 -> 9^2 = 81 -> 81^2 = 61 -> 61^2 = 21 -> 21^2 = 41 -> 41^2 = 81 -> ... (mod 100)
MATHEMATICA
LinearRecurrence[{1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1}, {3, 9, 11, 13, 17, 19, 21, 23, 27, 29, 31, 33, 37, 39, 41, 47, 53}, 60] (* Harvey P. Dale, Feb 25 2026 *)
PROG
(PARI) a(n)=(n-1)\16*50+[47, 3, 9, 11, 13, 17, 19, 21, 23, 27, 29, 31, 33, 37, 39, 41][n%16+1] \\ Charles R Greathouse IV, May 16 2026
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Martin Renner, Sep 25 2024
STATUS
approved