%I #18 Jul 06 2024 23:04:51
%S 1,3,9,5,5,1,5,7,9,9,1,9,3,7,7,1,7,5,3,3,1,3,9,5,5,1,5,7,9,9,1,9,3,7,
%T 7,1,7,5,3,3,1,3,9,5,5,1,5,7,9,9,1,9,3,7,7,1,7,5,3,3,1,3,9,5,5,1,5,7,
%U 9,9,1,9,3,7,7,1,7,5,3,3
%N Last digit of n*2^n + 1.
%C This is a cyclic sequence of 20 numbers, using only 1,3,5,7 and 9 (4 times each).
%D James Cullen, Question 15897, Educational Times, Vol. 58 (December 1905), p. 534.
%D Richard K. Guy (2004), Unsolved Problems in Number Theory (3rd ed.), New York: Springer Verlag, pp. section B20, ISBN 0-387-20860-7.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Cullen_number">Cullen number</a>.
%H <a href="/index/Rec#order_13">Index entries for linear recurrences with constant coefficients</a>, signature (0,1,0,-1,1,1,-1,-1,1,0,-1,0,1).
%F a(n) = A010879(A002064(n)).
%F From _Chai Wah Wu_, Jul 06 2024: (Start)
%F a(n) = a(n-2) - a(n-4) + a(n-5) + a(n-6) - a(n-7) - a(n-8) + a(n-9) - a(n-11) + a(n-13) for n > 12.
%F G.f.: (-3*x^12 - 3*x^11 - 2*x^10 - 4*x^9 + x^8 - 5*x^6 + 2*x^5 + 3*x^4 - 2*x^3 - 8*x^2 - 3*x - 1)/((x - 1)*(x^4 + x^3 + x^2 + x + 1)*(x^8 - x^6 + x^4 - x^2 + 1)). (End)
%p lastDigit := proc(n)
%p return (n * 2^n + 1) mod 10;
%p end proc:
%p # Example usage
%p minN := 1; maxN := 10;
%p lastDigits := [seq(lastDigit(n), n = minN .. maxN)];
%p print(lastDigits);
%t lastDigit[n_] := Mod[n * 2^n + 1, 10]
%t (* Example usage *)
%t minN = 1; maxN = 10;
%t lastDigits = Table[lastDigit[n], {n, minN, maxN}]
%t Print[lastDigits]
%o (Python)
%o def last_digit(n):
%o return (n * 2**n + 1) % 10
%o # Example usage
%o min_n, max_n = 1, 10
%o last_digits = [last_digit(n) for n in range(min_n, max_n + 1)]
%o print(last_digits)
%o (PARI) a(n) = lift(Mod(n*2^n + 1, 10))
%Y Cf. A010879, A002064.
%Y Cf. A373098, A373100.
%K nonn,base,easy
%O 0,2
%A _Javier Rodríguez Ríos_, May 23 2024