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 A232965 Number of circular n-bit strings that, when circularly shifted by 3 bits, do not have coincident 1's in any position. 1
 1, 3, 1, 7, 11, 27, 29, 47, 64, 123, 199, 343, 521, 843, 1331, 2207, 3571, 5832, 9349, 15127, 24389, 39603, 64079, 103823, 167761, 271443, 438976, 710647, 1149851, 1860867, 3010349, 4870847, 7880599, 12752043, 20633239, 33386248, 54018521 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS a(n) = L[n/gcd(n,3)]^gcd(n,3) where L[n] is the Lucas sequence (A000032). K[n;s] = L[n/gcd(n,s)]^gcd(n,s) counts circular n-bit strings that, when circularly shifted by s bits, do not have coincident 1's in any position. K[n,s] = #{x|((x<< 8, derived from the denominator polynomial (1+phi^(-1)*x)*(1-phi*x)*(1-phi^(-1)*x^3)*(1+phi*x^3) of the generating function, where phi = (1+sqrt(5)/2), the golden ratio. Empirical g.f.: -x*(x-1)*(8*x^6+15*x^5+9*x^4+4*x^3+3*x+1) / ((x^2+x-1)*(x^6-x^3-1)). - Colin Barker, Oct 10 2015 EXAMPLE K[1;3] = L[1] = 1; K[2;3] = L[2] = 3; K[3;3] = L[1] = 1; K[4;3] = L[4] = 7; K[5;3] = L[5] = 11; K[6;3] = L[2]^3 = 27; K[7;3] = L[7] = 29; K[8;3] = L[8] = 47. PROG (C) int gcd(int n, int s)//Return the gcd of n and s int raiseToPower(int n, int d)//Return n^d #define N 40 #define S 3 int Lucas[N+1]  = {2, 1, 3, 4, 7, 1, 18, ....} main() { int n; for(n = 1; n < N; n++) printf("%i: %i\n", n, raiseToPower(Lucas[n/gcd(n, S)], gcd(n, S)); return; } (PARI) L(n) = fibonacci(n-1) + fibonacci(n+1); a(n) = L(n/gcd(n, 3))^gcd(n, 3) \\ Rick L. Shepherd, Jan 23 2014 CROSSREFS Cf. A000032 (Lucas sequence), A001638 (Fielder sequence). Sequence in context: A279939 A286511 A307901 * A249401 A196845 A263446 Adjacent sequences:  A232962 A232963 A232964 * A232966 A232967 A232968 KEYWORD nonn,easy AUTHOR Gideon J. Kuhn, Dec 02 2013 EXTENSIONS More terms from Rick L. Shepherd, Jan 23 2014 STATUS approved

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Last modified August 18 04:50 EDT 2019. Contains 326072 sequences. (Running on oeis4.)