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A265408 Prime factorization representation of Spironacci polynomials: a(0) = 1, a(1) = 2, and for n > 1, a(n) = A003961(a(n-1)) * a(A265409(n)). 5

%I

%S 1,2,3,5,7,11,13,17,38,138,870,9765,213675,4309305,201226025,

%T 9679967297,810726926009,40855897091009,4259653632223561,

%U 380804291082185737,44319264099050115071,4644246052673250585913

%N Prime factorization representation of Spironacci polynomials: a(0) = 1, a(1) = 2, and for n > 1, a(n) = A003961(a(n-1)) * a(A265409(n)).

%C The polynomials encoded by these numbers could also be called "Fernandez spiral polynomials" after _Neil Fernandez_ who discovered sequence A078510 which is obtained when they are evaluated at X=1.

%C The polynomial recurrence uses the same composition rules as the Fibonacci polynomials (A206296), but with the neighborhood rules of A078510, where the other polynomial is taken from the nearest inner neighbor (A265409) when the polynomials are arranged as a spiral into a square grid. See A265409 for the illustration.

%F a(0) = 1, a(1) = 2, and for n >= 2, a(n) = A003961(a(n-1)) * a(A265409(n)).

%F Other identities. For all n >= 0:

%F A078510(n) = A001222(a(n)). [When each polynomial is evaluated at x=1.]

%F A265407(n) = A248663(a(n)). [at x=2 over the field GF(2).]

%e n a(n) prime factorization Spironacci polynomial

%e ------------------------------------------------------------

%e 0 1 (empty) S_0(x) = 0

%e 1 2 p_1 S_1(x) = 1

%e 2 3 p_2 S_2(x) = x

%e 3 5 p_3 S_3(x) = x^2

%e 4 7 p_4 S_4(x) = x^3

%e 5 11 p_5 S_5(x) = x^4

%e 6 13 p_6 S_6(x) = x^5

%e 7 17 p_7 S_7(x) = x^6

%e 8 38 p_8 * p_1 S_8(x) = x^7 + 1

%e 9 138 p_9 * p_2 * p_1 S_9(x) = x^8 + x + 1

%o (Scheme, with memoization-macro definec)

%o (definec (A265408 n) (cond ((<= n 1) (+ 1 n)) (else (* (A003961 (A265408 (- n 1))) (A265408 (A265409 n))))))

%Y Cf. A003961, A265407, A265409.

%Y Cf. also A078510, A206296 and A265752, A265753.

%K nonn

%O 0,2

%A _Antti Karttunen_, Dec 13 2015

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Last modified July 22 10:03 EDT 2019. Contains 325219 sequences. (Running on oeis4.)