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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)).

%I A265408 #19 Jun 01 2024 09:58:18
%S A265408 1,2,3,5,7,11,13,17,38,138,870,9765,213675,4309305,201226025,
%T A265408 9679967297,810726926009,40855897091009,4259653632223561,
%U A265408 380804291082185737,44319264099050115071,4644246052673250585913
%N 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)).
%C A265408 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 A265408 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 A265408 a(0) = 1, a(1) = 2, and for n >= 2, a(n) = A003961(a(n-1)) * a(A265409(n)).
%F A265408 Other identities. For all n >= 0:
%F A265408 A078510(n) = A001222(a(n)). [when each polynomial is evaluated at x=1]
%F A265408 A265407(n) = A248663(a(n)). [at x=2 over the field GF(2)]
%e A265408 n    a(n)   prime factorization    Spironacci polynomial
%e A265408 ------------------------------------------------------------
%e A265408 0       1   (empty)                S_0(x) = 0
%e A265408 1       2   p_1                    S_1(x) = 1
%e A265408 2       3   p_2                    S_2(x) = x
%e A265408 3       5   p_3                    S_3(x) = x^2
%e A265408 4       7   p_4                    S_4(x) = x^3
%e A265408 5      11   p_5                    S_5(x) = x^4
%e A265408 6      13   p_6                    S_6(x) = x^5
%e A265408 7      17   p_7                    S_7(x) = x^6
%e A265408 8      38   p_8 * p_1              S_8(x) = x^7 + 1
%e A265408 9     138   p_9 * p_2 * p_1        S_9(x) = x^8 + x + 1
%o A265408 (Scheme, with memoization-macro definec)
%o A265408 (definec (A265408 n) (cond ((<= n 1) (+ 1 n)) (else (* (A003961 (A265408 (- n 1))) (A265408 (A265409 n))))))
%Y A265408 Cf. A003961, A265407, A265409.
%Y A265408 Cf. also A078510, A206296 and A265752, A265753.
%K A265408 nonn
%O A265408 0,2
%A A265408 _Antti Karttunen_, Dec 13 2015