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)).

1, 2, 3, 5, 7, 11, 13, 17, 38, 138, 870, 9765, 213675, 4309305, 201226025, 9679967297, 810726926009, 40855897091009, 4259653632223561, 380804291082185737, 44319264099050115071, 4644246052673250585913

Offset: 0

Author

Antti Karttunen, Dec 13 2015

Keywords

• nonn

Comments

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.

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.

Examples

			n    a(n)   prime factorization    Spironacci polynomial
------------------------------------------------------------
0       1   (empty)                S_0(x) = 0
1       2   p_1                    S_1(x) = 1
2       3   p_2                    S_2(x) = x
3       5   p_3                    S_3(x) = x^2
4       7   p_4                    S_4(x) = x^3
5      11   p_5                    S_5(x) = x^4
6      13   p_6                    S_6(x) = x^5
7      17   p_7                    S_7(x) = x^6
8      38   p_8 * p_1              S_8(x) = x^7 + 1
9     138   p_9 * p_2 * p_1        S_9(x) = x^8 + x + 1
		

Crossrefs

Cf. A003961, A265407, A265409.

Cf. also A078510, A206296 and A265752, A265753.

Formula

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

Other identities. For all n >= 0:

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

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