A283393 a(n) = gcd(n^2-1, n^2+9).

1, 10, 1, 2, 5, 2, 5, 2, 1, 10, 1, 10, 1, 2, 5, 2, 5, 2, 1, 10, 1, 10, 1, 2, 5, 2, 5, 2, 1, 10, 1, 10, 1, 2, 5, 2, 5, 2, 1, 10, 1, 10, 1, 2, 5, 2, 5, 2, 1, 10, 1, 10, 1, 2, 5, 2, 5, 2, 1, 10, 1, 10, 1, 2, 5, 2, 5, 2, 1, 10, 1, 10, 1, 2, 5, 2, 5, 2, 1, 10, 1, 10, 1, 2, 5, 2, 5, 2, 1, 10

Offset: 0

Author

Bruno Berselli, Mar 07 2017

Keywords

• nonn
• easy

Comments

Periodic with period 10.

Similar sequences with formula gcd(n^2-1, n^2+k):

k= 1: 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, ... (A000034)

k= 3: 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, ... (A010685)

k= 5: 1, 6, 3, 2, 3, 6, 1, 6, 3, 2, 3, 6, 1, ... (A129203, start 6)

k= 7: 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, ... (A010689)

k= 9: 1, 10, 1, 2, 5, 2, 5, 2, 1, 10, 1, 10, 1, ... (this sequence)

k=11: 1, 12, 3, 4, 3, 12, 1, 12, 3, 4, 3, 12, 1, ... (A129197, start 12)

Connection between the values of a(n) and the last digit of n:

. if n ends with 0, 2 or 8, then a(n) = 1;

. if n ends with 1 or 9, then a(n) = 10;

. if n ends with 3, 5 or 7, then a(n) = 2;

. if n ends with 4 or 6, then a(n) = 5.

Also, continued fraction expansion of (57 + sqrt(4579))/114.

Links

• Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,1).

Crossrefs

Cf. A000034, A010685, A010689, A129197, A129203.

Programs

Magma

&cat [[1, 10, 1, 2, 5, 2, 5, 2, 1, 10]^^10];

Mathematica

Table[PolynomialGCD[n^2 - 1, n^2 + 9], {n, 0, 100}]
LinearRecurrence[{0, 0, 0, 0, 0, 0, 0, 0, 0, 1}, {1, 10, 1, 2, 5, 2, 5, 2, 1, 10}, 100]

Maxima

makelist(gcd(n^2-1, n^2+9), n, 0, 100);

PARI

Vec((1 + 10*x + x^2 + 2*x^3 + 5*x^4 + 2*x^5 + 5*x^6 + 2*x^7 + x^8 + 10*x^9)/(1 - x^10) + O(x^100)) \\ Colin Barker, Mar 08 2017

Python

[1, 10, 1, 2, 5, 2, 5, 2, 1, 10]*10

Sage

[gcd(n^2-1, n^2+9) for n in range(100)]

Formula

G.f.: (1 + 10*x + x^2 + 2*x^3 + 5*x^4 + 2*x^5 + 5*x^6 + 2*x^7 + x^8 + 10*x^9)/(1 - x^10).