A283393 - OEIS

%I #34 Dec 12 2023 07:46:56

%S 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,

%T 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,

%U 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

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

%C Periodic with period 10.

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

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

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

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

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

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

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

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

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

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

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

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

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

%H <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,0,0,0,0,0,1).

%F 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).

%t Table[PolynomialGCD[n^2 - 1, n^2 + 9], {n, 0, 100}]

%t LinearRecurrence[{0, 0, 0, 0, 0, 0, 0, 0, 0, 1}, {1, 10, 1, 2, 5, 2, 5, 2, 1, 10}, 100]

%o (Python) [1, 10, 1, 2, 5, 2, 5, 2, 1, 10]*10

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

%o (Magma) &cat [[1, 10, 1, 2, 5, 2, 5, 2, 1, 10]^^10];

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

%o (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

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

%K nonn,easy

%O 0,2

%A _Bruno Berselli_, Mar 07 2017