A111002 a(n) = gcd(f(n), f(n+1)) where f(n) = n^4 + n^2 + 1.

1, 3, 7, 91, 21, 31, 43, 57, 73, 91, 777, 133, 157, 183, 211, 241, 273, 2149, 343, 381, 421, 463, 507, 553, 4207, 651, 703, 757, 813, 871, 931, 6951, 1057, 1123, 1191, 1261, 1333, 1407, 10381, 1561, 1641, 1723, 1807, 1893, 1981, 14497, 2163, 2257, 2353

Offset: 0

Author

Pahikkala Jussi, Sep 30 2005

Keywords

• easy
• nonn

Examples

			a(10) = 7(10^2 + 10 + 1) = 777 because 10 is congruent to 3 modulo 7.
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000
• PlanetMath, Example of GCD

Programs

Magma

[Gcd(n^4+n^2+1, n^4+4*n^3+7*n^2+6*n+3): n in [0..50]]; // Vincenzo Librandi, Oct 07 2015

Mathematica

f[n_] := n^4 + n^2 + 1; Table[ GCD[f[n], f[n + 1]], {n, 0, 49}] (* Robert G. Wilson v, Oct 02 2005 *)
GCD[#[[1]],#[[2]]]&/@Partition[Table[n^4+n^2+1,{n,0,50}],2,1] (* Harvey P. Dale, Mar 07 2015 *)

PARI

m=50;a=3;for(k=2,m,b=k^4+k^2+1;print1(gcd(a,b),",");a=b) \\ Klaus Brockhaus, Oct 02 2005

Formula

a(n) = gcd(f(n), f(n+1)) for all n. a(n) = n^2 + n + 1, except when n congruent to 3 modulo 7 when a(n) = 7(n^2 + n + 1).

Conjectures from Colin Barker, Oct 06 2015: (Start)

a(n) = 3*a(n-7) - 3*a(n-14) + a(n-21) for n>20.

G.f.: -(x^20 +3*x^19 +7*x^18 +91*x^17 +21*x^16 +31*x^15 +43*x^14 +54*x^13 +64*x^12 +70*x^11 +504*x^10 +70*x^9 +64*x^8 +54*x^7 +43*x^6 +31*x^5 +21*x^4 +91*x^3 +7*x^2 +3*x +1) / ((x -1)^3*(x^6 +x^5 +x^4 +x^3 +x^2 +x +1)^3).

(End)

Extensions

More terms from Robert G. Wilson v and Klaus Brockhaus, Oct 02 2005