A106229 Least j > 1 for n > 0 such that j^2 = (n^2 + 1)*(k^2) + (n^2 + 1)*k + 1 where k sequence = A106230.

5, 19, 11, 35, 79, 149, 251, 391, 575, 809, 1099, 1451, 1871, 2365, 2939, 3599, 4351, 5201, 6155, 7219, 8399, 9701, 11131, 12695, 14399, 16249, 18251, 20411, 22735, 25229, 27899, 30751, 33791, 37025, 40459, 44099, 47951, 52021, 56315, 60839, 65599, 70601, 75851

Offset: 1

Author

Pierre CAMI, Apr 26 2005

Keywords

• nonn
• easy

Comments

For j^2 = (n^2 + 1)*(k^2) + (n^2 + 1)*k + 1, there is a sequence j(i,n) with a recurrence.

For n=1, j(1,1) = 1, j(2,1) = 5, j(i,1) = 6*j(i-1,1) - j(i-2,1).

For n=2, j(1,2) = 1, j(2,2) = 19, j(i,2) = 18*j(i-1,2) - j(i-2,2).

For n>2, j(1,n) = 1, j(2,n) = n^3 - 2*n^2 + n - 1, j(3,n) = n^3 + 2*n^2 + n + 1, j(4,n) = (4*n^2 + 2)*j(2,n) + 1 then j(i,n) = (4*n^2+2)*j(i-2,n) - j(i-4,n).

Links

• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. A003777, A005563, A005899, A106230.

Programs

Mathematica

LinearRecurrence[{4,-6,4,-1},{5,19,11,35,79,149},43] (* Georg Fischer, Oct 25 2020 *)

PARI

a(n) = if(n<3, 14*n-9, n^3-2*n^2+n-1); \\ Jinyuan Wang, Apr 07 2020

Formula

For n > 2, a(n) = n^3 - 2*n^2 + n - 1.

Extensions

More terms from Jinyuan Wang, Apr 07 2020