A249356 8*A200975(n)-7 where A200975 are the numbers on the diagonals in Ulam's spiral.

1, 17, 33, 49, 65, 97, 129, 161, 193, 241, 289, 337, 385, 449, 513, 577, 641, 721, 801, 881, 961, 1057, 1153, 1249, 1345, 1457, 1569, 1681, 1793, 1921, 2049, 2177, 2305, 2449, 2593, 2737, 2881, 3041, 3201, 3361, 3521, 3697, 3873, 4049, 4225, 4417, 4609, 4801

Offset: 1

Author

Todd Silvestri, Oct 27 2014

Keywords

• nonn
• easy

Comments

All elements are odd.

The pair (a(n), a(n+1)) is separated by A002265(n-1) elements in A158057.

Links

• Todd Silvestri, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (2,-1,0,1,-2,1).

Programs

Maple

seq(2*n*(n+2)+(-1)^n-4*sin((Pi*n)/2), n=1..100); # Robert Israel, Nov 04 2014

Mathematica

a[n_Integer/;n>0]:=2 n (n+2)+(-1)^n-4 Mod[n^2 (3 n+2),4,-1]
CoefficientList[Series[-(x^5 - x^4 + 15 x + 1) / ((x - 1)^3 (x^3 + x^2 + x + 1)), {x, 0, 40}], x] (* Vincenzo Librandi, Nov 16 2014 *)
Table[2 n (n + 2) - (1 - (-1)^n) (1 - 2 I^(n + 1)) + 1, {n, 1, 50}] (* Bruno Berselli, Nov 18 2014 *)
LinearRecurrence[{2,-1,0,1,-2,1},{1,17,33,49,65,97},50] (* Harvey P. Dale, Sep 29 2019 *)

PARI

a(n) = 2*n*(n+2)+(-1)^n-4*round(sin((Pi*n)/2)) \\ Charles R Greathouse IV, Nov 17 2014

Formula

a(n) = 2*n*(n+2)+(-1)^n-4*sin((Pi*n)/2).

G.f.: - x*(x^5-x^4+15*x+1)/((x-1)^3*(x^3+x^2+x+1)).

a(n) = 2*a(n-1) - a(n-2) + 16 if n == 2 mod 4, a(n) = 2*a(n-1) - a(n-2) otherwise. - Robert Israel, Nov 04 2014

a(n) = 2*n*(n+2) - (1-(-1)^n)*(1-2*i^(n+1)) + 1, where i=sqrt(-1). - Bruno Berselli, Nov 18 2014