A214393 Numbers of the form (4k+3)^2+4 or (4k+5)^2-8.

13, 17, 53, 73, 125, 161, 229, 281, 365, 433, 533, 617, 733, 833, 965, 1081, 1229, 1361, 1525, 1673, 1853, 2017, 2213, 2393, 2605, 2801, 3029, 3241, 3485, 3713, 3973, 4217, 4493, 4753, 5045, 5321, 5629, 5921, 6245, 6553, 6893, 7217, 7573, 7913, 8285, 8641

Offset: 0

Author

Yasir Karamelghani Gasmallah, Jul 15 2012

Keywords

• nonn
• easy

Comments

For every n=2k the triple (a(2k-1)^2, a(2k)^2 , a(2k+1)^2) is an arithmetic progression, i.e., 2*a(2k)^2 = a(2k-1)^2 + a(2k+1)^2.

In general a triple((x-y)^2,z^2,(x+y)^2) is an arithmetic progression if and only if x^2+y^2=z^2, e.g., (17^2, 53^2, 73^2).

The first differences of this sequence is the interleaved sequence 4,36,20,52,36,68,52,....

Examples

			a(5) = 2*a(4) - 2*a(2) + a(1) = 2*125 - 2*53 + 17 = 161.
		

Links

• Paolo Xausa, Table of n, a(n) for n = 0..10000
• Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Crossrefs

Cf. A178218, A214345.

Programs

Magma

I:=[13, 17, 53, 73]; [n le 4 select I[n] else 2*Self(n-1)-2*Self(n-3)+Self(n-4): n in [1..75]];

Mathematica

A214393[n_] := 4*n*(n+3) + 6*(-1)^n + 7; Array[A214393, 50, 0] (* or *)
LinearRecurrence[{2, 0, -2, 1}, {13, 17, 53, 73}, 50] (* Paolo Xausa, Feb 22 2024 *)

Maxima

A214393(n):=4*n*(n+3)+6*(-1)^n+7$
makelist(A214393(n),n,0,30); /* Martin Ettl, Nov 01 2012 */

Formula

a(n) = 2*a(n-1)-2*a(n-3)+a(n-4).

G.f.: (13-9*x+19*x^2-7*x^3)/((1+x)*(1-x)^3).

a(n) = 4*n*(n+3)+6*(-1)^n+7.

2*a(2n)^2 = a(2n-1)^2 + a(2n+1)^2.