A214405 Numbers of the form (4k+3)^2-8 or (4k+5)^2+4.

1, 29, 41, 85, 113, 173, 217, 293, 353, 445, 521, 629, 721, 845, 953, 1093, 1217, 1373, 1513, 1685, 1841, 2029, 2201, 2405, 2593, 2813, 3017, 3253, 3473, 3725, 3961, 4229, 4481, 4765, 5033, 5333, 5617, 5933, 6233, 6565, 6881, 7229, 7561, 7925, 8273, 8653

Offset: 1

Author

Yasir Karamelghani Gasmallah, Jul 15 2012

Keywords

• nonn
• easy

Comments

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

The first differences of this sequence is the interleaved sequence 28,12,44,28,60,44....

Examples

			a(4) = 2*a(3) - 2*a(1) + a(0) = 2*85 - 2*29 + 1 = 113.
		

Crossrefs

Cf. A178218, A214345.

Programs

Magma

I:=[1, 29, 41, 85]; [n le 4 select I[n] else 2*Self(n-1)-2*Self(n-3)+Self(n-4): n in [1..75]];

Maxima

A214405(n):=4*n*(n+3)-6*(-1)^n+7$
makelist(A214405(n),n,0,30); /* Martin Ettl, Nov 01 2012 */

Formula

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

O.G.f.: (1+27*x-17*x^2+5*x^3)/((1+x)*(1-x)^3).

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

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