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A214393 Numbers of the form (4k+3)^2+4 or (4k+5)^2-8. 9
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

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,....

LINKS

Table of n, a(n) for n=0..45.

Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1)

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.

EXAMPLE

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

PROG

(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]];

(Maxima) A214393(n):=4*n*(n+3)+6*(-1)^n+7$

makelist(A214393(n), n, 0, 30); /* Martin Ettl, Nov 01 2012 */

CROSSREFS

Cf. A178218, A214345.

Sequence in context: A180527 A076789 A089577 * A060569 A108265 A069853

Adjacent sequences:  A214390 A214391 A214392 * A214394 A214395 A214396

KEYWORD

nonn,easy

AUTHOR

Yasir Karamelghani Gasmallah, Jul 15 2012

STATUS

approved

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Last modified August 20 03:48 EDT 2019. Contains 326139 sequences. (Running on oeis4.)