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A158575 a(n) = 32*n^2 + 1. 4
1, 33, 129, 289, 513, 801, 1153, 1569, 2049, 2593, 3201, 3873, 4609, 5409, 6273, 7201, 8193, 9249, 10369, 11553, 12801, 14113, 15489, 16929, 18433, 20001, 21633, 23329, 25089, 26913, 28801, 30753, 32769, 34849, 36993, 39201, 41473, 43809, 46209 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

The identity (32*n^2+1)^2-(256*n^2+16)*(2*n)^2 = 1 can be written as a(n)^2-A158574(n)*A005843(n)^2 = 1. - Comment rewritten by R. J. Mathar, Oct 16 2009

Sequence found by reading the line segment from 1 to 33 together with the line from 33, in the direction 33, 129, ..., in the square spiral whose vertices are the generalized 18-gonal numbers A274979. - Omar E. Pol, Apr 21 2021

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..10000

Vincenzo Librandi, X^2-AY^2=1

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

FORMULA

G.f.: (1+30*x+33*x^2)/(1-x)^3.

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

For n>0 a(n) = sqrt(8*(A000217(4*n-1)^2 + A000217(4*n)^2) + 1). - J. M. Bergot, Sep 03 2015

EXAMPLE

a(n) = A244082(n) + 1. - Omar E. Pol, Apr 21 2021

MATHEMATICA

LinearRecurrence[{3, -3, 1}, {1, 33, 129}, 50] (* Vincenzo Librandi, Feb 15 2012 *)

32*Range[0, 40]^2+1 (* Harvey P. Dale, Jul 20 2021 *)

PROG

(MAGMA) I:=[1, 33, 129]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..50]]; // Vincenzo Librandi, Feb 15 2012

(PARI) for(n=0, 50, print1(32*n^2+1", ")); \\ Vincenzo Librandi, Feb 15 2012

CROSSREFS

Cf. A000217, A005843, A010021, A158563, A158574, A244082.

Cf. A274979 (generalized 18-gonal numbers).

Sequence in context: A235895 A010020 A007419 * A209981 A036546 A043502

Adjacent sequences:  A158572 A158573 A158574 * A158576 A158577 A158578

KEYWORD

nonn,easy,changed

AUTHOR

Vincenzo Librandi, Mar 21 2009

EXTENSIONS

a(0) added by R. J. Mathar, Oct 16 2009

STATUS

approved

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Last modified August 3 01:55 EDT 2021. Contains 346429 sequences. (Running on oeis4.)