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A084380 a(n) = n^3 + 2. 8
2, 3, 10, 29, 66, 127, 218, 345, 514, 731, 1002, 1333, 1730, 2199, 2746, 3377, 4098, 4915, 5834, 6861, 8002, 9263, 10650, 12169, 13826, 15627, 17578, 19685, 21954, 24391, 27002, 29793, 32770, 35939, 39306, 42877, 46658, 50655, 54874, 59321, 64002 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

This sequence contains no square numbers. A proof may be similar to the Hilliard link.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Dorin Andrica and Ovidiu Bagdasar, On k-partitions of multisets with equal sums, The Ramanujan J. (2021) Vol. 55, 421-435.

D. R. Heath-Brown, The largest prime factor of X^3 + 2, Proc. London Math. Soc. (3), 82:3 (2001), pp. 554-596.

Cino Hilliard, Proof that n^3+7 <> k^2 for all integers n,k. [broken link]

A. J. Irving, The largest prime factor of X^3+2, arXiv:1412.0024 [math.NT], 2014.

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

FORMULA

G.f.: (2 - 5*x + 10*x^2 - x^3) / (x-1)^4 . - R. J. Mathar, Feb 16 2011

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Vincenzo Librandi, Jun 24 2012

MATHEMATICA

f[n_]:=n^3+2; f[Range[60]] (* Offset 1. *) (* Vladimir Joseph Stephan Orlovsky, Feb 14 2011*)

CoefficientList[Series[(2-5*x+10*x^2-x^3)/(x-1)^4, {x, 0, 50}], x] (* Vincenzo Librandi, Jun 24 2012 *)

PROG

(PARI) n3pm(n, m=2) = { for(x=0, n, y=x^3+m; print1(y, ", ")) }

(MAGMA) I:=[2, 3, 10, 29]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3) -Self(n-4): n in [1..50]]; // Vincenzo Librandi, Jun 24 2012

CROSSREFS

Cf. sequences for n^3+7, n^3+17, n^3+3. Cf. A034324.

Sequence in context: A160909 A004980 A034324 * A286814 A338583 A121909

Adjacent sequences:  A084377 A084378 A084379 * A084381 A084382 A084383

KEYWORD

easy,nonn

AUTHOR

Cino Hilliard, Jun 23 2003

EXTENSIONS

Extended to offset 0 by R. J. Mathar, Feb 16 2011

STATUS

approved

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Last modified September 16 08:18 EDT 2021. Contains 347469 sequences. (Running on oeis4.)