A001093 a(n) = n^3 + 1.

0, 1, 2, 9, 28, 65, 126, 217, 344, 513, 730, 1001, 1332, 1729, 2198, 2745, 3376, 4097, 4914, 5833, 6860, 8001, 9262, 10649, 12168, 13825, 15626, 17577, 19684, 21953, 24390, 27001, 29792, 32769, 35938, 39305, 42876, 46657, 50654, 54873, 59320

Offset

-1,3

Comments

Nonnegative X values of solutions to the equation 1!*X^4 + 2!*(X - 1)^3 + 3!*(X - 2)^2 + (4^2)*(X - 3) + 5^2 = Y^3. To prove that X = n^3 + 1: Y^3 = 1!*X^4 + 2!*(X - 1)^3 + 3!*(X - 2)^2 + (4^2)*(X -3) + 5^2 = X^4 + 2*(X - 1)^3 + 6*(X - 2)^2 + 16(X -3) + 25 = X^4 + 2*X^3 - 2X - 1 = (X - 1)(X^3 + 3*X^2 + 3X + 1) = (X - 1)*(X + 1)^3 it means: (X - 1) must be a cube, so X = n^3 + 1 and Y = n(n^3 + 2). - Mohamed Bouhamida, Dec 04 2007

Where records occur in the (real) sequence ceiling(k^(1/3)) - k^(1/3), k = 1, 2, 3, ... . - John W. Layman, Sep 07 2010

Links

Nathaniel Johnston, Table of n, a(n) for n = -1..10000

K. Goldberg, Hadamard matrices of order cube plus one, Proc. Amer. Math. Soc. 17 (1966), 744-746.

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

Formula

G.f.: (1-2*x+7*x^2)/(1-x)^4. - Colin Barker, May 08 2012

E.g.f.: (1 + x + 3*x^2 + x^3)*exp(x). - Ilya Gutkovskiy, Apr 11 2016

a(n) = Sum_{k=0..n} A287326(n,k). - Kolosov Petro, Oct 22 2018

Sum_{n>=1} 1/a(n) = 1/2 + Sum_{n>=1} (-1)^(n+1) * (zeta(3*n) - 1) = 0.6865033423... - Amiram Eldar, Nov 06 2020

Product_{n>=1} (1 - 1/a(n)) = Pi*sech(sqrt(3)*Pi/2). - Amiram Eldar, Jan 20 2021

Maple

seq(n^3+1, n=-1..40); # Muniru A Asiru, Oct 22 2018

Mathematica

Table[n^3+1, {n, -1, 5!}] (* Vladimir Joseph Stephan Orlovsky, May 27 2010 *)
LinearRecurrence[{4, -6, 4, -1}, {0, 1, 2, 9}, 50] (* Harvey P. Dale, May 14 2017 *)

Prog

(Haskell)
a001093 = (+ 1) . (^ 3)  -- Reinhard Zumkeller, Sep 26 2014
(PARI) a(n)=n^3+1 \\ Charles R Greathouse IV, Sep 24 2015
(GAP) List([-1..40], n->n^3+1); # Muniru A Asiru, Oct 22 2018

Crossrefs

Cf. A000578, A287326.

Sequence in context: A340049 A256467 A303373 * A248658 A121643 A353017

Adjacent sequences: A001090 A001091 A001092 * A001094 A001095 A001096

Keyword

nonn,easy

Author

N. J. A. Sloane, Ray Wills (rwills(AT)vmprofs.estec.esa.nl)

Extensions

More terms from James A. Sellers, Sep 19 2000

Status

approved