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 A001093 a(n) = n^3 + 1. 35
 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 (list; graph; refs; listen; history; text; internal format)
 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 (bhmd95(AT)yahoo.fr), Dec 04 2007 Number of units of a(n) belongs to a periodic sequence: 0, 1, 2, 9, 8, 5, 6, 7, 4, 3. We conclude that a(n) and a(n+10) have the same number of units. - Mohamed Bouhamida (bhmd95(AT)yahoo.fr), Sep 05 2009 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 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. Sequence in context: A202679 A256467 A303373 * A248658 A121643 A183376 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

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Last modified October 22 04:25 EDT 2019. Contains 328315 sequences. (Running on oeis4.)