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%I #60 Mar 01 2024 05:25:12
%S 0,1,2,9,28,65,126,217,344,513,730,1001,1332,1729,2198,2745,3376,4097,
%T 4914,5833,6860,8001,9262,10649,12168,13825,15626,17577,19684,21953,
%U 24390,27001,29792,32769,35938,39305,42876,46657,50654,54873,59320
%N a(n) = n^3 + 1.
%C 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
%C Where records occur in the (real) sequence ceiling(k^(1/3)) - k^(1/3), k = 1, 2, 3, ... . - _John W. Layman_, Sep 07 2010
%H Nathaniel Johnston, <a href="/A001093/b001093.txt">Table of n, a(n) for n = -1..10000</a>
%H K. Goldberg, <a href="http://dx.doi.org/10.1090/S0002-9939-1966-0191832-6">Hadamard matrices of order cube plus one</a>, Proc. Amer. Math. Soc. 17 (1966), 744-746.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F G.f.: (1-2*x+7*x^2)/(1-x)^4. - _Colin Barker_, May 08 2012
%F E.g.f.: (1 + x + 3*x^2 + x^3)*exp(x). - _Ilya Gutkovskiy_, Apr 11 2016
%F a(n) = Sum_{k=0..n} A287326(n,k). - _Kolosov Petro_, Oct 22 2018
%F 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
%F Product_{n>=1} (1 - 1/a(n)) = Pi*sech(sqrt(3)*Pi/2). - _Amiram Eldar_, Jan 20 2021
%p seq(n^3+1,n=-1..40); # _Muniru A Asiru_, Oct 22 2018
%t Table[n^3+1,{n,-1,5!}] (* _Vladimir Joseph Stephan Orlovsky_, May 27 2010 *)
%t LinearRecurrence[{4,-6,4,-1},{0,1,2,9},50] (* _Harvey P. Dale_, May 14 2017 *)
%o (Haskell)
%o a001093 = (+ 1) . (^ 3) -- _Reinhard Zumkeller_, Sep 26 2014
%o (PARI) a(n)=n^3+1 \\ _Charles R Greathouse IV_, Sep 24 2015
%o (GAP) List([-1..40],n->n^3+1); # _Muniru A Asiru_, Oct 22 2018
%Y Cf. A000578, A287326.
%K nonn,easy
%O -1,3
%A _N. J. A. Sloane_, Ray Wills (rwills(AT)vmprofs.estec.esa.nl)
%E More terms from _James A. Sellers_, Sep 19 2000
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