

A232397


a(n) = ceiling(sqrt(n^4 + n^3 + n^2 + n + 1))^2  (n^4 + n^3 + n^2 + n + 1).


2



0, 4, 5, 0, 20, 3, 45, 8, 80, 15, 125, 24, 180, 35, 245, 48, 320, 63, 405, 80, 500, 99, 605, 120, 720, 143, 845, 168, 980, 195, 1125, 224, 1280, 255, 1445, 288, 1620, 323, 1805, 360, 2000, 399, 2205, 440, 2420, 483, 2645, 528, 2880, 575, 3125, 624, 3380, 675
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

0,2


COMMENTS

a(n) = 0 if and only if n^4 + n^3 + n^2 + n + 1 is a perfect square.
Using formula below, we immediately prove that a(n)=0 iff n=0 or n=3. This means that all nonnegative solutions of the Diophantine equation n^4 + n^3 + n^2 + n + 1 = m^2 are n=0, m=1 and n=3, m=11.
For m >=0, if we also consider negative values of n, we obtain only one more solution: n=1, m=1.
Indeed, if one considers sequence b(n) = ceiling(sqrt(n^4  n^3 + n^2  n + 1))^2  (n^4  n^3 + n^2  n +1 ), then, for even n, a(n) = b(n), while for odd n>=3, a(n) = b(n2).


LINKS

Robert Israel, Table of n, a(n) for n = 0..10001
Max Alekseyev, Re: oddness of iterations of sigma(), SeqFan Mailing List.


FORMULA

a(1) = 4, for other odd n, a(n) = ((n1)/2)^2  1; for even n>=0, a(n) = 5/4 * n^2.
a(n) = A068527(A053699(n)). [Straight from the description: Difference between smallest square >= (n^4 + n^3 + n^2 + n + 1) and (n^4 + n^3 + n^2 + n + 1)].  Antti Karttunen, Nov 28 2013
a(n) = (6*n^22*n3+(4*n^2+2*n+3)*(1)^n+20*(1(1)^(2^abs(n1))))/8.  Luce ETIENNE, Jan 30 2016
G.f.: 4*x+x^2*(x^53*x^35*x^25)/(x^21)^3.  Robert Israel, Feb 02 2016


MAPLE

0, 4, seq(op([5*k^2, k^21]), k=1..100); # Robert Israel, Feb 02 2016


MATHEMATICA

Table[Ceiling[Sqrt[n^4 + n^3 + n^2 + n + 1]]^2  (n^4 + n^3 + n^2 + n + 1), {n, 0, 60}] (* Vincenzo Librandi, Jan 31 2016 *)


PROG

(MAGMA) [Ceiling(Sqrt(n^4+n^3+n^2+n+1))^2(n^4+n^3+n^2+n+1): n in [0..60]]; // Vincenzo Librandi, Jan 31 2016


CROSSREFS

Cf. A053699, A068527, A232395, A232423.
Sequence in context: A200013 A178219 A322232 * A122753 A016714 A211799
Adjacent sequences: A232394 A232395 A232396 * A232398 A232399 A232400


KEYWORD

nonn,easy


AUTHOR

Vladimir Shevelev, Nov 23 2013


EXTENSIONS

More terms from Peter J. C. Moses


STATUS

approved



