A232395 (ceiling(sqrt(n^3 + n^2 + n + 1)))^2 - (n^3 + n^2 + n + 1).

0, 0, 1, 9, 15, 13, 30, 0, 40, 21, 45, 57, 51, 21, 70, 105, 120, 109, 66, 156, 43, 77, 81, 49, 216, 108, 217, 9, 36, 21, 293, 192, 31, 189, 309, 385, 411, 381, 289, 129, 408, 112, 281, 396, 451, 440, 357, 196, 624, 309, 613, 120, 276, 360, 366, 288, 120, 725

Offset: 0

Author

Vladimir Shevelev, Nov 23 2013

Keywords

• nonn

Comments

a(n)=0, iff 1 + n + n^2 + n^3 is a perfect square. For example, a(7)=0 and we have 1 + 7 + 7^2 + 7^3 = 20^2.

a(n) = Difference between smallest square >= (n^3 + n^2 + n + 1) and (n^3 + n^2 + n + 1) - Antti Karttunen, Nov 27 2013

Links

• Ivan Panchenko, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A053698, A068527.

Programs

PARI

a(n) = ceil(sqrt(n^3+n^2+n+1))^2 - (n^3+n^2+n+1); \\ Michel Marcus, Nov 23 2013

Formula

Contribution from Antti Karttunen, Nov 27 2013: (Start)

a(n) = A000290(⌈sqrt(A053698(n))⌉) - A053698(n). Where ⌈x⌉ stands for ceiling(x). This further reduces as:

a(n) = A000290(A135034(A053698(n))) - A053698(n).

a(n) = A048761(A053698(n)) - A053698(n).

a(n) = A068527(A053698(n)).

(End)

Extensions

More terms from Peter J. C. Moses