This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A232423 #32 Jul 30 2022 08:19:55 %S A232423 0,0,2,0,15,3,38,8,71,15,114,24,167,35,230,48,303,63,386,80,479,99, %T A232423 582,120,695,143,818,168,951,195,1094,224,1247,255,1410,288,1583,323, %U A232423 1766,360,1959,399,2162,440,2375,483,2598,528,2831,575,3074,624,3327,675 %N A232423 a(n) = ceiling(sqrt(n^4 - n^3 - n^2 + n + 1))^2 - (n^4 - n^3 - n^2 + n + 1). %C A232423 a(n)=0, iff n^4 - n^3 - n^2 + n + 1 is a perfect square. %C A232423 One can prove that a(n)=0 iff n=0, 1 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 or 1, m=1 and n=3, m=7. %C A232423 For m>=0, if we also consider negative values of n, we obtain only one more solution: n=-1, m=1. %C A232423 Indeed, if we consider sequence %C A232423 b(n) = ceiling(sqrt(n^4 + n^3 - n^2 - n + 1))^2 - (n^4 + n^3 - n^2 - n +1 ), %C A232423 then, for even n, b(n) = a(n) + 2*n, while for odd n, b(n) = a(n+2). %F A232423 For odd n>=3, a(n) = A232397(n); for even n>=2, a(n) = 5/4 * n^2 - n - 1. %o A232423 (Python) %o A232423 from math import isqrt %o A232423 def A232423(n): return (1+isqrt(m:=n*(n*(n*(n-1)-1)+1)))**2-m-1 # _Chai Wah Wu_, Jul 29 2022 %Y A232423 Cf. A232395, A232397. %K A232423 nonn %O A232423 0,3 %A A232423 _Vladimir Shevelev_, Nov 23 2013 %E A232423 More terms from _Peter J. C. Moses_