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 A370259 #17 Mar 14 2024 07:13:48
%S A370259 1,2,9,75,961,16900,380689,10498005,343323841,13007560326,
%T A370259 560789801881,27125634729375,1455389462287489,85805768251305992,
%U A370259 5515372218107327521,383931652351786775721,28778117694539885440129,2311202255914842794592010,198009919900727928789497641,18027589454633803742596931571
%N A370259 a(n) = (T(n,n+1) - 1)/n^3 for n >= 1, where T(n,x) is the n-th Chebyshev polynomial of the first kind.
%C A370259 It appears that a(2*n+1) is always a square, while a(2*n) = (n + 1) * a square. See A370260 and A370261.
%H A370259 Paolo Xausa, <a href="/A370259/b370259.txt">Table of n, a(n) for n = 1..350</a>
%F A370259 a(n) = Sum_{k = 1..n} (2^k)*n^(k-2)*binomial(n+k, 2*k)/(n + k) (shows that a(n) is an integer).
%F A370259 a(n) = (cos(n*arccos(n+1)) - 1)/n^3.
%F A370259 a(n) = (A342205(n) - 1)/n^3.
%F A370259 a(n) = ( (n + 1 + sqrt(n*(n+2)))^n + (n + 1 - sqrt(n*(n+2)))^n - 2 )/(2*n^3).
%p A370259 seq( simplify( (ChebyshevT(n, n+1) - 1)/n^3 ), n = 1..20);
%t A370259 Array[(ChebyshevT[#, #+1]-1)/#^3 &, 20] (* _Paolo Xausa_, Mar 14 2024 *)
%o A370259 (Python)
%o A370259 from sympy import chebyshevt
%o A370259 def A370259(n): return (chebyshevt(n,n+1)-1)//n**3 # _Chai Wah Wu_, Mar 13 2024
%Y A370259 Cf. A008310, A342205, A370260, A370261.
%K A370259 nonn,easy
%O A370259 1,2
%A A370259 _Peter Bala_, Mar 11 2024