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 A272870 #23 Jul 07 2024 15:54:41
%S A272870 1,-4,-7,28,161,476,1081,2108,3713,6076,9401,13916,19873,27548,37241,
%T A272870 49276,64001,81788,103033,128156,157601,191836,231353,276668,328321,
%U A272870 386876,452921,527068,609953,702236,804601,917756,1042433,1179388,1329401,1493276
%N A272870 Real part of (n + i)^4.
%C A272870 a(1) and a(2) are the only two negative terms in the sequence. Since (n + i)^2 = (n^2 - 1) + 2ni, it follows that (n + i)^4 = (n^2 - 1 + 2ni)^2 = (n^4 - 6n^2 + 1) + (4n^3 - 4n)i, so the real part of (n + i)^4 is n^4 - 6n^2 + 1. Then n^4 + 1 > 6n^2 for all n > 2, ensuring a(n) is positive for all n > 2. - _Alonso del Arte_, Jun 04 2016
%H A272870 Colin Barker, <a href="/A272870/b272870.txt">Table of n, a(n) for n = 0..1000</a>
%H A272870 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F A272870 a(n) = n^4 - 6*n^2 + 1.
%F A272870 a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n > 4.
%F A272870 G.f.: (1-9*x+23*x^2+13*x^3-4*x^4) / (1-x)^5.
%F A272870 E.g.f.: (1 - 5*x + x^2 + 6*x^3 + x^4)*exp(x). - _Ilya Gutkovskiy_, May 08 2016
%e A272870 a(5) = 476 because (5 + i)^4 = 476 + 480*i.
%t A272870 Table[Re[(n + I)^4], {n, 0, 35}] (* or *)
%t A272870 Table[n^4 - 6 n^2 + 1, {n, 0, 35}] (* or *)
%t A272870 CoefficientList[Series[(1 - 9 x + 23 x^2 + 13 x^3 - 4 x^4)/(1 - x)^5, {x, 0, 35}], x] (* _Michael De Vlieger_, May 08 2016 *)
%o A272870 (PARI) a(n) = n^4-6*n^2+1
%o A272870 (PARI) vector(50, n, n--; real((n+I)^4))
%o A272870 (PARI) Vec((1-9*x+23*x^2+13*x^3-4*x^4)/(1-x)^5 + O(x^50))
%Y A272870 Cf. A272871 (imaginary part).
%Y A272870 Cf. A121670 ((n+i)^3), A121672 ((n+i)^5).
%K A272870 sign,easy
%O A272870 0,2
%A A272870 _Colin Barker_, May 08 2016