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 A165568 #27 Jul 25 2024 04:40:44 %S A165568 -1,1,31,137,391,889,1751,3121,5167,8081,12079,17401,24311,33097, %T A165568 44071,57569,73951,93601,116927,144361,176359,213401,255991,304657, %U A165568 359951,422449,492751,571481,659287,756841,864839,984001,1115071,1258817,1416031,1587529,1774151 %N A165568 a(n) = -1 - 2*n + n^2 + 2*n^3 + n^4. %C A165568 Consider the Lyman spectrum of Hydrogen A005563(n)/A000290(n+1) = n*(n+2)/(n+1)^2 = 0/1, 3/4, 8/9, 15/16, ... %C A165568 The first differences of these fractions are 3/4, 5/36, 7/144, 9/400, 11/900, 13/1764, 15/3136, ... = (2n+1)/(n*(n+1))^2. %C A165568 Adding numerator and denominator of these first differences yields 1 + 2n + n^2 + 2n^3 + n^4 = A165563(n) = 3+4, 5+36, 7+144, ... = 1 + 2n + n^2*(n+1)^2 = A144396(n) + A035287(n+1) = A005408(n) + A035287(n+1). %C A165568 Subtracting numerator from denominator, on the other hand, yields this sequence here: a(n) = A035287(n+1) - A005408(n). %H A165568 Vincenzo Librandi, <a href="/A165568/b165568.txt">Table of n, a(n) for n = 0..1000</a> %H A165568 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1). %F A165568 a(-1-n) = A165563(n). A165563(-1-n) = a(n). %F A165568 a(n) = A165563(n) - 2 - 4*n = A165563(n) - A016825(n). %F A165568 a(n) + A165563 + a(n) = 2*n^2*(1+n)^2 = 2*A035287(n+1). %F A165568 a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). %F A165568 a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) + 24. %F A165568 G.f.: (-1 + 6*x + 16*x^2 + 2*x^3 + x^4)/(1-x)^5. %o A165568 (Magma) [-1 -2*n +n^2 +2*n^3 +n^4: n in [0..40]]; // _Vincenzo Librandi_, May 21 2011 %K A165568 sign,easy %O A165568 0,3 %A A165568 _Paul Curtz_, Sep 22 2009 %E A165568 Edited and extended by _R. J. Mathar_, Feb 02 2010