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 A086224 #64 Apr 28 2025 21:55:46 %S A086224 6,13,27,55,111,223,447,895,1791,3583,7167,14335,28671,57343,114687, %T A086224 229375,458751,917503,1835007,3670015,7340031,14680063,29360127, %U A086224 58720255,117440511,234881023,469762047,939524095,1879048191,3758096383,7516192767,15032385535,30064771071 %N A086224 a(n) = 7*2^n - 1. %C A086224 a(n) = A164874(n+2,2); subsequence of A030130. - _Reinhard Zumkeller_, Aug 29 2009 %C A086224 Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=-3, A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=1, a(n-1)=(-1)^n*charpoly(A,-5). - _Milan Janjic_, Jan 27 2010 %H A086224 Michael De Vlieger, <a href="/A086224/b086224.txt">Table of n, a(n) for n = 0..3319</a> %H A086224 Gennady Eremin, <a href="https://arxiv.org/abs/2405.16143">Partitioning the set of natural numbers into Mersenne trees and into arithmetic progressions; Natural Matrix and Linnik's constant</a>, arXiv:2405.16143 [math.CO], 2024. See pp. 3-5, 14. %H A086224 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2). %F A086224 a(n+1) = 2*a(n) + 1. %F A086224 G.f.: (6-5*x)/((1-x)*(1-2*x)). - _Jaume Oliver Lafont_, Sep 14 2009 %F A086224 a(n-1)^2 + a(n) = (7*2^(n-1))^2. - _Vincenzo Librandi_, Aug 08 2010 %F A086224 a(n) = (A052940(n+1) + A000225(n+3))/2. - _Gennady Eremin_, Aug 31 2023 %F A086224 From _Elmo R. Oliveira_, Apr 22 2025: (Start) %F A086224 E.g.f.: exp(x)*(7*exp(x) - 1). %F A086224 a(n) = 3*a(n-1) - 2*a(n-2). (End) %t A086224 7*2^Range[0,30]-1 (* _Harvey P. Dale_, May 09 2018 *) %o A086224 (PARI) a(n)=7<<n-1 \\ _Charles R Greathouse IV_, Sep 24 2015 %Y A086224 Other sequences with recurrence a(n+1) = 2*a(n) + 1 are: %Y A086224 a(0) = 2 gives A153893, a(0)=3 essentially A126646. %Y A086224 a(0) = 4 gives A153894, a(0)=5 essentially A153893. %Y A086224 a(0) = 7 gives essentially A000225. %Y A086224 a(0) = 8 gives A052996 except for some initial terms, %Y A086224 a(0) = 9 is essentially A153894. %Y A086224 a(0) = 10 gives A086225, %Y A086224 a(0) = 11 is essentially A153893. %Y A086224 a(0) = 13 is essentially A086224. %Y A086224 Cf. A030130, A052940, A164874. %K A086224 nonn,easy %O A086224 0,1 %A A086224 _Marco Matosic_, Jul 27 2003 %E A086224 More terms from _David Wasserman_, Feb 22 2005