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 A133654 #26 Mar 23 2022 07:39:51 %S A133654 1,3,9,23,57,139,337,815,1969,4755,11481,27719,66921,161563,390049, %T A133654 941663,2273377,5488419,13250217,31988855,77227929,186444715, %U A133654 450117361,1086679439,2623476241,6333631923,15290740089,36915112103,89120964297,215157040699 %N A133654 a(n) = 2*A000129(n) - 1. %C A133654 a(n)/a(n-1) tends to (1 + sqrt(2)). %C A133654 Define a triangle by T(n,1) = n*(n-1)+1 and T(n,n) = 1, n >= 1. Let interior terms be T(r,c) = T(r-1,c) + T(r-1,c-1) + T(r-2,c-1). The triangle is 1; 3,1; 7,5,1; 13,15,7,1; etc. The row sums are 1, 4, 13, 36, 93, ... and the differences (sum of terms in row(n) minus those in row(n-1)) are a(n). - _J. M. Bergot_, Mar 10 2013 %H A133654 Colin Barker, <a href="/A133654/b133654.txt">Table of n, a(n) for n = 1..1000</a> %H A133654 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-1,-1). %F A133654 a(n) = 2*A000129(n) - 1, where A000129 = the Pell sequence. a(1) = 1, a(2) = 3, then for n>2, a(n) = 2*a(n-1) + a(n-2) + 2. %F A133654 G.f.: x*(1+x^2)/( (x-1)*(x^2+2*x-1) ). - _R. J. Mathar_, Nov 14 2007 %F A133654 a(n) = -1+(-(1-sqrt(2))^n+(1+sqrt(2))^n)/sqrt(2). - _Colin Barker_, Mar 16 2016 %e A133654 a(3) = 2*A000129(3) - 1 = 2*5 - 1. %e A133654 a(5) = 57 = 2*a(4) + a(3) + 2 = 2*23 + 9 + 2. %o A133654 (PARI) Vec(x*(1+x^2)/((x-1)*(x^2+2*x-1)) + O(x^50)) \\ _Colin Barker_, Mar 16 2016 %Y A133654 Cf. A000129. %K A133654 nonn,easy %O A133654 1,2 %A A133654 _Gary W. Adamson_, Sep 19 2007 %E A133654 More terms from _Philippe Deléham_, Oct 16 2007, corrected by _R. J. Mathar_, Mar 12 2013