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 A100227 #53 Nov 27 2024 17:56:02
%S A100227 1,1,5,13,33,81,197,477,1153,2785,6725,16237,39201,94641,228485,
%T A100227 551613,1331713,3215041,7761797,18738637,45239073,109216785,263672645,
%U A100227 636562077,1536796801,3710155681,8957108165,21624372013,52205852193,126036076401,304278004997
%N A100227 Main diagonal of triangle A100226.
%C A100227 Specify that a triangle has T(n,0) = T(n,n) = (n+1)*(n+2)/2. The interior terms T(r,c) = T(r-1,c) + T(r-1,c-1) + T(r-2,c-1). The difference between the sum of the terms in row(n+1) and those in row(n) is a(n+2). - _J. M. Bergot_, Mar 15 2013
%C A100227 Starting with offset 1 the sequence is A001333: (1, 3, 7, 17, 41, ...), convolved with (1, 2, 0, 2, 0, 2, ...). - _Gary W. Adamson_, Aug 10 2016
%C A100227 Number of ways to tile a bracelet of length n with single-color squares, and two colors of k-ominoes for k > 1. Compare to A001333 as mentioned in the previous comment: A001333 can be thought of as the number of ways to tile a strip of length n with single-color squares and two-color k-ominoes for k > 1. - _Greg Dresden_, Feb 26 2020
%H A100227 Colin Barker, <a href="/A100227/b100227.txt">Table of n, a(n) for n = 0..1000</a>
%H A100227 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-1,-1).
%F A100227 a(n) = A002203(n) - 1.
%F A100227 a(n) = 2*a(n-1) + a(n-2) + 2 for n > 1, with a(0)=1, a(1)=1.
%F A100227 G.f.: Sum_{n>=1} a(n)*x^n/n = log((1-x)/(1-2*x-x^2)).
%F A100227 G.f.: (1-2*x+3*x^2)/((1-x)*(1-2*x-x^2)). - _Paul D. Hanna_, Feb 22 2005
%F A100227 a(n) = n*Sum_{k=1..n} (1/k)*Sum_{i=0..n-k} binomial(k, i)*binomial(n-i-1, k-1), n > 0, a(0)=1. - _Vladimir Kruchinin_, May 13 2011
%F A100227 a(n) = -1 + (1-sqrt(2))^n + (1+sqrt(2))^n. - _Colin Barker_, Mar 16 2016
%F A100227 E.g.f.: (2*cosh(sqrt(2)*x) - 1)*exp(x). - _Ilya Gutkovskiy_, Aug 22 2016
%F A100227 a(n) = A000129(n+1) + A000129(n-1)-1 for n > 0. - _Rigoberto Florez_, Jul 12 2020
%F A100227 a(n) = 3*a(n-1) - a(n-2) - a(n-3). - _Wesley Ivan Hurt_, Jul 13 2020
%t A100227 LucasL[Range[0,35], 2] - 1 (* _G. C. Greubel_, Feb 26 2020 *)
%o A100227 (PARI) a(n)=if(n==0,1,n*polcoeff(log((1-x)/(1-2*x-x^2)+x*O(x^n)),n))
%o A100227 (PARI) a(n)=polcoeff((1-2*x+3*x^2)/(1-3*x+x^2+x^3)+x*O(x^n),n)
%o A100227 (Maxima)
%o A100227 a(n):=if n=0 then 1 else n*sum(sum(binomial(k, i)*binomial(n-i-1, k-1),i,0,n-k)/k,k,1,n); /* _Vladimir Kruchinin_, May 13 2011 */
%Y A100227 Cf. A000129, A001333, A002203, A100225, A100226.
%K A100227 nonn,easy
%O A100227 0,3
%A A100227 _Paul D. Hanna_, Nov 29 2004