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A100227
Main diagonal of triangle A100226.
4
1, 1, 5, 13, 33, 81, 197, 477, 1153, 2785, 6725, 16237, 39201, 94641, 228485, 551613, 1331713, 3215041, 7761797, 18738637, 45239073, 109216785, 263672645, 636562077, 1536796801, 3710155681, 8957108165, 21624372013, 52205852193, 126036076401, 304278004997
OFFSET
0,3
COMMENTS
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
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
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
FORMULA
a(n) = A002203(n) - 1.
a(n) = 2*a(n-1) + a(n-2) + 2 for n > 1, with a(0)=1, a(1)=1.
G.f.: Sum_{n>=1} a(n)*x^n/n = log((1-x)/(1-2*x-x^2)).
G.f.: (1-2*x+3*x^2)/((1-x)*(1-2*x-x^2)). - Paul D. Hanna, Feb 22 2005
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
a(n) = -1 + (1-sqrt(2))^n + (1+sqrt(2))^n. - Colin Barker, Mar 16 2016
E.g.f.: (2*cosh(sqrt(2)*x) - 1)*exp(x). - Ilya Gutkovskiy, Aug 22 2016
a(n) = A000129(n+1) + A000129(n-1)-1 for n > 0. - Rigoberto Florez, Jul 12 2020
a(n) = 3*a(n-1) - a(n-2) - a(n-3). - Wesley Ivan Hurt, Jul 13 2020
MATHEMATICA
LucasL[Range[0, 35], 2] - 1 (* G. C. Greubel, Feb 26 2020 *)
PROG
(PARI) a(n)=if(n==0, 1, n*polcoeff(log((1-x)/(1-2*x-x^2)+x*O(x^n)), n))
(PARI) a(n)=polcoeff((1-2*x+3*x^2)/(1-3*x+x^2+x^3)+x*O(x^n), n)
(Maxima)
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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Paul D. Hanna, Nov 29 2004
STATUS
approved