A100227 Main diagonal of triangle A100226.

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

Author

Paul D. Hanna, Nov 29 2004

Keywords

• nonn
• easy

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

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (3,-1,-1).

Crossrefs

Cf. A000129, A001333, A002203, A100225, A100226.

Programs

Mathematica

LucasL[Range[0,35], 2] - 1 (* G. C. Greubel, Feb 26 2020 *)

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 */

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)

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