A077859 Expansion of g.f. (1 - x)^(-1)/(1 - 2*x + 2*x^2 + x^3).
1, 3, 5, 6, 6, 6, 7, 9, 11, 12, 12, 12, 13, 15, 17, 18, 18, 18, 19, 21, 23, 24, 24, 24, 25, 27, 29, 30, 30, 30, 31, 33, 35, 36, 36, 36, 37, 39, 41, 42, 42, 42, 43, 45, 47, 48, 48, 48, 49, 51, 53, 54, 54, 54, 55, 57, 59, 60, 60, 60, 61, 63, 65, 66, 66, 66, 67, 69, 71, 72, 72, 72, 73, 75
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, signature (3,-4,3,-1).
Programs
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Magma
I:=[1, 3, 5, 6]; [n le 4 select I[n] else 3*Self(n-1)-4*Self(n-2)+3*Self(n-3)-Self(n-4): n in [1..100]]
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Maple
A010892 := proc(n) op(1+(n mod 6),[1,1,0,-1,-1,0]) ; end proc: A077859 := proc(n) n+2+A010892(n+4) ; end proc: seq(A077859(n),n=0..50) ; # R. J. Mathar, Mar 22 2011
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Mathematica
CoefficientList[Series[(1 - x)^(-1)/(1 - 2 x + 2 x^2 - x^3), {x, 0, 100}], x] (* Vincenzo Librandi, Feb 04 2014 *) LinearRecurrence[{3,-4,3,-1},{1,3,5,6},80] (* Harvey P. Dale, Apr 21 2023 *) -
PARI
Vec(1/(1-x)/(1-2*x+2*x^2-x^3)+O(x^99)) \\ Charles R Greathouse IV, Sep 24 2012
Formula
G.f.: 1/((1-x)^2*(1-x+x^2)).
a(n) = Sum_{k=0..n} (k+1)*2*sin(Pi(n-k)/3 + Pi/3)/sqrt(3). - Paul Barry, May 18 2004
a(n) = Sum_{k=0..n} binomial(n-2k, n-k-1). - Paul Barry, Jan 15 2005
a(n) = n + 2 + (-1 + n - 3*floor(n/3))*(-1)^floor(n/3). - Tani Akinari, Jun 27 2013
a(n) = n + 1 + a(n-1) - a(n-2), with a(-1) = a(-2) = 0. - Richard R. Forberg, Jul 11 2013
a(n) = 3*a(n-1) - 4*a(n-2) + 3*a(n-3) - 1*a(n-4). - Joerg Arndt, Jul 12 2013
a(n) = Sum_{k=0..n} (-1)^k*(n+1-k)*b(k), where b(n) = A049347(n). - Mircea Merca, Feb 04 2014
E.g.f.: exp(x)*(2 + x) + exp(x/2)*(sqrt(3)*sin(sqrt(3)*x/2) - 3*cos(sqrt(3)*x/2))/3. - Stefano Spezia, Feb 11 2023
Sum_{n>=0} (-1)^n/a(n) = log(2)/3 + log(3)/2 = log(108)/6. - Amiram Eldar, Feb 14 2023
Comments