OFFSET
0,2
COMMENTS
Number of binary strings of length n with no substrings equal to 0001, 1001, or 1011. - R. H. Hardin, Aug 14 2009
Degree sequence d(n) of recursion x(n+1)+x(n)+x(n-1) = b + c(n)/x(n) where c(n) = c(n-1) + c(n-2) - c(n-3) and x(n) = u(n)/f(n) and x(n-1) = v(n)/f(n) in homogeneous coordinates (projectivization). Denoted by sigma_1 on page 32 of Hiertarinta and Viallet (2000). - Michael Somos, Jan 04 2022
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..10000
Jarmo Hietarinta and Claude Viallet, Discrete Painlevé I and singularity confinement in projective space, Chaos, Solitons and Fractals 11 (2000), pp. 29-32.
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).
FORMULA
a(n) = (6*n^2 + 9 - (-1)^n)/8. - Charles R Greathouse IV, Sep 10 2014
G.f.: ( 1+2*x^3 ) / ( (1+x)*(1-x)^3 ). - R. J. Mathar, Sep 11 2014
a(n) = 2*a(n-1)-2*a(n-3)+a(n-4). - Colin Barker, Sep 11 2014
a(n) = a(-n) for all n in Z. - Michael Somos, Feb 08 2015
a(n) - a(n-1) = A001651(n), a(n+1) - a(n-1) = 3*n for all n in Z. - Michael Somos, Feb 08 2015
(a(n) - a(n+1))^2 - (2*a(n) + a(n+1)) + 4 = 3*n/2 + 1 for all even n in Z. - Michael Somos, Feb 08 2015
0 = -4 + a(n)*(-a(n+1) + a(n+2)) + a(n+1)*(+3 + a(n+1) - a(n+2)) for all n in Z. - Michael Somos, Feb 08 2015
A122958(n-1) = p(-1) where p(x) is the unique degree-n polynomial such that p(k) = a(k) for k = 0, 1, ..., n for all n>1. - Michael Somos, Feb 08 2015
a(n) = 2*a(n-1) - 3*A002620(n-2) for all n in Z. - Michael Somos, Dec 27 2021
a(n) = 3*(a(n-1) + a(n-4)) - 2*(a(n-2) + a(n-3)) - a(n-5) for all n in Z. - Michael Somos, Jan 04 2022
EXAMPLE
G.f. = 1 + 2*x + 4*x^2 + 8*x^3 + 13*x^4 + 20*x^5 + 28*x^6 + 38*x^7 + ...
MATHEMATICA
a[ n_] := Quotient[ 3*n^2 + 6, 4]; (* Michael Somos, Feb 08 2015 *)
LinearRecurrence[{2, 0, -2, 1}, {1, 2, 4, 8}, 70] (* Harvey P. Dale, Jul 21 2021 *)
PROG
(PARI) a(n)=(6*n^2 + 9 - (-1)^n)/8 \\ Charles R Greathouse IV, Sep 10 2014
(PARI) {a(n) = (n^2 + 2)*3 \ 4}; /* Michael Somos, Feb 08 2015 */
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Jul 16 2003
EXTENSIONS
More terms from Charles R Greathouse IV, Sep 10 2014
Edited by N. J. A. Sloane, Jan 04 2022
STATUS
approved