A084684 Degrees of certain maps (see Comments and Formulas for more precise definitions).

1, 2, 4, 8, 13, 20, 28, 38, 49, 62, 76, 92, 109, 128, 148, 170, 193, 218, 244, 272, 301, 332, 364, 398, 433, 470, 508, 548, 589, 632, 676, 722, 769, 818, 868, 920, 973, 1028, 1084, 1142, 1201, 1262, 1324, 1388, 1453, 1520, 1588, 1658, 1729, 1802, 1876, 1952, 2029, 2108, 2188, 2270, 2353, 2438, 2524, 2612, 2701, 2792, 2884, 2978, 3073, 3170, 3268, 3368, 3469, 3572

Offset: 0

Author

N. J. A. Sloane, Jul 16 2003

Keywords

• nonn
• easy

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

Examples

			G.f. = 1 + 2*x + 4*x^2 + 8*x^3 + 13*x^4 + 20*x^5 + 28*x^6 + 38*x^7 + ...
		

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).

Crossrefs

Cf. A064863, A056107 (bisection), A077588 (bisection).

Cf. also A001651, A002620, A122958.

Programs

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 *)

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 */

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

Extensions

More terms from Charles R Greathouse IV, Sep 10 2014

Edited by N. J. A. Sloane, Jan 04 2022