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

0,2

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

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

Cf. also A001651, A002620, A122958.

Sequence in context: A030503 A245094 A164486 * A011907 A056133 A172131

Adjacent sequences: A084681 A084682 A084683 * A084685 A084686 A084687

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