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

%I A084684 #45 Sep 09 2023 10:09:55
%S A084684 1,2,4,8,13,20,28,38,49,62,76,92,109,128,148,170,193,218,244,272,301,
%T A084684 332,364,398,433,470,508,548,589,632,676,722,769,818,868,920,973,1028,
%U A084684 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
%N A084684 Degrees of certain maps (see Comments and Formulas for more precise definitions).
%C A084684 Number of binary strings of length n with no substrings equal to 0001, 1001, or 1011. - _R. H. Hardin_, Aug 14 2009
%C A084684 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
%H A084684 Alois P. Heinz, <a href="/A084684/b084684.txt">Table of n, a(n) for n = 0..10000</a>
%H A084684 Jarmo Hietarinta and Claude Viallet, <a href="http://dx.doi.org/10.1016/S0960-0779(98)00266-5">Discrete Painlevé I and singularity confinement in projective space</a>, Chaos, Solitons and Fractals 11 (2000), pp. 29-32.
%H A084684 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-2,1).
%F A084684 a(n) = (6*n^2 + 9 - (-1)^n)/8. - _Charles R Greathouse IV_, Sep 10 2014
%F A084684 G.f.: ( 1+2*x^3 ) / ( (1+x)*(1-x)^3 ). - _R. J. Mathar_, Sep 11 2014
%F A084684 a(n) = 2*a(n-1)-2*a(n-3)+a(n-4). - _Colin Barker_, Sep 11 2014
%F A084684 a(n) = a(-n) for all n in Z. - _Michael Somos_, Feb 08 2015
%F A084684 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
%F A084684 (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
%F A084684 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
%F A084684 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
%F A084684 a(n) = 2*a(n-1) - 3*A002620(n-2) for all n in Z. - _Michael Somos_, Dec 27 2021
%F A084684 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
%e A084684 G.f. = 1 + 2*x + 4*x^2 + 8*x^3 + 13*x^4 + 20*x^5 + 28*x^6 + 38*x^7 + ...
%t A084684 a[ n_] := Quotient[ 3*n^2 + 6, 4]; (* _Michael Somos_, Feb 08 2015 *)
%t A084684 LinearRecurrence[{2,0,-2,1},{1,2,4,8},70] (* _Harvey P. Dale_, Jul 21 2021 *)
%o A084684 (PARI) a(n)=(6*n^2 + 9 - (-1)^n)/8 \\ _Charles R Greathouse IV_, Sep 10 2014
%o A084684 (PARI) {a(n) = (n^2 + 2)*3 \ 4}; /* _Michael Somos_, Feb 08 2015 */
%Y A084684 Cf. A064863, A056107 (bisection), A077588 (bisection).
%Y A084684 Cf. also A001651, A002620, A122958.
%K A084684 nonn,easy
%O A084684 0,2
%A A084684 _N. J. A. Sloane_, Jul 16 2003
%E A084684 More terms from _Charles R Greathouse IV_, Sep 10 2014
%E A084684 Edited by _N. J. A. Sloane_, Jan 04 2022