A236294 - OEIS

%I #22 Sep 08 2022 08:46:06

%S 1,1,2,3,3,5,6,7,9,10,12,14,15,18,20,22,25,27,30,33,35,39,42,45,49,52,

%T 56,60,63,68,72,76,81,85,90,95,99,105,110,115,121,126,132,138,143,150,

%U 156,162,169,175,182,189,195,203,210,217,225,232,240,248,255

%N a(n) = max( a(n-1) + a(n-3), 2*a(n-2) ) - a(n-4), with a(0)=1, a(1)=1, a(2)=2, a(3)=3.

%C Tropical version of Somos-4 sequence A006720.

%C Second difference is period 8 sequence [1, 0, -1, 2, -1, 0, 1, -1, ...]

%C The numerator of the g.f. is the reciprocal polynomial of the numerator of the g.f. of A220838.

%H G. C. Greubel, <a href="/A236294/b236294.txt">Table of n, a(n) for n = 0..2500</a>

%H <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,0,0,0,0,0,1,-2,1).

%F G.f.: (1 - x + x^2 - x^4 + 2*x^5 - x^6) / ( (1 - x)^2 * (1 - x^8) ).

%F a(n) = a(-8 - n) = A220838(n + 5) for all n in Z.

%F 0 = (a(n+5) - 2*a(n+3) + a(n+1)) * (a(n+4) - 2*a(n+2) * a(n)) for all n in Z.

%e G.f. = 1 + x + 2*x^2 + 3*x^3 + 3*x^4 + 5*x^5 + 6*x^6 + 7*x^7 + 9*x^8 + ...

%t CoefficientList[Series[(1-x+x^2-x^4+2*x^5-x^6)/((1-x)^2*(1-x^8)), {x, 0, 50}], x] (* _G. C. Greubel_, Aug 07 2018 *)

%o (PARI) {a(n) = if( n<-4, n = -8-n); if( n<0, -(n==-4), polcoeff( (1 - x + x^2 - x^4 + 2*x^5 - x^6) / ( (1 - x)^2 * (1 - x^8) ) + x * O(x^n), n))};

%o (Magma) m:=25; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-x+x^2-x^4+2*x^5-x^6)/((1-x)^2*(1-x^8)))); // _G. C. Greubel_, Aug 07 2018

%Y Cf. A006720, A220838.

%K nonn,easy

%O 0,3

%A _Michael Somos_, Jan 21 2014