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A131300 a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3) + a(n-4) with n>3, a(0)=1, a(1)=2, a(2)=3, a(3)=7.

%I A131300 #35 Dec 09 2024 06:00:40
%S A131300 1,2,3,7,14,27,49,86,147,247,410,675,1105,1802,2931,4759,7718,12507,
%T A131300 20257,32798,53091,85927,139058,225027,364129,589202,953379,1542631,
%U A131300 2496062,4038747,6534865,10573670,17108595,27682327,44790986,72473379,117264433,189737882
%N A131300 a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3) + a(n-4) with n>3, a(0)=1, a(1)=2, a(2)=3, a(3)=7.
%C A131300 Row sums of A131299 and A131301.
%C A131300 a(n)/a(n-1) tends to phi.
%H A131300 Nathaniel Johnston, <a href="/A131300/b131300.txt">Table of n, a(n) for n = 0..1000</a>
%H A131300 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2,-1,1).
%F A131300 From _Bruno Berselli_, May 03 2012: (Start)
%F A131300 G.f.: (1-x-x^2+3*x^3)/((1-x-x^2)*(1-x)^2).
%F A131300 a(n) = (3*A131269(n)-n-1)/2.
%F A131300 a(n) = 3*((1+sqrt(5))^(n+2)-(1-sqrt(5))^(n+2))/(2^(n+2)*sqrt(5))-2*(n+1). (End)
%F A131300 a(n) = 3*A000045(n+2)-2*(n+1). - _R. J. Mathar_, Mar 24 2018
%e A131300 a(4) = 14 = (1 + 1 + 7 + 4 + 1).
%p A131300 seq(add(3*binomial(floor((n+k)/2),k)-2,k=0..n),n=0..50); # _Nathaniel Johnston_, Jun 29 2011
%t A131300 LinearRecurrence[{3, -2, -1, 1}, {1, 2, 3, 7}, 38] (* _Bruno Berselli_, May 03 2012 *)
%o A131300 (PARI) Vec((1-x-x^2+3*x^3)/((1-x-x^2)*(1-x)^2)+O(x^38)) \\ _Bruno Berselli_, May 03 2012
%o A131300 (Magma) /* By first comment: */ [&+[3*Binomial(n-Floor((k+1)/2), Floor(k/2))-2: k in [0..n]]: n in [0..37]]; // _Bruno Berselli_, May 03 2012
%o A131300 (Maxima) makelist(expand(3*((1+sqrt(5))^(n+2)-(1-sqrt(5))^(n+2))/(2^(n+2)*sqrt(5))-2*(n+1)), n, 0, 37); /* _Bruno Berselli_, May 03 2012 */
%Y A131300 Cf. A131269, A131299, A131301.
%K A131300 nonn,easy
%O A131300 0,2
%A A131300 _Gary W. Adamson_, Jun 27 2007
%E A131300 Terms after a(9) from _Nathaniel Johnston_, Jun 29 2011
%E A131300 New definition from _Bruno Berselli_, May 03 2012