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A135364 First column of a triangle - see Comments lines.

%I A135364 #34 Sep 08 2022 08:45:32
%S A135364 1,2,3,7,17,40,93,216,502,1167,2713,6307,14662,34085,79238,184206,
%T A135364 428227,995507,2314273,5380032,12507057,29075380,67592058,157132471,
%U A135364 365288677,849193147,1974134558,4589306057,10668842202
%N A135364 First column of a triangle - see Comments lines.
%C A135364 ...1;
%C A135364 ...2,...1;
%C A135364 ...3,...3,...1;
%C A135364 ...7,...5,...4,...1;
%C A135364 ..17,..10,...7,...5,...1;
%C A135364 ..40,..24,..13,...9,...6,...1;
%C A135364 ..93,..57,..31,..16,..11,...7,...1;
%C A135364 From the second, the sum of a row gives the first term of the following one. Diagonal differences are the first term upon. First column is a(n).
%H A135364 G. C. Greubel, <a href="/A135364/b135364.txt">Table of n, a(n) for n = 0..1000</a>
%H A135364 Richard Choulet, <a href="https://www.apmep.fr/IMG/pdf/curtz1.pdf">Transformation à la Curtz. Curtz like Transformation</a>, March 2008.
%H A135364 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2,1).
%F A135364 From _Richard Choulet_, Jan 06 2008: (Start)
%F A135364 a(n+1) = a(n) + a(n-1) + (n-1)*a(1) + (n-2)*a(2) + ... + 2*a(n-2) for n>=3.
%F A135364 O.g.f.: 1 + x*(2 - 3*x + 2*x^2) / (1 - 3*x + 2*x^2 - x^3).
%F A135364 a(n+3) = 3*a(n+2) - 2*a(n+1) + a(n). (End)
%F A135364 a(n) = A034943(n) + A034943(n+1). - _R. J. Mathar_, Apr 09 2008
%F A135364 a(0) = 1, a(n) = term (1,3) in the 1 X 3 matrix [7,3,2].[3,1,0; -2,0,1; 1,0,0]^(n-1) (n>0). - _Alois P. Heinz_, Jul 24 2008
%F A135364 a(n) = 2*A095263(n-1) -3*A095263(n-2) +2*A095263(n-3) with a(0) = 1. - _G. C. Greubel_, Apr 19 2021
%p A135364 a:= n-> `if`(n=0, 1, (<<7|3|2>> .<<3|1|0>, <-2|0|1>, <1|0|0>>^(n-1))[1, 3]):
%p A135364 seq(a(n), n=0..50);  # _Alois P. Heinz_, Jul 24 2008
%t A135364 LinearRecurrence[{3,-2,1}, {1,2,3,7,17}, 51] (* _G. C. Greubel_, Oct 11 2016; Apr 19 2021 *)
%o A135364 (Magma) I:=[3,7,17]; [1,2] cat [n le 3 select I[n] else 3*Self(n-1) -2*Self(n-2) +Self(n-3): n in [1..51]]; // _G. C. Greubel_, Apr 19 2021
%o A135364 (Sage)
%o A135364 @CachedFunction
%o A135364 def A095263(n): return sum( binomial(n+j+2, 3*j+2) for j in (0..n//2) )
%o A135364 def A135364(n): return 1 if n==0 else 2*A095263(n-1) -3*A095263(n-2) +2*A095263(n-3)
%o A135364 [A135364(n) for n in (0..50)] # _G. C. Greubel_, Apr 19 2021
%Y A135364 Cf. A034943, A097550, A136302, A136303, A136304, A136305, A137229, A137234, A137249.
%K A135364 nonn
%O A135364 0,2
%A A135364 _Paul Curtz_, Dec 09 2007
%E A135364 More terms from _Richard Choulet_, Jan 06 2008