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A106523 Diagonal sums of number triangle A106522.

%I A106523 #13 Sep 08 2022 08:45:18
%S A106523 1,1,3,4,10,14,33,49,109,170,362,586,1207,2011,4037,6878,13536,23464,
%T A106523 45475,79891,153011,271612,515460,922372,1738101,3129565,5865063,
%U A106523 10611336,19802382,35960970,66888917,121820229,226016385,412547222
%N A106523 Diagonal sums of number triangle A106522.
%H A106523 G. C. Greubel, <a href="/A106523/b106523.txt">Table of n, a(n) for n = 0..1000</a>
%H A106523 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (0,3,1,0,-1).
%F A106523 G.f.: (1+x)/((1+x-x^2)*(1-x-x^2-x^3)).
%F A106523 a(n) = 3*a(n-2) + a(n-3) - a(n-5).
%F A106523 a(n) = Sum_{k=0..floor(n/2)} A106522(n-k, k)
%F A106523 a(n) = (1/11)*( 10*T(n+2) + 5*T(n+1) + 3*T(n) + (-1)^n*( F(n+1) + 3*F(n) ) ), where T(n) = A000073, and F(n) = A000045. - _G. C. Greubel_, Aug 10 2021
%t A106523 T[n_]:= T[n]= If[n<2, 0, If[n==2, 1, T[n-1] + T[n-2] + T[n-3]]]; (* A000073 *)
%t A106523 a[n_]:= (1/11)*((-1)^n*(Fibonacci[n+2] +2*Fibonacci[n]) +10*T[n+2] +5*T[n+1] + 3*T[n]);
%t A106523 Table[a[n], {n, 0, 40}] (* _G. C. Greubel_, Aug 10 2021 *)
%o A106523 (Magma) I:=[1,1,3,4,10]; [n le 5 select I[n] else 3*Self(n-2) + Self(n-3) -Self(n-5): n in [1..41]]; // _G. C. Greubel_, Aug 10 2021
%o A106523 (Sage)
%o A106523 @CachedFunction
%o A106523 def T(n):
%o A106523     if (n<2): return 0
%o A106523     elif (n==2): return 2
%o A106523     else: return T(n-1) + T(n-2) + T(n-3)
%o A106523 def a(n): return (1/11)*((-1)^n*(fibonacci(n+2) +2*fibonacci(n)) +10*T(n+2) +5*T(n+1) + 3*T(n))
%o A106523 [a(n) for n in (0..40)] # _G. C. Greubel_, Aug 10 2021
%Y A106523 Cf. A106522, A000045, A000073.
%K A106523 easy,nonn
%O A106523 0,3
%A A106523 _Paul Barry_, May 06 2005