|
%I #16 Jun 16 2022 04:05:31
%S 1,1,2,7,11,22,35,61,98,163,263,430,695,1129,1826,2959,4787,7750,
%T 12539,20293,32834,53131,85967,139102,225071,364177,589250,953431,
%U 1542683,2496118,4038803,6534925,10573730,17108659,27682391,44791054,72473447,117264505
%N Diagonal sums of the triangle A132047.
%H Colin Barker, <a href="/A144707/b144707.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-1,-1).
%F G.f.: (1 - x^2 + 4*x^3 + 2*x^4) / ((1 - x^2)*(1 - x - x^2)).
%F a(n) = 3*Fibonacci(n+1) - 3 - (-1)^n + 2*0^n.
%F a(n) = a(n-1) + 2*a(n-2) - a(n-3) - a(n-4) for n>4. - _Philippe Deléham_, Dec 16 2008
%F From _Colin Barker_, Jul 12 2017: (Start)
%F a(n) = (3*2^(-n-1)*((1 + sqrt(5))^(n+1) - (1-sqrt(5))^(n+1))) / sqrt(5) - 4 for n>0 and even.
%F a(n) = (3*2^(-n-1)*((1+sqrt(5))^(n+1) - (1-sqrt(5))^(n+1)))/sqrt(5) - 2 for n odd.
%F (End)
%t Table[3*Fibonacci[n+1] -3 -(-1)^n +2*Boole[n==0], {n,0,40}] (* _G. C. Greubel_, Jun 16 2022 *)
%o (PARI) Vec((1-x^2+4*x^3+2*x^4) / ((1-x^2)*(1-x-x^2)) + O(x^50)) \\ _Colin Barker_, Jul 12 2017
%o (SageMath) [3*fibonacci(n+1) -2 -2*((n+1)%2) +2*bool(n==0) for n in (0..40)] # _G. C. Greubel_, Jun 16 2022
%Y Cf. A000045.
%K easy,nonn
%O 0,3
%A _Paul Barry_, Sep 19 2008
|
