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A227300 Rising diagonal sums of triangle of Fibonacci polynomials (rows displayed as centered text).

%I A227300 #69 Jul 07 2024 01:46:30
%S A227300 1,2,2,3,7,11,16,28,48,77,126,211,349,573,947,1568,2588,4271,7058,
%T A227300 11661,19256,31804,52538,86779,143329,236744,391046,645900,1066850,
%U A227300 1762163,2910634,4807590,7940870,13116238,21664568,35784145,59105987,97627533,161254953,266350689
%N A227300 Rising diagonal sums of triangle of Fibonacci polynomials (rows displayed as centered text).
%C A227300 Rising diagonal sums of triangle A011973, taken with rows as centered text.
%H A227300 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,2,0,0,-1).
%F A227300 a(n) = Sum_{k=0..floor((n-1)/3)} (binomial(2*n-2-5*k,k) + binomial(2*n-3-5*k,k)) for n >= 2; a(1)=1. - _John Molokach_, Jul 11 2013
%F A227300 a(n) = a(n-1) + 2*a(n-3) - a(n-6), starting with {1, 2, 2, 3, 7, 11}. - _T. D. Noe_, Jul 11 2013
%F A227300 G.f.: x*(1+x-x^3)/(1-x-2*x^3+x^6) - _John Molokach_, Jul 15 2013
%F A227300 a(n) = Sum_{k=0..floor((2n-1)/3)} binomial(2n-k-2-3*floor(k/2),floor(k/2)). - _John Molokach_, Jul 29 2013
%e A227300 a(1)  = 1;
%e A227300 a(2)  = 1 +  1;
%e A227300 a(3)  = 1 +  1;
%e A227300 a(4)  = 1 +  1 +  1;
%e A227300 a(5)  = 1 +  1 +  3 +  2;
%e A227300 a(6)  = 1 +  1 +  5 +  4;
%e A227300 a(7)  = 1 +  1 +  7 +  6 +  1;
%e A227300 a(8)  = 1 +  1 +  9 +  8 +  6 +  3;
%e A227300 a(9)  = 1 +  1 + 11 + 10 + 15 + 10;
%e A227300 a(10) = 1 +  1 + 13 + 12 + 28 + 21 +  1.
%t A227300 LinearRecurrence[{1, 0, 2, 0, 0, -1}, {1, 2, 2, 3, 7, 11}, 40] (* _T. D. Noe_, Jul 11 2013 *)
%o A227300 (PARI) a(n) = if(n<=1, 1, sum(k=0, floor((n-1)/3), binomial(2*n-2-5*k,k)+binomial(2*n-1-5*k,k)) ); \\ _Joerg Arndt_, Jul 11 2013
%Y A227300 Cf. A011973 (triangle), A000045 (row sums of triangle), A005314 (falling diagonal sums of triangle).  Expansion of terms begin with A055624 at a(1) and adds A016813 at a(4), A016754 at a(7), and A100157 at a(10).
%K A227300 nonn
%O A227300 1,2
%A A227300 _John Molokach_, Jul 09 2013