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

1, 2, 2, 3, 7, 11, 16, 28, 48, 77, 126, 211, 349, 573, 947, 1568, 2588, 4271, 7058, 11661, 19256, 31804, 52538, 86779, 143329, 236744, 391046, 645900, 1066850, 1762163, 2910634, 4807590, 7940870, 13116238, 21664568, 35784145, 59105987, 97627533, 161254953, 266350689

Offset: 1

Author

John Molokach, Jul 09 2013

Keywords

• nonn

Comments

Rising diagonal sums of triangle A011973, taken with rows as centered text.

Examples

			a(1)  = 1;
a(2)  = 1 +  1;
a(3)  = 1 +  1;
a(4)  = 1 +  1 +  1;
a(5)  = 1 +  1 +  3 +  2;
a(6)  = 1 +  1 +  5 +  4;
a(7)  = 1 +  1 +  7 +  6 +  1;
a(8)  = 1 +  1 +  9 +  8 +  6 +  3;
a(9)  = 1 +  1 + 11 + 10 + 15 + 10;
a(10) = 1 +  1 + 13 + 12 + 28 + 21 +  1.
		

Links

• Index entries for linear recurrences with constant coefficients, signature (1,0,2,0,0,-1).

Crossrefs

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).

Programs

Mathematica

LinearRecurrence[{1, 0, 2, 0, 0, -1}, {1, 2, 2, 3, 7, 11}, 40] (* T. D. Noe, Jul 11 2013 *)

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

Formula

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

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

G.f.: x*(1+x-x^3)/(1-x-2*x^3+x^6) - John Molokach, Jul 15 2013

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