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A227300 Rising diagonal sums of triangle of Fibonacci polynomials (rows displayed as centered text). 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Rising diagonal sums of triangle A011973, taken with rows as centered text.
LINKS
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
EXAMPLE
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.
MATHEMATICA
LinearRecurrence[{1, 0, 2, 0, 0, -1}, {1, 2, 2, 3, 7, 11}, 40] (* T. D. Noe, Jul 11 2013 *)
PROG
(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
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).
Sequence in context: A291102 A208494 A036060 * A065383 A087384 A179283
KEYWORD
nonn
AUTHOR
John Molokach, Jul 09 2013
STATUS
approved

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Last modified March 28 22:04 EDT 2024. Contains 371254 sequences. (Running on oeis4.)