A110198 Diagonal sums of number triangle A110197.

1, 2, 4, 9, 20, 46, 109, 262, 638, 1569, 3886, 9680, 24225, 60856, 153368, 387573, 981742, 2491934, 6336721, 16139616, 41166912, 105139773, 268841100, 688157430, 1763206441, 4521749642, 11605580290, 29809644693, 76621733444

Offset

0,2

Comments

Partial sums of A051286.

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Formula

G.f.: 1/((1-x)*sqrt((1+x+x^2)*(1-3x+x^2))); a(n) = sum{k=0..floor(n/2), sum{i=0..n-2k, binomial(i+k, k)^2}}.

a(n) = sum{i=0..2n, A202411(i)}. - Peter Luschny, Jan 16 2012

Conjecture: n*a(n) +(-3*n+1)*a(n-1) +n*a(n-2) +(-n+2)*a(n-3) +(3*n-5)*a(n-4) +(-n+2)*a(n-5)=0. - R. J. Mathar, Nov 15 2012

a(n) ~ sqrt(100+45*sqrt(5)) * ((sqrt(5)+3)/2)^n / (10*sqrt(Pi*n)). - Vaclav Kotesovec, Feb 08 2014

Equivalently, a(n) ~ phi^(2*n + 3) / (2 * 5^(1/4) * sqrt(Pi*n)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Dec 07 2021

Mathematica

CoefficientList[Series[1/((1-x)*Sqrt[(1+x+x^2)*(1-3*x+x^2)]), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 08 2014 *)

Crossrefs

Cf. A201631.

Sequence in context: A052806 A329672 A218552 * A104508 A368830 A036623

Adjacent sequences: A110195 A110196 A110197 * A110199 A110200 A110201

Keyword

easy,nonn

Author

Paul Barry, Jul 15 2005

Status

approved