A081207 Main diagonal of number square A081206.

1, 2, 3, 7, 16, 37, 89, 216, 529, 1307, 3248, 8111, 20339, 51176, 129143, 326717, 828374, 2104361, 5354979, 13647682, 34830191, 89000157, 227674188, 583017657, 1494365341, 3833592212, 9842373849, 25287895051, 65016153154, 167264946727

Offset

0,2

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Formula

a(n) = Sum_{k=0..n} (binomial(floor((n+k)/2), k))^2.

G.f.: (1+x)/sqrt(1-2x-x^2-2x^3+x^4) - Paul Barry, Jun 04 2005

Conjecture: n*(n-2)*a(n) +(-2*n^2+5*n-1)*a(n-1) +(-n^2+3*n-4)*a(n-2) +(-2*n^2+7*n-4)*a(n-3) +(n-1)*(n-3)*a(n-4)=0. - R. J. Mathar, Nov 12 2012

a(n) ~ (5-sqrt(5)) * ((3+sqrt(5))/2)^n / (2*sqrt(14*sqrt(5)-30) * sqrt(Pi*n)). - Vaclav Kotesovec, Feb 04 2014

Equivalently, a(n) ~ 5^(1/4) * phi^(2*n + 1) / (2 * sqrt(Pi*n)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Dec 08 2021

Mathematica

Table[Sum[Binomial[Floor[(n+k)/2], k]^2, {k, 0, n}], {n, 0, 30}] (* Harvey P. Dale, Oct 02 2011 *)
CoefficientList[Series[(1+x)/Sqrt[1-2x-x^2-2x^3+x^4], {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 04 2014 *)

Prog

(PARI) for(n=0, 25, print1(sum(k=0, n, (binomial(floor((n+k)/2), k))^2), ", ")) \\ G. C. Greubel, Feb 16 2017
(PARI) x='x+O('x^25); Vec((1+x)/sqrt(1-2*x-x^2-2*x^3+x^4)) \\ G. C. Greubel, Feb 16 2017

Crossrefs

Cf. A051286, A081206.

Sequence in context: A289844 A153056 A235112 * A353580 A334398 A027118

Adjacent sequences: A081204 A081205 A081206 * A081208 A081209 A081210

Keyword

easy,nonn

Author

Paul Barry, Mar 11 2003

Status

approved