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
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000
Programs
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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 *) -
PARI
for(n=0,25, print1(sum(k=0,n, (binomial(floor((n+k)/2), k))^2), ", ")) \\ G. C. Greubel, Feb 16 2017
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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
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