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A081207
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Main diagonal of number square A081206.
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2
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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
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OFFSET
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0,2
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LINKS
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FORMULA
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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
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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