A089508 Solution to a binomial problem together with companion sequence A081016(n-1).

1, 14, 103, 713, 4894, 33551, 229969, 1576238, 10803703, 74049689, 507544126, 3478759199, 23843770273, 163427632718, 1120149658759, 7677619978601, 52623190191454, 360684711361583, 2472169789339633, 16944503814015854

Offset: 1

Author

Wolfdieter Lang, Dec 01 2003

Keywords

• nonn
• easy

Comments

a(n) and b(n) := A081016(n-1) are the solutions to the Diophantine equation binomial(a,b) = binomial(a+1,b-1). The first few binomials are given by A090162(n).

Examples

			n = 2: a(2) = 14, b(2) = A081016(1) = 6 satisfy binomial(14,6) = 3003 = binomial(15,5). 3003 = A090162(2).
		

References

• A. I. Shirshov: On the equation binomial(n,m)=binomial(n+1,m-1), pp. 83-86, in: Kvant Selecta: Algebra and Analysis, I, ed. S. Tabachnikov, Am.Math.Soc., 1999.

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• Wikipedia, Singmaster's conjecture
• Index entries for linear recurrences with constant coefficients, signature (8, -8, 1).

Crossrefs

Equals A081018 - 1.

Programs

Magma

[Fibonacci(2*n)*Fibonacci(2*n+1) - 1: n in [1..30]]; // G. C. Greubel, Dec 18 2017

Mathematica

Rest[CoefficientList[Series[x*(1 + 6*x - x^2)/((1 - x)*(1 - 7*x + x^2)), {x, 0, 50}], x]] (* G. C. Greubel, Dec 18 2017 *)

PARI

x='x+O('x^30); Vec(x*(1 + 6*x - x^2)/((1 - x)*(1 - 7*x + x^2))) \\ G. C. Greubel, Dec 18 2017

Formula

G.f.: x*(1+6*x-x^2)/((1-x)*(1-7*x+x^2)).

a(n) = A081018(n) - 1 = F(2*n)*F(2*n+1) - 1, n>=1; with F(n) := A000045(n) (Fibonacci).