A358464 a(n) is the greatest m such that Sum_{k = 1..m} 1/(1 + n*k) <= 1.

2, 6, 16, 42, 110, 288, 761, 2020, 5388, 14417, 38681, 103994, 280032, 755031, 2037848, 5504884, 14880978, 40250609, 108926101, 294902398, 798703663, 2163878141

Offset

1,1

Comments

This sequence coincidences with 2*Fibonacci(2*n) (A025169) for the first 6 terms.

Links

Table of n, a(n) for n=1..22.

David Borwein and Jonathan M. Borwein, Some Remarkable Properties of Sinc and Related Integrals, The Ramanujan Journal, 5 (2001) 73 - 89.

Hanspeter Schmid, Two curious integrals and a graphic proof, Elemente der Mathematik, 69 (2014) 11 - 17.

Formula

ceiling(digamma(a(n)+(1/n)+1) - digamma((1/n)+1)) = n.

Integral_{x=0..oo} Product_{k=0..m} sinc(x/(n*k+1)) dx = Pi for 0 <= m <= a(n). See links Schmid and Borwein.

ceiling(Sum_{m = 0..oo} ( 1/(m+1) * Sum_{k = 0..m} (-1)^k*binomial(m, k)*log( (a(n)+(1/n)+1+k) / ((1/n)+1+k) ) )) = n.

a(n) ~ floor(exp(n + digamma(1+(1/n))) - (1/2) - (1/n)). This appears to be accurate for at least n < 22.

Example

a(2) = 6 because Sum_{m = 1..a(2)} 1/(1+2*m) = 43024/45045 < 1, but Sum_{m = 1..a(2)+1} 1/(1+2*m) = 46027/45045 > 1.

Prog

(PARI) a(n) = {my(k = 2*fibonacci(2*n)-1); my(b = (psi(k+(1/n))-psi(1+(1/n)))/n); until(b > 1, b = b+(1/(1+n*k)); k=k+1 ); k-2}

Crossrefs

Cf. A002387, A025169, A064169.

Sequence in context: A156664 A025169 A111282 * A115730 A191694 A224232

Adjacent sequences: A358461 A358462 A358463 * A358465 A358466 A358467

Keyword

nonn,more

Author

Thomas Scheuerle, Nov 18 2022

Status

approved