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Number Series

Source: R/number_series.R
number_series.Rd

Functions to compute Bernoulli numbers, Tangent numbers, Prime numbers, and Fibonacci numbers. The library provides efficient implementations using table lookups for smaller indices and advanced algorithms for larger values.

Bernoulli Numbers B(2n):

The Bernoulli numbers are a sequence of rational numbers useful for Taylor series expansions, the Euler-Maclaurin formula, and the Riemann zeta function.

  • bernoulli_b2n(n): Returns the (2n)-th Bernoulli number \(B_{2n}\). Note that odd Bernoulli numbers are 0 (except B_1 = -1/2).

  • max_bernoulli_b2n(): Returns the largest n such that \(B_{2n}\) can be represented in the return type.

  • unchecked_bernoulli_b2n(n): A faster version without overflow checks.

  • bernoulli_b2n(start_index, number_of_bernoullis_b2n): Computes a range of Bernoulli numbers.

Tangent Numbers T(n):

Tangent numbers (or zag functions) appear in the Maclaurin series of tan(x).

  • tangent_t2n(n): Returns the n-th Tangent number.

  • tangent_t2n(start_index, number_of_tangent_t2n): Computes a range of Tangent numbers.

Prime Numbers:

Fast table lookup for the first 10,000 prime numbers.

  • prime(n): Returns the n-th prime number (0-indexed, so prime(0) = 2).

  • max_prime(): Returns the maximum index n supported (currently 10,000).

Fibonacci Numbers F(n):

Computes Fibonacci numbers defined by \(F_n = F_{n-1} + F_{n-2}\) with \(F_0 = 0, F_1 = 1\).

  • fibonacci(n): Returns the n-th Fibonacci number.

  • unchecked_fibonacci(n): A faster version without overflow checks.

Implementation uses table lookup for small values and iterative algorithms for larger values.

Usage

bernoulli_b2n(n = NULL, start_index = NULL, number_of_bernoullis_b2n = NULL)

max_bernoulli_b2n()

unchecked_bernoulli_b2n(n)

tangent_t2n(n = NULL, start_index = NULL, number_of_tangent_t2n = NULL)

prime(n)

max_prime()

fibonacci(n)

unchecked_fibonacci(n)

Arguments

n

Index of the number to compute (must be a non-negative integer).

start_index

The starting index for computing a range of numbers.

number_of_bernoullis_b2n

The number of Bernoulli numbers to compute in the range.

number_of_tangent_t2n

The number of Tangent numbers to compute in the range.

Value

A single numeric or integer value for scalar inputs, or a vector for range computations. For max_ functions, returns the maximum supported index.

See also

Boost Documentation

Examples

if (FALSE) { # \dontrun{
# 10th Bernoulli number B_20 (index is doubled)
bernoulli_b2n(10)
# Maximum supported index for Bernoulli numbers
max_bernoulli_b2n()
# Range of Bernoulli numbers B_0, B_2, ..., B_18 (10 numbers)
bernoulli_b2n(start_index = 0, number_of_bernoullis_b2n = 10)

# 10th Tangent number
tangent_t2n(10)

# 10th Prime number (0-indexed)
prime(10)

# 10th Fibonacci number
fibonacci(10)
} # }