Lower Wythoff sequence calculator
The 10th term of the lower Wythoff sequence is 16, and the sequence opens 1, 3, 4, 6, 8. Enter a position counted from one to get that exact term without listing all the earlier ones. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- N
- 10
- Lower wythoff number
- 16
find the n-th term
What the lower Wythoff sequence represents
Because phi is approximately 1.618, the terms increase steadily but not by a constant amount: consecutive differences are either one or two. The first positions therefore produce 1, 3, 4, 6, 8, and so on. This calculator uses one-based indexing, so n equal to one returns the first term rather than a zeroth placeholder.
That convention matches the mathematical definition and matters when comparing results with reference tables. The word lower distinguishes this sequence from its companion, the upper Wythoff sequence. Together they form complementary Beatty sequences, meaning every positive integer occurs in exactly one of the two and neither sequence repeats an integer.
The returned object includes both the requested position and the lower Wythoff number, making the result easy to retain in a worksheet, program, or reproducible calculation record.
Using the result in sequence and game problems
Use the calculator when you need one term rather than a generated list. For example, a lesson can request selected positions and ask students to compare their gaps, while a test suite can check an implementation against stable reference points. In combinatorial game theory, lower and upper Wythoff pairs describe cold positions in Wythoff Nim, so an exact term is useful when constructing or validating a position without generating all earlier terms.
The function also supports experiments with Beatty partitions: calculate a lower term, compare it with the corresponding upper term from another method, and examine how the two sequences distribute positive integers. Invalid positions fail explicitly. Zero, negative values, fractions, nonnumeric text, and integers beyond the documented bound are rejected instead of being rounded or coerced.
In either channel, the same pure solver supplies the result and gives identical output for identical input.
What you can do with it
Check an integer-sequence implementation
Compare selected positions with exact lower Wythoff terms when testing a sequence generator or mathematics library.
Build Wythoff Nim examples
Calculate the lower coordinate used when constructing or validating cold positions in Wythoff Nim.
Explore complementary Beatty sequences
Retrieve individual lower-sequence terms for lessons, proofs, worksheets, and computational experiments.
FAQ
Does indexing start at zero or one?
Indexing starts at one. An n value of 1 returns the first lower Wythoff number, which is 1.
What is the 10th lower Wythoff number?
It is 16. Positions are counted from one, so asking for position 10 gives the tenth term of the sequence directly, without any zeroth placeholder.
What are the first lower Wythoff numbers?
The sequence begins 1, 3, 4, 6, 8. Because phi is about 1.618, consecutive terms always differ by either one or two, never by a constant amount.
How is the lower Wythoff sequence related to the upper one?
They complement each other: every positive whole number appears in exactly one of the two sequences, and neither repeats a number. Together they split the positive integers.
What is the lower Wythoff sequence used for in Wythoff Nim?
Its terms give the lower coordinate of the cold positions in Wythoff Nim. With one exact term you can build or check such a position without generating every earlier one.
Related tools
For developers — API access
Everything on this page is available programmatically. This section is for teams who want to wire it into their own systems; everyone else can just use the tool above.
API endpoint
Prefer to automate it? One authenticated POST creates the task; the result comes back by webhook or a signed link. The same capability also runs here on the web, by email and from Telegram — and soon from our app too.
Call it from your stack
curl -X POST https://api.kit.forhosting.com/numth/wythoff-lower \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"n":10}'const res = await fetch("https://api.kit.forhosting.com/numth/wythoff-lower", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"n": 10
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/numth/wythoff-lower",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"n": 10
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/numth/wythoff-lower", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"n":10}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"n":10}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/numth/wythoff-lower", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"n": 10
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "numth.wythoff_lower",
"status": "queued",
"_links": {
"result": "/tasks/tsk_…/result"
}
}The API is asynchronous: the call returns a task_id immediately and the result arrives by webhook. Polling is capped at 1 req/s per task.
Pricing
Published price — no tokens, no invented credits. A failed task is never charged.
Errors
| HTTP | Code | Meaning |
|---|---|---|
401 | unauthorized | Missing or invalid API key. |
402 | insufficient_balance | Your balance doesn't cover the task price. |
404 | unknown_type | That task type doesn't exist. |
429 | rate_limited | Too many requests. Use the webhook instead of polling. |
Technical notes
Every request is deterministic: the same valid position always produces the same integer.
How the term is calculated exactly
Since the square root of five is irrational, the transformation preserves the required floor exactly for every accepted input. The calculation uses a bounded Newton iteration on integers, contains no random choices, and does not depend on network services, clocks, locale settings, or machine-specific decimal formatting. Inputs must be safe JavaScript integers from one through five quadrillion. That upper bound keeps both the supplied position and the returned term exactly representable as JSON numbers, so callers do not receive a value that merely looks integral while silently losing precision.
Using the result in sequence and game problems
Supply n as an integer, then read lower_wythoff_number from the response. Browser use is convenient for occasional checks, while automated calls cost $0.002 per request.
Technical questions
What formula does this capability use?
It returns floor(n times the golden ratio), evaluated through equivalent integer arithmetic to avoid decimal-rounding errors.
What happens when n is below one?
The request fails with an invalid_input error because the sequence position must be at least 1.
Can n be a fraction or numeric string?
No. The input must be a JSON integer; fractions and strings are rejected rather than rounded or coerced.
What does an API request cost?
Each API request costs $0.002. The browser calculator can run the same deterministic solver locally.