{"success":true,"version":"v1","request":{"tool":"get_calculator_schema","calculator_id":"fibonacci"},"result":{"entity_type":"calculator","id":"fibonacci","calculator_id":"fibonacci","canonical_url":"https://tttkmbb.com/math/fibonacci","name":"Fibonacci Number Calculator","title":"Fibonacci Calculator – Exact F(n) up to n = 500, Neighbouring Terms, Golden Ratio and Fibonacci Test","category":"math","category_name":"Math","tool_name":"calculate_fibonacci_number","featured_mcp_tool":false,"description":"Computes the n-th Fibonacci number exactly with arbitrary-precision integers, together with F(n−1), F(n+1), the ratio F(n)/F(n−1) compared with the golden ratio, and optionally tests whether a given integer is a Fibonacci number.","use_when":"You need a specific Fibonacci number (exact, even with hundreds of digits), the terms around it, or want to check whether a number belongs to the sequence.","do_not_use_when":"You need a general geometric or arithmetic progression (use geometric-sequence / arithmetic-sequence) or factorials and binomial coefficients (use factorial, combinations-permutations).","inputs":[{"name":"n","label":"Index n","type":"integer","required":true,"min":0,"max":500,"description":"Position in the sequence with F(0) = 0, F(1) = 1, F(2) = 1, F(3) = 2, …","example":50},{"name":"check_number","label":"Number to test","type":"string","required":false,"description":"Optional non-negative integer (as digits, any size) to test for membership in the Fibonacci sequence.","example":"144"}],"outputs":[{"name":"fibonacci","label":"F(n)","type":"string","decimals":4,"description":"The n-th Fibonacci number as an exact decimal string."},{"name":"fibonacci_number","label":"F(n) as number","type":"number","decimals":0,"description":"The same value as a number; omitted from F(74) on (above 2^50), where a number would no longer carry every digit."},{"name":"previous","label":"F(n − 1)","type":"string","decimals":4,"description":"The preceding Fibonacci number (omitted for n = 0)."},{"name":"next","label":"F(n + 1)","type":"string","decimals":4,"description":"The following Fibonacci number."},{"name":"ratio","label":"F(n) / F(n − 1)","type":"number","decimals":10,"description":"Ratio of consecutive terms, which converges to the golden ratio (omitted for n ≤ 1)."},{"name":"golden_ratio","label":"Golden ratio φ","type":"number","decimals":10,"description":"(1 + √5) / 2 = 1.6180339887…, the limit of the ratio."},{"name":"digits","label":"Number of digits","type":"integer","decimals":0,"description":"Decimal digits of F(n)."},{"name":"is_fibonacci","label":"check_number is Fibonacci","type":"boolean","decimals":4,"description":"true when 5·N² + 4 or 5·N² − 4 is a perfect square (only when check_number is given)."},{"name":"check_index","label":"Index of check_number","type":"integer","decimals":0,"description":"The smallest k with F(k) = check_number (only when it is a Fibonacci number)."}],"input_schema":{"type":"object","properties":{"n":{"description":"Position in the sequence with F(0) = 0, F(1) = 1, F(2) = 1, F(3) = 2, …","type":"integer","minimum":0,"maximum":500,"examples":[50]},"check_number":{"description":"Optional non-negative integer (as digits, any size) to test for membership in the Fibonacci sequence.","type":"string","examples":["144"]}},"additionalProperties":false,"required":["n"]},"output_schema":{"type":"object","properties":{"fibonacci":{"description":"The n-th Fibonacci number as an exact decimal string.","type":"string"},"fibonacci_number":{"description":"The same value as a number; omitted from F(74) on (above 2^50), where a number would no longer carry every digit.","type":"number"},"previous":{"description":"The preceding Fibonacci number (omitted for n = 0).","type":"string"},"next":{"description":"The following Fibonacci number.","type":"string"},"ratio":{"description":"Ratio of consecutive terms, which converges to the golden ratio (omitted for n ≤ 1).","type":"number"},"golden_ratio":{"description":"(1 + √5) / 2 = 1.6180339887…, the limit of the ratio.","type":"number"},"digits":{"description":"Decimal digits of F(n).","type":"integer"},"is_fibonacci":{"description":"true when 5·N² + 4 or 5·N² − 4 is a perfect square (only when check_number is given).","type":"boolean"},"check_index":{"description":"The smallest k with F(k) = check_number (only when it is a Fibonacci number).","type":"integer"}}},"formula":"F(0) = 0, F(1) = 1, F(n) = F(n − 1) + F(n − 2); φ = (1 + √5)/2; N is a Fibonacci number ⇔ 5N² + 4 or 5N² − 4 is a perfect square (Gessel's test)","sources":[{"name":"Wikipedia – Fibonacci sequence","url":"https://en.wikipedia.org/wiki/Fibonacci_sequence","type":"reference","retrieved_at":"2026-09-24"},{"name":"OEIS A000045 – Fibonacci numbers","url":"https://oeis.org/A000045","type":"reference","retrieved_at":"2026-09-24"},{"name":"Wolfram MathWorld – Fibonacci Number","url":"https://mathworld.wolfram.com/FibonacciNumber.html","type":"reference","retrieved_at":"2026-09-24"}],"freshness":{"type":"static","max_age_seconds":null,"note":"Deterministic formula with fixed constants; results never go stale. Inputs supplied by the caller determine the output."},"examples":[{"name":"F(50), is 144 Fibonacci?","inputs":{"n":50,"check_number":"144"},"expected":{"fibonacci":"12586269025","fibonacci_number":12586269025,"previous":"7778742049","next":"20365011074","ratio":1.6180339887,"golden_ratio":1.6180339887,"digits":11,"is_fibonacci":true,"check_index":12},"url":"https://tttkmbb.com/api/v1/calculate/fibonacci?n=50&check_number=144"},{"name":"F(10)","inputs":{"n":10},"expected":{"fibonacci":"55","fibonacci_number":55,"previous":"34","next":"89","ratio":1.6176470588,"digits":2},"url":"https://tttkmbb.com/api/v1/calculate/fibonacci?n=10"}],"faq":[{"q":"Does the sequence start with 0 or 1?","a":"This calculator uses the standard indexing F(0) = 0, F(1) = 1 (OEIS A000045), so F(10) = 55. Some texts start at F(1) = 1, F(2) = 1, which shifts the index by one."},{"q":"How large can n be?","a":"Up to 500: F(500) has 105 digits and is returned exactly as a string. The number output is omitted from F(74) on, because values above 2^50 cannot be returned as exact numbers."}],"tags":["fibonacci","fibonacci number","fibonacci sequence","golden ratio","nth fibonacci","is fibonacci"],"related":[{"calculator_id":"geometric-sequence","reason":"Fibonacci numbers grow like a geometric sequence with ratio φ."},{"calculator_id":"arithmetic-sequence","reason":"Sequences with a constant difference."},{"calculator_id":"factorial","reason":"Another fast-growing integer sequence."}],"links":{"html":"https://tttkmbb.com/math/fibonacci","markdown":"https://tttkmbb.com/math/fibonacci.md","json":"https://tttkmbb.com/math/fibonacci.json","api":"https://tttkmbb.com/api/v1/calculate/fibonacci","schema":"https://tttkmbb.com/api/v1/calculators/fibonacci","openapi":"https://tttkmbb.com/openapi.json","mcp":"https://tttkmbb.com/mcp"},"version":"v1","updated_at":"2026-09-24"},"timestamp":"2026-09-24T01:51:02Z"}