# OCME ยท Knowledge & practice explorer 302 MKO / 1,441 families / 360 batches / 36,000 questions. 36,000 second-context math/bilingual reviewed; 0 display/symbol-checked candidates. Source questions: candidate. English: ai_candidate. No new proof, Lean, SREG or training. 100 results; page 3/4. ## Question 22751: The Recurrence and Both Closed Forms Use the Same Zero-Based Index - ID: `q-20261014-advance-20261111-r05-catalan-numbers-0051`; number: 22751 - Processing: second_context_reviewed - MKO: [mko-catalan-numbers](/en/mko/mko-catalan-numbers/) - Family: [qf-mod42-catalan-recurrence-closed-cross-check](/en/explore/?family=qf-mod42-catalan-recurrence-closed-cross-check) - Batch: [batch-20261014-advance-20261111-r05-catalan-numbers-foundations](/en/explore/?batch=batch-20261014-advance-20261111-r05-catalan-numbers-foundations) - [Derivation](/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0051/) / [Markdown](/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0051/index.md) / [JSON](/data/question-derivations/questions/q-20261014-advance-20261111-r05-catalan-numbers-0051.json) ## Question 22752: The Recurrence and Both Closed Forms Use the Same Zero-Based Index - ID: `q-20261014-advance-20261111-r05-catalan-numbers-0052`; number: 22752 - Processing: second_context_reviewed - MKO: [mko-catalan-numbers](/en/mko/mko-catalan-numbers/) - Family: [qf-mod42-catalan-recurrence-closed-cross-check](/en/explore/?family=qf-mod42-catalan-recurrence-closed-cross-check) - Batch: [batch-20261014-advance-20261111-r05-catalan-numbers-foundations](/en/explore/?batch=batch-20261014-advance-20261111-r05-catalan-numbers-foundations) - [Derivation](/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0052/) / [Markdown](/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0052/index.md) / [JSON](/data/question-derivations/questions/q-20261014-advance-20261111-r05-catalan-numbers-0052.json) ## Question 22753: The Recurrence and Both Closed Forms Use the Same Zero-Based Index - ID: `q-20261014-advance-20261111-r05-catalan-numbers-0053`; number: 22753 - Processing: second_context_reviewed - MKO: [mko-catalan-numbers](/en/mko/mko-catalan-numbers/) - Family: [qf-mod42-catalan-recurrence-closed-cross-check](/en/explore/?family=qf-mod42-catalan-recurrence-closed-cross-check) - Batch: [batch-20261014-advance-20261111-r05-catalan-numbers-foundations](/en/explore/?batch=batch-20261014-advance-20261111-r05-catalan-numbers-foundations) - [Derivation](/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0053/) / [Markdown](/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0053/index.md) / [JSON](/data/question-derivations/questions/q-20261014-advance-20261111-r05-catalan-numbers-0053.json) ## Question 22754: The Recurrence and Both Closed Forms Use the Same Zero-Based Index - ID: `q-20261014-advance-20261111-r05-catalan-numbers-0054`; number: 22754 - Processing: second_context_reviewed - MKO: [mko-catalan-numbers](/en/mko/mko-catalan-numbers/) - Family: [qf-mod42-catalan-recurrence-closed-cross-check](/en/explore/?family=qf-mod42-catalan-recurrence-closed-cross-check) - Batch: [batch-20261014-advance-20261111-r05-catalan-numbers-foundations](/en/explore/?batch=batch-20261014-advance-20261111-r05-catalan-numbers-foundations) - [Derivation](/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0054/) / [Markdown](/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0054/index.md) / [JSON](/data/question-derivations/questions/q-20261014-advance-20261111-r05-catalan-numbers-0054.json) ## Question 22755: The Recurrence and Both Closed Forms Use the Same Zero-Based Index - ID: `q-20261014-advance-20261111-r05-catalan-numbers-0055`; number: 22755 - Processing: second_context_reviewed - MKO: [mko-catalan-numbers](/en/mko/mko-catalan-numbers/) - Family: [qf-mod42-catalan-recurrence-closed-cross-check](/en/explore/?family=qf-mod42-catalan-recurrence-closed-cross-check) - Batch: [batch-20261014-advance-20261111-r05-catalan-numbers-foundations](/en/explore/?batch=batch-20261014-advance-20261111-r05-catalan-numbers-foundations) - [Derivation](/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0055/) / [Markdown](/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0055/index.md) / [JSON](/data/question-derivations/questions/q-20261014-advance-20261111-r05-catalan-numbers-0055.json) ## Question 22756: The Recurrence and Both Closed Forms Use the Same Zero-Based Index - ID: `q-20261014-advance-20261111-r05-catalan-numbers-0056`; number: 22756 - Processing: second_context_reviewed - MKO: [mko-catalan-numbers](/en/mko/mko-catalan-numbers/) - Family: [qf-mod42-catalan-recurrence-closed-cross-check](/en/explore/?family=qf-mod42-catalan-recurrence-closed-cross-check) - Batch: [batch-20261014-advance-20261111-r05-catalan-numbers-foundations](/en/explore/?batch=batch-20261014-advance-20261111-r05-catalan-numbers-foundations) - [Derivation](/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0056/) / [Markdown](/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0056/index.md) / [JSON](/data/question-derivations/questions/q-20261014-advance-20261111-r05-catalan-numbers-0056.json) ## Question 22757: The Recurrence and Both Closed Forms Use the Same Zero-Based Index - ID: `q-20261014-advance-20261111-r05-catalan-numbers-0057`; number: 22757 - Processing: second_context_reviewed - MKO: [mko-catalan-numbers](/en/mko/mko-catalan-numbers/) - Family: [qf-mod42-catalan-recurrence-closed-cross-check](/en/explore/?family=qf-mod42-catalan-recurrence-closed-cross-check) - Batch: [batch-20261014-advance-20261111-r05-catalan-numbers-foundations](/en/explore/?batch=batch-20261014-advance-20261111-r05-catalan-numbers-foundations) - [Derivation](/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0057/) / [Markdown](/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0057/index.md) / [JSON](/data/question-derivations/questions/q-20261014-advance-20261111-r05-catalan-numbers-0057.json) ## Question 22758: The Recurrence and Both Closed Forms Use the Same Zero-Based Index - ID: `q-20261014-advance-20261111-r05-catalan-numbers-0058`; number: 22758 - Processing: second_context_reviewed - MKO: [mko-catalan-numbers](/en/mko/mko-catalan-numbers/) - Family: [qf-mod42-catalan-recurrence-closed-cross-check](/en/explore/?family=qf-mod42-catalan-recurrence-closed-cross-check) - Batch: [batch-20261014-advance-20261111-r05-catalan-numbers-foundations](/en/explore/?batch=batch-20261014-advance-20261111-r05-catalan-numbers-foundations) - [Derivation](/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0058/) / [Markdown](/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0058/index.md) / [JSON](/data/question-derivations/questions/q-20261014-advance-20261111-r05-catalan-numbers-0058.json) ## Question 22759: The Recurrence and Both Closed Forms Use the Same Zero-Based Index - ID: `q-20261014-advance-20261111-r05-catalan-numbers-0059`; number: 22759 - Processing: second_context_reviewed - MKO: [mko-catalan-numbers](/en/mko/mko-catalan-numbers/) - Family: [qf-mod42-catalan-recurrence-closed-cross-check](/en/explore/?family=qf-mod42-catalan-recurrence-closed-cross-check) - Batch: [batch-20261014-advance-20261111-r05-catalan-numbers-foundations](/en/explore/?batch=batch-20261014-advance-20261111-r05-catalan-numbers-foundations) - [Derivation](/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0059/) / [Markdown](/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0059/index.md) / [JSON](/data/question-derivations/questions/q-20261014-advance-20261111-r05-catalan-numbers-0059.json) ## Question 22760: The Recurrence and Both Closed Forms Use the Same Zero-Based Index - ID: `q-20261014-advance-20261111-r05-catalan-numbers-0060`; number: 22760 - Processing: second_context_reviewed - MKO: [mko-catalan-numbers](/en/mko/mko-catalan-numbers/) - Family: [qf-mod42-catalan-recurrence-closed-cross-check](/en/explore/?family=qf-mod42-catalan-recurrence-closed-cross-check) - Batch: [batch-20261014-advance-20261111-r05-catalan-numbers-foundations](/en/explore/?batch=batch-20261014-advance-20261111-r05-catalan-numbers-foundations) - [Derivation](/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0060/) / [Markdown](/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0060/index.md) / [JSON](/data/question-derivations/questions/q-20261014-advance-20261111-r05-catalan-numbers-0060.json) ## Question 22761: The Recurrence and Both Closed Forms Use the Same Zero-Based Index - ID: `q-20261014-advance-20261111-r05-catalan-numbers-0061`; number: 22761 - Processing: second_context_reviewed - MKO: [mko-catalan-numbers](/en/mko/mko-catalan-numbers/) - Family: [qf-mod42-catalan-recurrence-closed-cross-check](/en/explore/?family=qf-mod42-catalan-recurrence-closed-cross-check) - Batch: [batch-20261014-advance-20261111-r05-catalan-numbers-foundations](/en/explore/?batch=batch-20261014-advance-20261111-r05-catalan-numbers-foundations) - [Derivation](/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0061/) / [Markdown](/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0061/index.md) / [JSON](/data/question-derivations/questions/q-20261014-advance-20261111-r05-catalan-numbers-0061.json) ## Question 22762: The Recurrence and Both Closed Forms Use the Same Zero-Based Index - ID: `q-20261014-advance-20261111-r05-catalan-numbers-0062`; number: 22762 - Processing: second_context_reviewed - MKO: [mko-catalan-numbers](/en/mko/mko-catalan-numbers/) - Family: [qf-mod42-catalan-recurrence-closed-cross-check](/en/explore/?family=qf-mod42-catalan-recurrence-closed-cross-check) - Batch: [batch-20261014-advance-20261111-r05-catalan-numbers-foundations](/en/explore/?batch=batch-20261014-advance-20261111-r05-catalan-numbers-foundations) - [Derivation](/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0062/) / [Markdown](/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0062/index.md) / [JSON](/data/question-derivations/questions/q-20261014-advance-20261111-r05-catalan-numbers-0062.json) ## Question 22763: The Recurrence and Both Closed Forms Use the Same Zero-Based Index - ID: `q-20261014-advance-20261111-r05-catalan-numbers-0063`; number: 22763 - Processing: second_context_reviewed - MKO: [mko-catalan-numbers](/en/mko/mko-catalan-numbers/) - Family: [qf-mod42-catalan-recurrence-closed-cross-check](/en/explore/?family=qf-mod42-catalan-recurrence-closed-cross-check) - Batch: [batch-20261014-advance-20261111-r05-catalan-numbers-foundations](/en/explore/?batch=batch-20261014-advance-20261111-r05-catalan-numbers-foundations) - [Derivation](/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0063/) / [Markdown](/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0063/index.md) / [JSON](/data/question-derivations/questions/q-20261014-advance-20261111-r05-catalan-numbers-0063.json) ## Question 22764: The Recurrence and Both Closed Forms Use the Same Zero-Based Index - ID: `q-20261014-advance-20261111-r05-catalan-numbers-0064`; number: 22764 - Processing: second_context_reviewed - MKO: [mko-catalan-numbers](/en/mko/mko-catalan-numbers/) - Family: [qf-mod42-catalan-recurrence-closed-cross-check](/en/explore/?family=qf-mod42-catalan-recurrence-closed-cross-check) - Batch: [batch-20261014-advance-20261111-r05-catalan-numbers-foundations](/en/explore/?batch=batch-20261014-advance-20261111-r05-catalan-numbers-foundations) - [Derivation](/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0064/) / [Markdown](/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0064/index.md) / [JSON](/data/question-derivations/questions/q-20261014-advance-20261111-r05-catalan-numbers-0064.json) ## Question 22765: The Recurrence and Both Closed Forms Use the Same Zero-Based Index - ID: `q-20261014-advance-20261111-r05-catalan-numbers-0065`; number: 22765 - Processing: second_context_reviewed - MKO: [mko-catalan-numbers](/en/mko/mko-catalan-numbers/) - Family: [qf-mod42-catalan-recurrence-closed-cross-check](/en/explore/?family=qf-mod42-catalan-recurrence-closed-cross-check) - Batch: [batch-20261014-advance-20261111-r05-catalan-numbers-foundations](/en/explore/?batch=batch-20261014-advance-20261111-r05-catalan-numbers-foundations) - [Derivation](/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0065/) / [Markdown](/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0065/index.md) / [JSON](/data/question-derivations/questions/q-20261014-advance-20261111-r05-catalan-numbers-0065.json) ## Question 22766: The Recurrence and Both Closed Forms Use the Same Zero-Based Index - ID: `q-20261014-advance-20261111-r05-catalan-numbers-0066`; number: 22766 - Processing: second_context_reviewed - MKO: [mko-catalan-numbers](/en/mko/mko-catalan-numbers/) - Family: [qf-mod42-catalan-recurrence-closed-cross-check](/en/explore/?family=qf-mod42-catalan-recurrence-closed-cross-check) - Batch: [batch-20261014-advance-20261111-r05-catalan-numbers-foundations](/en/explore/?batch=batch-20261014-advance-20261111-r05-catalan-numbers-foundations) - [Derivation](/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0066/) / [Markdown](/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0066/index.md) / [JSON](/data/question-derivations/questions/q-20261014-advance-20261111-r05-catalan-numbers-0066.json) ## Question 22767: The Recurrence and Both Closed Forms Use the Same Zero-Based Index - ID: `q-20261014-advance-20261111-r05-catalan-numbers-0067`; number: 22767 - Processing: second_context_reviewed - MKO: [mko-catalan-numbers](/en/mko/mko-catalan-numbers/) - Family: [qf-mod42-catalan-recurrence-closed-cross-check](/en/explore/?family=qf-mod42-catalan-recurrence-closed-cross-check) - Batch: [batch-20261014-advance-20261111-r05-catalan-numbers-foundations](/en/explore/?batch=batch-20261014-advance-20261111-r05-catalan-numbers-foundations) - [Derivation](/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0067/) / [Markdown](/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0067/index.md) / [JSON](/data/question-derivations/questions/q-20261014-advance-20261111-r05-catalan-numbers-0067.json) ## Question 22768: The Recurrence and Both Closed Forms Use the Same Zero-Based Index - ID: `q-20261014-advance-20261111-r05-catalan-numbers-0068`; number: 22768 - Processing: second_context_reviewed - MKO: [mko-catalan-numbers](/en/mko/mko-catalan-numbers/) - Family: [qf-mod42-catalan-recurrence-closed-cross-check](/en/explore/?family=qf-mod42-catalan-recurrence-closed-cross-check) - Batch: [batch-20261014-advance-20261111-r05-catalan-numbers-foundations](/en/explore/?batch=batch-20261014-advance-20261111-r05-catalan-numbers-foundations) - [Derivation](/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0068/) / [Markdown](/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0068/index.md) / [JSON](/data/question-derivations/questions/q-20261014-advance-20261111-r05-catalan-numbers-0068.json) ## Question 22769: The Recurrence and Both Closed Forms Use the Same Zero-Based Index - ID: `q-20261014-advance-20261111-r05-catalan-numbers-0069`; number: 22769 - Processing: second_context_reviewed - MKO: [mko-catalan-numbers](/en/mko/mko-catalan-numbers/) - Family: [qf-mod42-catalan-recurrence-closed-cross-check](/en/explore/?family=qf-mod42-catalan-recurrence-closed-cross-check) - Batch: [batch-20261014-advance-20261111-r05-catalan-numbers-foundations](/en/explore/?batch=batch-20261014-advance-20261111-r05-catalan-numbers-foundations) - [Derivation](/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0069/) / [Markdown](/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0069/index.md) / [JSON](/data/question-derivations/questions/q-20261014-advance-20261111-r05-catalan-numbers-0069.json) ## Question 22770: The Recurrence and Both Closed Forms Use the Same Zero-Based Index - ID: `q-20261014-advance-20261111-r05-catalan-numbers-0070`; number: 22770 - Processing: second_context_reviewed - MKO: [mko-catalan-numbers](/en/mko/mko-catalan-numbers/) - Family: [qf-mod42-catalan-recurrence-closed-cross-check](/en/explore/?family=qf-mod42-catalan-recurrence-closed-cross-check) - Batch: [batch-20261014-advance-20261111-r05-catalan-numbers-foundations](/en/explore/?batch=batch-20261014-advance-20261111-r05-catalan-numbers-foundations) - [Derivation](/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0070/) / [Markdown](/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0070/index.md) / [JSON](/data/question-derivations/questions/q-20261014-advance-20261111-r05-catalan-numbers-0070.json) ## Question 22771: The Recurrence and Both Closed Forms Use the Same Zero-Based Index - ID: `q-20261014-advance-20261111-r05-catalan-numbers-0071`; number: 22771 - Processing: second_context_reviewed - MKO: [mko-catalan-numbers](/en/mko/mko-catalan-numbers/) - Family: [qf-mod42-catalan-recurrence-closed-cross-check](/en/explore/?family=qf-mod42-catalan-recurrence-closed-cross-check) - Batch: [batch-20261014-advance-20261111-r05-catalan-numbers-foundations](/en/explore/?batch=batch-20261014-advance-20261111-r05-catalan-numbers-foundations) - [Derivation](/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0071/) / [Markdown](/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0071/index.md) / [JSON](/data/question-derivations/questions/q-20261014-advance-20261111-r05-catalan-numbers-0071.json) ## Question 22772: The Recurrence and Both Closed Forms Use the Same Zero-Based Index - ID: `q-20261014-advance-20261111-r05-catalan-numbers-0072`; number: 22772 - Processing: second_context_reviewed - MKO: [mko-catalan-numbers](/en/mko/mko-catalan-numbers/) - Family: [qf-mod42-catalan-recurrence-closed-cross-check](/en/explore/?family=qf-mod42-catalan-recurrence-closed-cross-check) - Batch: [batch-20261014-advance-20261111-r05-catalan-numbers-foundations](/en/explore/?batch=batch-20261014-advance-20261111-r05-catalan-numbers-foundations) - [Derivation](/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0072/) / [Markdown](/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0072/index.md) / [JSON](/data/question-derivations/questions/q-20261014-advance-20261111-r05-catalan-numbers-0072.json) ## Question 22773: The Recurrence and Both Closed Forms Use the Same Zero-Based Index - ID: `q-20261014-advance-20261111-r05-catalan-numbers-0073`; number: 22773 - Processing: second_context_reviewed - MKO: [mko-catalan-numbers](/en/mko/mko-catalan-numbers/) - Family: [qf-mod42-catalan-recurrence-closed-cross-check](/en/explore/?family=qf-mod42-catalan-recurrence-closed-cross-check) - Batch: [batch-20261014-advance-20261111-r05-catalan-numbers-foundations](/en/explore/?batch=batch-20261014-advance-20261111-r05-catalan-numbers-foundations) - [Derivation](/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0073/) / [Markdown](/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0073/index.md) / [JSON](/data/question-derivations/questions/q-20261014-advance-20261111-r05-catalan-numbers-0073.json) ## Question 22774: The Recurrence and Both Closed Forms Use the Same Zero-Based Index - ID: `q-20261014-advance-20261111-r05-catalan-numbers-0074`; number: 22774 - Processing: second_context_reviewed - MKO: [mko-catalan-numbers](/en/mko/mko-catalan-numbers/) - Family: [qf-mod42-catalan-recurrence-closed-cross-check](/en/explore/?family=qf-mod42-catalan-recurrence-closed-cross-check) - Batch: [batch-20261014-advance-20261111-r05-catalan-numbers-foundations](/en/explore/?batch=batch-20261014-advance-20261111-r05-catalan-numbers-foundations) - [Derivation](/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0074/) / [Markdown](/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0074/index.md) / [JSON](/data/question-derivations/questions/q-20261014-advance-20261111-r05-catalan-numbers-0074.json) ## Question 22775: The Recurrence and Both Closed Forms Use the Same Zero-Based Index - ID: `q-20261014-advance-20261111-r05-catalan-numbers-0075`; number: 22775 - Processing: second_context_reviewed - MKO: [mko-catalan-numbers](/en/mko/mko-catalan-numbers/) - Family: [qf-mod42-catalan-recurrence-closed-cross-check](/en/explore/?family=qf-mod42-catalan-recurrence-closed-cross-check) - Batch: [batch-20261014-advance-20261111-r05-catalan-numbers-foundations](/en/explore/?batch=batch-20261014-advance-20261111-r05-catalan-numbers-foundations) - [Derivation](/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0075/) / [Markdown](/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0075/index.md) / [JSON](/data/question-derivations/questions/q-20261014-advance-20261111-r05-catalan-numbers-0075.json) [Next](/en/explore/index.md?mko=mko-catalan-numbers&page=4)