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Quadrangle was playing number theory games when his mother called.

"Quaddy!" she shouted from the counting room. "Head down to the Numeraland Grocery Store and get me some new numbers! Our primes are getting rotten and the perfect squares aren't much better."

Quadrangle sighed and headed downstairs, grabbing the grocery list on his way out, not bothering to look at it. He hopped on his pentacycle and pedaled down the street, which was parallel in a non-Euclidean way with the adjacent street, so eventually he crossed over and found himself at the Numeraland Grocery Store.

He ditched his bike in the Klein bottle storage receptacle and headed inside. He glanced around at the intimidating aisles. "So many sets!" he muttered. "And not enough intersection." He passed right by the Irrational Numbers section (his family weren't big pi fans) and the Imaginary Aisle (his little sister was already part of the new so-called 'imaginary generation', she didn't need any more imaginary influences in the house), as well as the pesky Infinitesimals row (you had to be of at least Hyperreal status to gain access to those after the notorious Epsilon Incident).

Finally, the familiar Natural Numbers section: fond childhood memories rolled over him, before the days of the discovery of Zero and the Negatives, known as 'puberty'.

He browsed the aisle absentmindedly until remembering the grocery list. His mother was making a very specific function for dinner tonight, and all the ingredients fit together like a pattern. Quadrangle read them off out loud:

"Zero and One," he started. "Of course. All series start with those. Duh. So the next one is 'negative one'. Uh-oh, I'll need to find the Negatives Freezer for that. The fourth item is two. Ah, that's right over here, perfect! Then -1 again! Yikes! What is mom making tonight? Now what? 5... that shouldn't be too hard. Then -4, then 29, then -13. Then...... what's that supposed to be? The last number is smudged out! Gah!"

He thought for a moment, remembering that it formed a sensible series. "Aha!" he said at last. When he got home on his pentacycle, he had all of the ingredients for his mother to make a delicious Whole-Number-Grain Wheat Bread & Soup. What was the last number that Quadrangle bought?

Edited by unreality
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Quadrangle was playing number theory games when his mother called.

"Quaddy!" she shouted from the counting room. "Head down to the Numeraland Grocery Store and get me some new numbers! Our primes are getting rotten and the perfect squares aren't much better."

Quadrangle sighed and headed downstairs, grabbing the grocery list on his way out, not bothering to look at it. He hopped on his pentacycle and pedaled down the street, which was parallel in a non-Euclidean way with the adjacent street, so eventually he crossed over and found himself at the Numeraland Grocery Store.

He ditched his bike in the Klein bottle storage receptacle and headed inside. He glanced around at the intimidating aisles. "So many sets!" he muttered. "And not enough intersection." He passed right by the Irrational Numbers section (his family weren't big pi fans) and the Imaginary Aisle (his little sister was already part of the new so-called 'imaginary generation', she didn't need any more imaginary influences in the house), as well as the pesky Infinitesimals row (you had to be of at least Hyperreal status to gain access to those after the notorious Epsilon Incident).

Finally, the familiar Natural Numbers section: fond childhood memories rolled over him, before the days of the discovery of Zero and the Negatives, known as 'puberty'.

He browsed the aisle absentmindedly until remembering the grocery list. His mother was making a very specific function for dinner tonight, and all the ingredients fit together like a pattern. Quadrangle read them off out loud:

"Zero and One," he started. "Of course. All series start with those. Duh. So the next one is 'negative one'. Uh-oh, I'll need to find the Negatives Freezer for that. The fourth item is two. Ah, that's right over here, perfect! Then -1 again! Yikes! What is mom making tonight? Now what? 5... that shouldn't be too hard. Then -4, then 29, then -13. Then...... what's that supposed to be? The last number is smudged out! Gah!"

He thought for a moment, remembering that it formed a sensible series. "Aha!" he said at last. When he got home on his pentacycle, he had all of the ingredients for his mother to make a delicious Whole-Number-Grain Wheat Bread & Soup. What was the last number that Quadrangle bought?

sequence = 0, 1, -1, 2, -1, 5, -4, 29, -13, ?

so.... 0-1=-1, 1 - -1 (1+1) = 2, -1 + 2 = -1, but now I'm stuck... maybe this is a hint to other solvers...

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S(0)=0, S(1)=1 are given ("Zero and One," he started. "Of course. All series start with those. Duh.)

then for n>1,

S(n)=S(n-2)*S(n-2)-S(n-1)

Thus, making the final ingredient = 29*29-(-13) = 854.

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