Deduction process:
1+2+3+ ... the numbers of these triangles. If this is developed into three dimensions, it is a problem of a triangular crib. The number of stacks is1+(1+2)+(1+2+3)+...+(1+2+3+...+n).
For1+(1+2)+(1+2+3)+...+(1+2+3+...+n), please refer to the following figure. Take n = 4 as an example, and arrange1+(1+2)+(1+2+3)+(1+2+3+4) into a triangle. Then rotate this triangle by 120 degrees to get two other triangles with these numbers.
Accumulate the corresponding positions, and the sum of each position is 6, so the sum of the three rotated numbers is 6*( 1+2+3+4), so the sum of a pile is (4*5*6)/(2*3). When this is extended to layer N, the formula N (n+1) (n+2)/(3 * 2 *1) can be obtained.
Stacking:
Shen Kuo, a great scientist in the Northern Song Dynasty, initiated the "gap product technique" in Meng Qian Bi Tan, and began to study the problem of finding the total number of certain items piled up according to certain rules, that is, the sum of high-order arithmetic progression, and calculated the rectangular formula of platform stack.
Yang Hui, a mathematician in the Southern Song Dynasty, enriched and developed the achievements of Shen Kuo's gap product method in Nine Chapters' Detailed Explanation of Algorithms and Origin and Variation of Algorithms, and put forward some new superposition formulas.
The sequence discussed by Shen Kuo and Yang Hui is different from the general arithmetic progression. The difference between the two terms is not equal, but the difference of item by item or the difference of higher order is equal. This kind of high-order arithmetic progression's research is generally called "piling up" after Yang Hui.
Shen Kuo's gap product and Yang Hui's accumulation product;
Gap products in Shen Kuo:
In the Northern Song Dynasty, Shen Kuo initiated the skill of gap product in Meng Qian Bi Tan, Volume 18, Skills: gap product, tired chess player, people who are in the altar, and unscrupulous people.
Although it looks like a fight, it is killing on all sides, and there are gaps and gaps on the side. If you want it in a child's way, you will often lose a few. I thought about it and understood that I used the method of fighting for children as the upper position; Lower order: the lower order is widely reduced, the rest is multiplied by the upper order, and the six are combined into the upper order. Pseudomonas piled up poppies: there are two poppies in the top row and twelve poppies in the bottom row, with the same number of rows. Comparing the first two lines, the rate is twelve, and it is eleven lines.
Ask in a child's way, four times as long as the upper platoon leader, sixteen times as long as the lower platoon leader, and thirty-two times as long as the upper platoon leader; If you multiply the length of downline by 24, you will get 26; if you multiply it by downline, you will get 312; And the two got 344, multiplied by the height, and got 3,784.
Re-list the following 12, generalize the upper one, multiply the remaining 10 by the height to get 1 10, and merge them to the upper position to get 3894; Six plus one, you get 649, which is the number of traders. Children seek to see the real products, while gap products seek to see endless angles, which is conducive to envy products.
Yang Hui stacking technique:
Yang Hui elaborated on square crib, haystack, haystack and triangular crib in the chapters of Detailed Explanation of Algorithms in Nine Chapters and Commerce.