Yang Hui Triangle, also known as Jia Xian Triangle, Pascal Triangle and Pascal Triangle, is a geometric arrangement of binomial coefficients in triangles. Yang Hui's triangle also corresponds to the coefficient of binomial theorem. The binomial coefficient of degree n corresponds to the n+ 1 line of Yang Hui triangle. For example, the picture reference: upload.wikimedia/Math/2/d/4/2D4ED0ECCF4a35668a436fb1620b, and the quadratic binomial just corresponds to the coefficient of the third line of Pascal's triangle 1 2 1. Image reference: upload.wikimedia/ * * */Mons/Thumb /e/EA/ Yang Hui _ Triangle /200 px- Yang Hui _ Triangle Image reference: zh. * * */Skins-1.5/Mon/Images/Magnify-Clip Yang Hui's Painting Seven Paintings of Ancient Law1121331464/Kloc. The first six lines of the triangle are symmetrical, and the number of each line gradually increases from 1, then decreases and returns to 1. The number of numbers in the nth row is n. The sum of the numbers in the nth row is 2n? 1。 Each number is equal to the sum of the left and right numbers in the previous line. (because of the picture reference: upload.wikimedia/math/2/6/d/26D3ad31268a0d2187bb85aff853d). This property can be used to write the whole Pascal triangle. Connect the number 1 in the 2n+ 1 line with the number 3 in the 2n+2 line and the number 5 in the 2n+2 line, and the sum of these numbers is 2n Fibonacci number. The sum of the number 2 in 2n row, the number 4 in 2n+ 1 row and the number 6 in 2n+2 row is 2n- 1 Fibonacci number. [Editor] The earliest record of this triangle in history was in ancient India. Indian mathematician Bingaro mentioned this "staircase of Mount Sumi" in his Sanskrit poetry anthology (about 450 years). He also pointed out the relationship between Fibonacci sequence and this triangle. Karadji, a Persian mathematician, and omar khayyam, an astronomer and poet, both discovered this triangle, and Karadji also knew that we could use this triangle to find the root of n degree and its relationship with binomial. In Iran, this triangle is called "khayyam Triangle". Italians call it the "tartaglia Triangle" to commemorate Tattaglia's discovery of the solution of a cubic equation. /kloc-in the 3rd century, Yang Hui, a mathematician in the Song Dynasty, discussed this numerical table form in Nine Chapters of Arithmetic, explaining that this table was quoted from Jia Xian's Unlocking Arithmetic, and drew the Seven Square Diagram of Ancient Method. Blaise Pascal's book The Arithmetic Trajectory of Triangle (1655) introduced this triangle. Pascal collected several results about it, solved some problems in probability theory and had a wide influence. Pierre Raymond demont (1708) and Abraham de Moivre (1730) both called this triangle Pascal. Mathematicians who have drawn this picture in history: Yang Hui's contribution recorded in Nine Chapters' Detailed Explanation of Algorithms in Southern Song Dynasty 126 1 Zhu Shijie Yuan Dynasty 1299' Meet Siyuan' series summation formula Al Cassie * * *. 1427 The Key to Arithmetic Appian Germany 1527 Stifel Germany 1544 Comprehensive Arithmetic Binomial Expansion Coefficient Chebel France 1545b Pascal France 1654 On Arithmetic Triangle [Editor] The number of times a number appears in Yang Hui Triangle is 65438.
1
2
2
2
three
2
2
2
four
2
2
2
2
four
... (OEIS:A0030 16). The minimum number that is greater than 1 and appears at least n times in Yang Hui triangle is 2.
three
six
10
120
120
3003
3003
... (OEIS:A062527) Except 1, all positive integers appear a finite number of times. Only 2 occurs exactly 1 time. six
20
70 and so on. Appeared three times. There are many people who appear twice and four times. I haven't found the number that appears exactly five times. 120
2 10
1540 and so on only appeared six times. (OEIS:A098565) Because the Diophantine equation picture reference: upload.wikimedia/Math/0/3/8/03862 AF4F4 A8C7 392 Babb8B00846EF2 has infinite solutions [1], and there are infinite numbers that appear at least 6 times. The answer is the picture reference: upload.wikimedia/math/4/6/2/462e3ff167664d1c8028b715a5369 picture reference: upload.wikimedia/math/b/4/9/.b49f3/ Kloc-0/aa384ad944cb77db4d2816562b where Fn stands for the nth Fibonacci number (f 6543333003 is the first number to appear eight times. [Editor] Reference = Singmaster
David
"Repetitive Binomial Coefficient and Fibonacci Number"
Fibonacci quarterly
Volume 13
fourth
Page 296-298
1975.
Reference: zh. *** /w/index？ Title =% E6% 9d% A8% E8% Be% 89% E4% B8% 89% E8% A7% 92% E5% BD% A2&; Variant =zh-