Hexagonal number

src: i.ytimg.com

hexagonal number is the number of figurates. The hexagonal number n h _n is the number of different points in the dot pattern consisting of the outline from the usual hexagon with the sides up to n points, when the hexagon is stacked so that they share a single point.

Rumus untuk angka hexagonal n

${\ displaystyle h_ {n} = 2n2 -n = n (2n-1) = {{2n} \ kali {(2n-1) )} \ over 2}. \, \!} Â Â$

Some of the first hexagonal numbers (sequence A000384 in OEIS) are:

1, 6, 15, 28, 45, 66, 91, 120, 153, 190, 231, 276, 325, 378, 435, 496, 561, 630, 703, 780, 861, 946.

Every hexagonal number is a triangular number, but only every triangular number other (1, 3, 5, 7, etc.) is a hexagonal number. Like the number of triangles, the digital roots in base 10 of the hexagonal numbers can only be 1, 3, 6, or 9. The digital root pattern, repeating every nine quarters, is "1 6 6 1 9 3 1 3 9".

Setiap angka yang sempalana adalah heksagonal, yang diberikan oleh rumus

${\ displaystyle M {p} 2 ^ {p-1} = M {{p} (M {p} 1)/2 = h _ {(M_ {p} 1)/2} = h_ {2 ^ {p-1}}} Â Â$

di mana M _p adalah Perdana Mersenne. Tidak ada angka sempurna aneh yang diketahui, maka semua angka sempurna yang diketahui adalah heksagonal.

Misalnya, angka hexagonal ke-2 adalah 2ÃÆ' â € "3 = 6; yang ke-4 adalah 4ÃÆ' â € "7 = 28; 16 adalah 16ÃÆ' â € "31 = 496; dan yang ke 64 adalah 64ÃÆ' â € "127 = 8128.

The largest number that can not be written as the sum of at most four hexagonal numbers is 130. Adrien-Marie Legendre proves in 1830 that any integer greater than 1791 can be expressed in this way.

The hexagonal numbers can be rearranged into rectangles with sizes n by (2 n -1).

Hexagonal numbers should not be equated with centralized hexagonal numbers, which model Viennese sausage standard packaging. To avoid ambiguity, hexagonal numbers are sometimes called "hexagonal numbers cornered".

Video Hexagonal number

Test for hexagonal numbers

Seseorang dapat secara efisien menguji apakah bilangan bulat positif x adalah angka heksagonal oleh computputasi

${\ displaystyle n = {\ frac {{\ sqrt {8x 1}} 1} {4}}.} Â Â$

If n is an integer, then x is the hexagonal number n . If n is not an integer, then x is not hexagonal.

Maps Hexagonal number

Other properties

Expression using sigma notation

The n nomor th urutan heksagonal juga dapat dinyatakan dengan menggunakan notasi Sigma sebagai

${\ displaystyle h_ {n} = \ jumlah _ {i = 0} ^ {n-1} {(4i 1)}} Â Â$

where the empty amount is taken to 0.

Number of inversions of hexagonal numbers

Source of the article : Wikipedia