Truncated Tetrahedral Number

A figurate number constructed by taking the th tetrahedral number and removing the th tetrahedral number from each of the four corners,

			(1)
			(2)

The first few are 1, 16, 68, 180, 375, ... (Sloane's A005906). The generating function for the truncated tetrahedral numbers is

(3)

Ref: mathworld.wolfram.com

Gnomonic Number

A figurate number of the form  giving the area of the square gnomon obtained by removing a square of side  from a square of side ,

			(1)
			(2)

The gnomonic numbers are therefore equivalent to the odd numbers, and the first few are 1, 3, 5, 7, 9, 11, ... (Sloane's A005408). The generating function for the gnomonic numbers is

Ref: mathworld.wolfram.com

(3)

Decagonal Number

A figurate number of the form . The first few are 1, 10, 27, 52, 85, ... (Sloane's A001107). The generating function giving the decagonal numbers is

The first few odd decagonal numbers are 1, 27, 85, 175, 297, ... (Sloane's A028993), and the first few even decagonal numbers are 10, 52, 126, 232, 370, 540, ... (Sloane's A028994).

Ref: mathworld.wolfram.com

Centered Triangular Number

A centered triangular number is a centered polygonal number consisting of a central dot with three dots around it, and then additional dots in the gaps between adjacent dots. The th term is , and the first few such numbers for , 1, 2, ... are 1, 4, 10, 19, 31, 46, 64, ... (Sloane's A005448). The generating function giving the centered triangular numbers is

Ref: mathworld.wolfram.com

Centered Square Number

A centered polygonal number consisting of a central dot with four dots around it, and then additional dots in the gaps between adjacent dots. The general term is , and the first few such numbers are 1, 5, 13, 25, 41, ... (Sloane's A001844). Centered square numbers are the sum of two consecutive square numbers and are congruent to 1 (mod 4). The generating function giving the centered square numbers is

Ref: mathworld.wolfram.com

Centered Pentagonal Number

A centered polygonal number consisting of a central dot with five dots around it, and then additional dots in the gaps between adjacent dots. The general term is , and the first few such numbers are 1, 6, 16, 31, 51, 76, ... (Sloane's A005891). The generating function of the centered pentagonal numbers is

Ref: mathworld.wolfram.com

Centered Cube Number

A figurate number of the form,

The first few are 1, 9, 35, 91, 189, 341, ... (Sloane's A005898). The generating function for the centered cube numbers is

Ref: mathworld.wolfram.com

Biquadratic Number

A biquadratic number is a fourth power, . The first few biquadratic numbers are 1, 16, 81, 256, 625, ... (Sloane's A000583). The minimum number of biquadratic numbers needed to represent the numbers 1, 2, 3, ... are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 1, 2, 3, 4, 5, ... (Sloane's A002377), and the number of distinct ways to represent the numbers 1, 2, 3, ... in terms of biquadratic numbers are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, .... A brute-force algorithm for enumerating the biquadratic permutations of  is repeated application of the greedy algorithm.

Every positive integer is expressible as a sum of (at most)  biquadratic numbers (Waring's problem). Davenport (1939) showed that , meaning that all sufficiently large integers require only 16 biquadratic numbers. It is also known that every integer is a sum of at most 10 signed biquadrates (; although it is not known if 10 can be reduced to 9). The following table gives the first few numbers which require 1, 2, 3, ..., 19 biquadratic numbers to represent them as a sum, with the sequences for 17, 18, and 19 being finite.

#	Sloane	numbers
1	A000290	1, 16, 81, 256, 625, 1296, 2401, 4096, ...
2	A003336	2, 17, 32, 82, 97, 162, 257, 272, ...
3	A003337	3, 18, 33, 48, 83, 98, 113, 163, ...
4	A003338	4, 19, 34, 49, 64, 84, 99, 114, 129, ...
5	A003339	5, 20, 35, 50, 65, 80, 85, 100, 115, ...
6	A003340	6, 21, 36, 51, 66, 86, 96, 101, 116, ...
7	A003341	7, 22, 37, 52, 67, 87, 102, 112, 117, ...
8	A003342	8, 23, 38, 53, 68, 88, 103, 118, 128, ...
9	A003343	9, 24, 39, 54, 69, 89, 104, 119, 134, ...
10	A003344	10, 25, 40, 55, 70, 90, 105, 120, 135, ...
11	A003345	11, 26, 41, 56, 71, 91, 106, 121, 136, ...
12	A003346	12, 27, 42, 57, 72, 92, 107, 122, 137, ...
13	A046044	13, 28, 43, 58, 73, 93, 108, 123, 138, ...
14	A046045	14, 29, 44, 59, 74, 94, 109, 124, 139, ...
15	A046046	15, 30, 45, 60, 75, 95, 110, 125, 140, ...
16	A046047	31, 46, 61, 76, 111, 126, 141, 156, ...
17	A046048	47, 62, 77, 127, 142, 157, 207, 222, ...
18	A046049	63, 78, 143, 158, 223, 238, 303, 318, ...
19	A046050	79, 159, 239, 319, 399

The following table gives the numbers which can be represented in  different ways as a sum of  biquadrates.

		Sloane	numbers
1	1	A000290	1, 16, 81, 256, 625, 1296, 2401, 4096, ...
2	2	A018786	635318657, 3262811042, 8657437697, ...

The numbers 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 18, 19, 20, 21, ... (Sloane's A046039) cannot be represented using distinct biquadrates.

Ref: mathworld.wolfram.com

Figurate Number

A figurate number, also (but mostly in texts from the 1500 and 1600s) known as a figural number (Simpson and Weiner 1992, p. 587), is a number that can be represented by a regular geometrical arrangement of equally spaced points. If the arrangement forms a regular polygon, the number is called a polygonal number. The polygonal numbers illustrated above are called triangular, square, pentagonal, and hexagonal numbers, respectively. Figurate numbers can also form other shapes such as centered polygons, L-shapes, three-dimensional solids, etc.

The th regular -polytopic number is given by

			(1)
			(2)
			(3)

where  is the multichoose function,  is a binomial coefficient, and  is a rising factorial. Special cases therefore include the triangular numbers

(4)

tetrahedral numbers

(5)

pentatope numbers

(6)

and so on (Dickson 2005, p. 7).

The following table lists the most common types of figurate numbers.

figurate number	formula
biquadratic number
centered cube number
centered pentagonal number
centered square number
centered triangular number
cubic number
decagonal number
gnomonic number
Haűy octahedral number
Haűy rhombic dodecahedral number
heptagonal number
hex number
heptagonal pyramidal number
hexagonal number
hexagonal pyramidal number
octagonal number
octahedral number
pentagonal number
pentagonal pyramidal number
pentatope number
pronic number
rhombic dodecahedral number
square number
square pyramidal number
stella octangula number
tetrahedral number
triangular number
truncated octahedral number
truncated tetrahedral number

Ref: mathworld.wolfram.com