Decagonal number

A decagonal number is a figurate number that extends the concept of triangular and square numbers to the decagon (a ten-sided polygon). However, unlike the triangular and square numbers, the patterns involved in the construction of decagonal numbers are not rotationally symmetrical. Specifically, the nth decagonal numbers counts the number of dots in a pattern of n nested decagons, all sharing a common corner, where the ith decagon in the pattern has sides made of i dots spaced one unit apart from each other. The n-th decagonal number is given by the formula

Dn=4n2−3n.displaystyle D_n=4n^2-3n.

The first few decagonal numbers are:

0, 1, 10, 27, 52, 85, 126, 175, 232, 297, 370, 451, 540, 637, 742, 855, 976, 1105, 1242, 1387, 1540, 1701, 1870, 2047, 2232, 2425, 2626, 2835, 3052, 3277, 3510, 3751, 4000, 4257, 4522, 4795, 5076, 5365, 5662, 5967, 6280, 6601, 6930, 7267, 7612, 7965, 8326 (sequence A001107 in the OEIS)

The n-th decagonal number can also be calculated by adding the square of n to thrice the (n—1)-th pronic number or, to put it algebraically, as

Dn=n2+3(n2−n).displaystyle D_n=n^2+3(n^2-n).

Properties

• Decagonal numbers consistently alternate parity.

v t e Classes of natural numbers
Powers and related numbers	Achilles Power of 2 Power of 10 Square Cube Fourth power Fifth power Sixth power Seventh power Perfect power Powerful Prime power
Of the form a × 2b ± 1	Cullen Double Mersenne Fermat Mersenne Proth Thabit Woodall
Other polynomial numbers	Carol Hilbert Idoneal Kynea Leyland Loeschian Lucky numbers of Euler Repunit
Recursively defined numbers	Fibonacci Jacobsthal Leonardo Lucas Padovan Pell Perrin
Possessing a specific set of other numbers	Knödel Riesel Sierpinski
Expressible via specific sums	Nonhypotenuse Polite Practical Primary pseudoperfect Ulam Wolstenholme
Generated via a sieve	Lucky
Code related	Meertens
Figurate numbers	2-dimensional centered Centered triangular Centered square Centered pentagonal Centered hexagonal Centered heptagonal Centered octagonal Centered nonagonal Centered decagonal Star non-centered Triangular Square Square triangular Pentagonal Hexagonal Heptagonal Octagonal Nonagonal Decagonal Dodecagonal 3-dimensional centered Centered tetrahedral Centered cube Centered octahedral Centered dodecahedral Centered icosahedral non-centered Tetrahedral Octahedral Dodecahedral Icosahedral Stella octangula pyramidal Square pyramidal Pentagonal pyramidal Hexagonal pyramidal Heptagonal pyramidal 4-dimensional centered Centered pentachoric Squared triangular non-centered Pentatope
Pseudoprimes	Carmichael number Catalan pseudoprime Elliptic pseudoprime Euler pseudoprime Euler–Jacobi pseudoprime Fermat pseudoprime Frobenius pseudoprime Lucas pseudoprime Somer–Lucas pseudoprime Strong pseudoprime
Combinatorial numbers	Bell Cake Catalan Dedekind Delannoy Euler Fuss–Catalan Lazy caterer's sequence Lobb Motzkin Narayana Ordered Bell Schröder Schröder–Hipparchus
Arithmetic functions	By properties of σ(n) Abundant Almost perfect Arithmetic Colossally abundant Descartes Hemiperfect Highly abundant Highly composite Hyperperfect Multiply perfect Perfect Practical number Primitive abundant Quasiperfect Refactorable Sublime Superabundant Superior highly composite Superperfect Untouchable By properties of Ω(n) Almost prime Semiprime By properties of φ(n) Highly cototient Highly totient Noncototient Nontotient Perfect totient Sparsely totient By properties of s(n) Amicable Betrothed Deficient Semiperfect Euclid Fortunate
Dividing a quotient	Wieferich Wall–Sun–Sun Wolstenholme prime Wilson
Other prime factor or divisor related numbers	Blum Erdős–Nicolas Erdős–Woods Friendly Frugal Giuga Harmonic divisor Lucas–Carmichael Pronic Regular Rough Smooth Sociable Sphenic Størmer Super-Poulet Zeisel
Recreational mathematics	Base-dependent numbers Automorphic Cyclic Digit-reassembly Dudeney Equidigital Extravagant Factorion Friedman Happy Harshad Kaprekar Keith Lychrel Missing-digit sum Narcissistic Palindromic Pandigital Parasitic Pernicious Polydivisible Primeval Repdigit Repunit Self Self-descriptive Smarandache–Wellin Strictly non-palindromic Strobogrammatic Sum-product Transposable Trimorphic Undulating Vampire Aronson's sequence Ban Pancake number

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