Centered cube number

Centered cube number
35 points in a body-centered cubic lattice, forming two cubical layers around a central point.
Total no. of terms	Infinity
Subsequence of	Polyhedral numbers
Formula	$n^{3}+(n+1)^{3}$
First terms	1, 9, 35, 91, 189, 341, 559
OEIS index	A005898 Centered cube

A centered cube number is a centered figurate number that counts the points in a three-dimensional pattern formed by a point surrounded by concentric cubical layers of points, with $i 2$ points on the square faces of the $i$ th layer. Equivalently, it is the number of points in a body-centered cubic pattern within a cube that has $n + 1$ points along each of its edges.

The first few centered cube numbers are

1, 9, 35, 91, 189, 341, 559, 855, 1241, 1729, 2331, 3059, 3925, 4941, 6119, 7471, 9009, ... (sequence A005898 in the OEIS).

Formulas

The centered cube number for a pattern with $n$ concentric layers around the central point is given by the formula^[1]

n^{3}+(n+1)^{3}=(2n+1)\left(n^{2}+n+1\right).

The same number can also be expressed as a trapezoidal number (difference of two triangular numbers), or a sum of consecutive numbers, as^[2]

{\binom {(n+1)^{2}+1}{2}}-{\binom {n^{2}+1}{2}}=(n^{2}+1)+(n^{2}+2)+\cdots +(n+1)^{2}.

Properties

Because of the factorization $(2 n + 1)(n 2 + n + 1)$ , it is impossible for a centered cube number to be a prime number.^[3] The only centered cube numbers which are also the square numbers are 1 and 9,^[4]^[5] which can be shown by solving $x 2 = y 3 + 3 y$ , the only integer solutions being (x,y) from {(0,0), (1,2), (3,6), (12,42)}, By substituting a=(x-1)/2 and b=y/2, we obtain x^2=2y^3+3y^2+3y+1. This gives only (a,b) from {(-1/2,0), (0,1), (1,3), (11/2,21)} where a,b are half-integers.

References

^ Deza, Elena; Deza, Michel (2012), Figurate Numbers, World Scientific, pp. 121–123, ISBN 9789814355483
^ Lanski, Charles (2005), Concepts in Abstract Algebra, American Mathematical Society, p. 22, ISBN 9780821874288.
^ Sloane, N. J. A. (ed.). "Sequence A005898". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
^ Stroeker, R. J. (1995), "On the sum of consecutive cubes being a perfect square", Compositio Mathematica, 97 (1–2): 295–307, MR 1355130.
^ O'Shea, Owen; Dudley, Underwood (2007), The Magic Numbers of the Professor, MAA Spectrum, Mathematical Association of America, p. 17, ISBN 9780883855577.

External links

Weisstein, Eric W. "Centered Cube Number". MathWorld.

Figurate numbers

2-dimensional

centered	Centered triangular numbers Centered square numbers Centered pentagonal numbers Centered hexagonal numbers Centered heptagonal numbers Centered octagonal numbers Centered nonagonal numbers Centered decagonal numbers Star numbers
non-centered	Triangular numbers Square numbers Pentagonal numbers Hexagonal numbers Heptagonal numbers Octagonal numbers Nonagonal numbers Decagonal numbers Dodecagonal numbers

3-dimensional

centered	Centered tetrahedral numbers Centered cube numbers Centered octahedral numbers Centered dodecahedral numbers Centered icosahedral numbers
non-centered	Cube numbers Octahedral numbers Dodecahedral numbers Icosahedral numbers Stella octangula numbers
pyramidal	Tetrahedral numbers Square pyramidal numbers

4-dimensional

non-centered	Pentatope numbers Squared triangular numbers Tesseractic numbers

Higher dimensional

non-centered	5-hypercube numbers 6-hypercube numbers 7-hypercube numbers 8-hypercube numbers

Classes of natural numbers

Powers and related numbers
Achilles Power of 2 Power of 3 Power of 10 Square Cube Fourth power Fifth power Sixth power Seventh power Eighth power Perfect power Powerful Prime power

Of the form a × 2^b ± 1
Cullen Double Mersenne Fermat Mersenne Proth Thabit Woodall

Other polynomial numbers
Hilbert Idoneal Leyland Loeschian Lucky numbers of Euler

Recursively defined numbers
Fibonacci Jacobsthal Leonardo Lucas Narayana Padovan Pell Perrin

Possessing a specific set of other numbers
Amenable Congruent Knödel Riesel Sierpiński

Expressible via specific sums
Nonhypotenuse Polite Practical Primary pseudoperfect Ulam Wolstenholme

Figurate numbers

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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 Cubic Octahedral Dodecahedral Icosahedral Stella octangula
pyramidal	Square pyramidal

4-dimensional

non-centered	Pentatope Squared triangular Tesseractic

Combinatorial numbers
Bell Cake Catalan Dedekind Delannoy Euler Eulerian Fuss–Catalan Lah Lazy caterer's sequence Lobb Motzkin Narayana Ordered Bell Schröder Schröder–Hipparchus Stirling first Stirling second Telephone number Wedderburn–Etherington

Primes
Wieferich Wall–Sun–Sun Wolstenholme prime Wilson

Pseudoprimes
Carmichael number Catalan pseudoprime Elliptic pseudoprime Euler pseudoprime Euler–Jacobi pseudoprime Fermat pseudoprime Frobenius pseudoprime Lucas pseudoprime Lucas–Carmichael number Perrin pseudoprime Somer–Lucas pseudoprime Strong pseudoprime

Arithmetic functions and dynamics

Divisor functions	Abundant Almost perfect Arithmetic Betrothed Colossally abundant Deficient Descartes Hemiperfect Highly abundant Highly composite Hyperperfect Multiply perfect Perfect Practical Primitive abundant Quasiperfect Refactorable Semiperfect Sublime Superabundant Superior highly composite Superperfect
Prime omega functions	Almost prime Semiprime
Euler's totient function	Highly cototient Highly totient Noncototient Nontotient Perfect totient Sparsely totient
Aliquot sequences	Amicable Perfect Sociable Untouchable
Primorial	Euclid Fortunate

Other prime factor or divisor related numbers
Blum Cyclic Erdős–Nicolas Erdős–Woods Friendly Giuga Harmonic divisor Jordan–Pólya Lucas–Carmichael Pronic Regular Rough Smooth Sphenic Størmer Super-Poulet

Numeral system-dependent numbers

Arithmetic functions
and dynamics

Persistence
- Additive
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Digit sum	Digit sum Digital root Self Sum-product
Digit product	Multiplicative digital root Sum-product
Coding-related	Meertens
Other	Dudeney Factorion Kaprekar Kaprekar's constant Keith Lychrel Narcissistic Perfect digit-to-digit invariant Perfect digital invariant Happy

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