## Figurate Number in JavaA
Figurate numbers can also form other geometric shapes such as centered polygons, L-shapes, three-dimensional (and multidimensional) solids, etc. ## Figurate NumberIt is the number of triangles (all of whose vertices lie inside the circle) formed when n points in general position on a circle are joined by straight lines. A
Let' see their pictorial representation of the figurate number. The following figure illustrates the same. The polygonal numbers illustrated above are called It is based on a 6-dimensional regular simplex. It is an OEIS sequence According to ## Properties- a(n) is the number of terms in the expansion of (a_1 + a_2 + a_3 + a_4 + a_5 + a_6 + a_7)^n. The only prime in this sequence is 7.
- 6-dimensional triangular numbers, sixth partial sums of binomial transform of [1, 0, 0, 0, ...].
- a(n) = fallfac(n, 6)/6! is also the number of independent components of an antisymmetric tensor of rank 6 and dimension n >= 1. Here,
**fallfac**is the falling factorial. - Number of orbits of Aut(Z^7) as a function of the infinity norm n of the representative integer lattice point of the orbit, when the cardinality of the orbit is equal to 645120.
- For a set of integers {1,2,...,n}, a(n) is the sum of the 2 smallest elements of each subset with 5 elements that is 3*C(n+1,6) (for n>=5), hence a(n) = 3*C(n+1,6) = 3*A000579(n+1).
- a(n) = A000292(n-5)*A000292(n-2)/20
## Formulas to Calculate Figurate Number**General Formula:**x^6/(1-x)^7**Exponential Formula:**exp(x)*x^6/720**Conjecture:**a(n+3) = Sum{0 <= k, l, m <= n; k + l + m <= n} k*l*m
Some other formulas are: - a(n) = (n^6 - 15*n^5 + 85*n^4 - 225*n^3 + 274*n^2 - 120*n)/720.
- a(n) = 3*C(n+1, 6)
Tn=n(n+1)/2
S _{n}=n^{2}P _{n}=n(3n-1)/2H _{n}=n(4n-2)/2HP _{n}= n(5n-3)/2O _{n}= n(6n-4)/2NO _{n}= n(7n-5)/2
The above formulas lead to a conjecture a formula for a general N-agonal number:
N
_{n}=n((N-2)n-(N-4))/2
## Note: The above formula works only for N=2, 3, and 4.## Figurate Number ExampleFor example, consider a set of integers From the given set Z, create subsets that must have {1, 2, 3, 4, 5}, {1, 2, 3, 4, 6}, {1, 2, 3, 5, 6}, {1, 2, 4, 5, 6}, {1, 3, 4, 5, 6}, {2, 3, 4, 5, 6} From the above subsets, determine the a(6) = (1+2) + (1+2) + (1+2) + (1+2) + (1+3) + (2+3) = Similarly, we can use the formula a(6) = 21 = 3*C(6+1,6) Let's use another formula to check whether the number 84 is a figurate number or not. a(9) = (1, 3, 3, 1). (1, 6, 15, 20) = (1 + 18 + 45 + 20) = Hence, the 9 First few figurate numbers are: 0, 0, 0, 0, 0, 0, 1, 7, 28, 84, 210, 462, 924, 1716, 3003, 5005, 8008, 12376, 18564, 27132, 38760, 54264, 74613, 100947, 134596, 177100, 230230, 296010, 376740, 475020, 593775, 736281, 906192, 1107568, 1344904, 1623160, 1947792, 2324784, 2760681, 3262623 ## Using Successive Difference MethodWe can calculate the figurate number using successive differences. For example, consider the following nonagonal numbers. But the above method is not always helpful. ## Types of Figurate Number## 4-dimensional Figurate Numbers (A002417)It can be calculated by using the following formulas:
a(n)=n*binomial (n+2, 3).
It is an OEIS sequence First few 1, 8, 30, 80, 175, 336, 588, 960, 1485, 2200, 3146, 4368, 5915, 7840, 10200, 13056, 16473, 20520, 25270, 30800, 37191, 44528, 52900, 62400, 73125, 85176, 98658, 113680, 130355, 148800, 169136, 191488, 215985, 242760, 271950, 303696, 338143, 375440, 415740.
- a(n) is 1/6 the number of colorings of a 2 X 2 hexagonal array with n+2 colors.
a(n) = n^2*(n+1)*(n+2)/6
- a(n) is the sum of all numbers that cannot be written as t*(n+1) + u*(n+2) for nonnegative integers t, u.
- a(n) is the total number of rectangles (including squares) contained in a stepped pyramid of n rows (or of base 2n-1) of squares.
- a(n) equals -1 times the coefficient of x^3 of the characteristic polynomial of the (n + 2) X (n + 2) matrix with 2's along the main diagonal and 1's everywhere else.
- a(n) is the n-th antidiagonal sum of the convolution array.
- Also, the number of 3-cycles in the (n+2)-triangular graph.
**General Formula:**x*(1+3*x)/(1-x)^5- a(n) = C(n+2, 2)*n^2/3
- a(n) = C(n+3, n)*C(n+1, 1)
- a(n) = (binomial(n+3, n-1) - binomial(n+2,n-2))*(binomial(n+1,n-1) - binomial(n,n-2))
- a(n) = 5*a(n-1) -10*a(n-2) +10*a(n-3) -5*a(n-4) +a(n-5), n>5
- a(n) = A080852(4, n-1)
- a(n) = A000332(n+3) + 3*A000332(n+2)
## 4-Dimensional Figurate Number (A002418)There is another formula to calculate
(5*n-1)*binomial(n+2,3)/4
It is an OEIS sequence First few figurate numbers of the sequence 0, 1, 9, 35, 95, 210, 406, 714, 1170, 1815, 2695, 3861, 5369, 7280, 9660, 12580, 16116, 20349, 25365, 31255, 38115, 46046, 55154, 65550, 77350, 90675, 105651, 122409, 141085, 161820, 184760, 210056, 237864, 268345, 301665, 337995.
- The sequence
**A002418**is the partial sum of A002413. - Principal diagonal of the convolution array A213550, for n>0.
- Convolution of A000027 with A000566.
**General Formula:**x*(1+4*x)/(1-x)^5**Exponential Formula:**x*(24 + 84*x + 44*x^2 + 5*x^3)*exp(x)/4!
Other formulas are: - a(n) = n*C((n+3),4)) - (n-1)*C((n+2),4)) or a(n) = A128064* A000332.
For example,**a(5) = 5*C(8,4) - 4*C(7,4) = 5*70 - 4*35 =210**. - a(0)=0, a(1)=1, a(2)=9, a(3)=35, a(4)=95 then a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5)
- a(n) = A080852(5, n-1)
## 4-Dimensional Figurate Number (A002419)There is another formula to calculate
(6*n-2)*binomial(n+2,3)/4.
First few figurate numbers of the sequence 1, 10, 40, 110, 245, 476, 840, 1380, 2145, 3190, 4576, 6370, 8645, 11480, 14960, 19176, 24225, 30210, 37240, 45430, 54901, 65780, 78200, 92300, 108225, 126126, 146160, 168490, 193285, 220720, 250976, 284240, 320705, 360570, 404040, 451326
- a(n) is the n-th antidiagonal sum of the convolution array A213761.
- a(n) = the sum of all the ways of adding the k-tuples of A016777(0) to A016777(n-1).
Suppose, we have to calculate n=4. The terms are
- a(n) = (3*n-1)*binomial(n+2, 3)/2
**General Formula:**x*(1+5*x)/(1-x)^5**Exponential Formula:**x*(12 + 48*x + 26*x^2 + 3*x^3)*exp(x)/12.- a(n) = (3*n^4 + 8*n^3 + 3*n^2 - 2*n)/12
- a(n) = A080852(6, n-1)
## Regular Figurative Number (A090466)The sorted k-gonal numbers of order greater than 2. If one were to include either the rank 2 or the 2-gonal numbers, then every number would appear. It is also known as Number of terms less than or equal to 10^k for k = 1, 2, 3, ...: 3, 57, 622, 6357, 63889, 639946, 6402325, 64032121, 640349979, 6403587409, 64036148166, 640362343980, and so on. For all squares (A001248) of primes p >= 5 at least one a(n) exists with
For For all primes Numbers m such that r = (2*m/d - 2)/(d - 1) is an integer for some d, where 2 < d < m is a divisor of 2*m. If r is an integer, then m is the d-th (r+2)-gonal number. First few numbers of the sequence A090466 are: 6, 9, 10, 12, 15, 16, 18, 21, 22, 24, 25, 27, 28, 30, 33, 34, 35, 36, 39, 40, 42, 45, 46, 48, 49, 51, 52, 54, 55, 57, 58, 60, 63, 64, 65, 66, 69, 70, 72, 75, 76, 78, 81, 82, 84, 85, 87, 88, 90, 91, 92, 93, 94, 95, 96, 99, 100, 102, 105, 106, 108, 111, 112, 114, 115, 117, 118. ## Figurate Number Java ProgramThe following Java program check whether the give number is a square number or not.
Enter a number: 100 100.0 is a square number. Let's create another Java program for a figurate number called nonagonal number.
Enter the term you want to find: 4 The 4 rd/th nonagonal number is: 46 |

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