A centered triangular number is a centered figurate number that represents a triangle with a dot in the center and all other dots surrounding the center in successive triangular layers. The centered polygonal numbers are a class of series of Figurate numbers each formed by a central dot surrounded by polygonal layers with a constant number of sides A figurate number is a number that can be represented as a regular and discrete geometric pattern (e A triangle is one of the basic Shapes of Geometry: a Polygon with three corners or vertices and three sides or edges which are Line The centered triangular number for n is given by the formula
The following image shows the building of the centered triangular numbers using the associated figures: at each step the previous figure, shown in red, is surrounded by a triangle of new points, in blue.
The first few centered triangular numbers (sequence A005448 in OEIS) are
1, 4, 10, 19, 31, 46, 64, 85, 109, 136, 166, 199, 235, 274, 316, 361, 409, 460, 514, 571, 631, 694, 760, 829, 901, 976, 1054, 1135, 1219, 1306, 1396, 1489, 1585, 1684, 1786, 1891, 1999, 2110, 2224, 2341, 2461, 2584, 2710, 2839, 2971
Each centered triangular number from 10 onwards is the sum of three consecutive regular triangular numbers. The On-Line Encyclopedia of Integer Sequences ( OEIS) also cited simply as Sloane's, is an extensive searchable Database of Integer sequences Mathematics For any number x: x ·1 = 1· x = x (1 is the multiplicative identity In mathematics Four is the smallest Composite number, its proper Divisors being and. 19 ( nineteen) is the Natural number following 18 and preceding 20. 31 ( thirty-one) is the Natural number following 30 and preceding 32. 46 ( forty-six) is the Natural number following 45 and preceding 47. 64 ( sixty-four) is the Natural number following 63 and preceding 65. 85 ( eighty-five) is the Natural number following 84 and preceding 86. A triangular number is the sum of the n Natural numbers from 1 to n. Also each centred triangular number has a remainder of 1 when divided by three and the quotient (if positive) is the previous regular triangular number.
The sum of the first n centered triangular numbers is the magic constant for an n by n normal magic square for n > 2. The magic constant or magic sum of a Magic square is the sum of numbers in any row column and diagonal of the magic square In Recreational mathematics, a magic square of order n is an arrangement of n ² numbers usually distinct Integers in a square, such
Centered triangular prime
A centered triangular prime is a centered triangular number that is prime. In Mathematics, a prime number (or a prime) is a Natural number which has exactly two distinct natural number Divisors 1 The first few centered triangular primes are (sequence A125602 in OEIS)
19, 31, 109, 199, 409, . The On-Line Encyclopedia of Integer Sequences ( OEIS) also cited simply as Sloane's, is an extensive searchable Database of Integer sequences . .
(corresponding to n: 3, 4, 8, 11, 16, . . . )
References
- Lancelot Hogben: Mathematics for the Million. Lancelot Thomas Hogben ( 9 December[[ 895]] - 22 August 1975) was a versatile British experimental Zoologist, and medical Statistician (1936), republished by W. W. Norton & Company (September 1993), ISBN 978-0393310719
- Eric W. Weisstein, Centered Triangular Number at MathWorld. Eric W Weisstein (born March 18, 1969, in Bloomington Indiana) is an Encyclopedist who created and maintains MathWorld MathWorld is an online Mathematics reference work created and largely written by Eric W
- On-Line Encyclopedia of Integer Sequences, sequence A005448 and A125602. The On-Line Encyclopedia of Integer Sequences ( OEIS) also cited simply as Sloane's, is an extensive searchable Database of Integer sequences
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