Centered triangular number

A centered (or centred) triangular number is a centered figurate number that represents an equilateral triangle with a dot in the center and all its other dots surrounding the center in successive equilateral triangular layers. This is also the number of points of a hexagonal lattice with nearest-neighbor coupling whose distance from a given point is less than or equal to n . The following image shows the building of the centered triangular numbers by using the associated figures: at each step, the previous triangle (shown in red) is surrounded by a triangular layer of new dots (in blue).

A centered (or centred) triangular number is a centered figurate number that represents an equilateral triangle with a dot in the center and all its other dots surrounding the center in successive equilateral triangular layers. This is also the number of points of a hexagonal lattice with nearest-neighbor coupling whose distance from a given point is less than or equal to n . The following image shows the building of the centered triangular numbers by using the associated figures: at each step, the previous triangle (shown in red) is surrounded by a triangular layer of new dots (in blue). ]] ==Properties== The gnomon of the n-th centered triangular number, corresponding to the (n + 1)-th triangular layer, is: C_{3,n+1} - C_{3,n} = 3(n+1). The n-th centered triangular number, corresponding to n layers plus the center, is given by the formula: C_{3,n} = 1 + 3 \frac{n(n+1)}{2} = \frac{3n^2 + 3n + 2}{2}. Each centered triangular number has a remainder of 1 when divided by 3, and the quotient (if positive) is the previous regular triangular number. Each centered triangular number from 10 onwards is the sum of three consecutive regular triangular numbers. For n > 2, the sum of the first n centered triangular numbers is the magic constant for an n by n normal magic square. ===Relationship with centered square numbers=== The centered triangular numbers can be expressed in terms of the centered square numbers: C_{3,n} = \frac{3C_{4,n} + 1}{4}, where C_{4,n} = n^{2} + (n+1)^{2}. ==Lists of centered triangular numbers== The first centered triangular numbers (C 3,n 3,n = T N 9 ) are: 1, 10, 136, 1 891, 26 335, 366 796, 5 108 806, 71 156 485, 991 081 981, … . ==The generating function== If the centered triangular numbers are treated as the coefficients of the McLaurin series of a function, that function converges for all |x| , in which case it can be expressed as the meromorphic generating function 1 + 4x + 10x^2 + 19x^3 + 31x^4 +~... = \frac{1-x^3}{(1-x)^4} = \frac{x^2+x+1}{(1-x)^3} ~. ==References== Lancelot Hogben: Mathematics for the Million (1936), republished by W. W. Norton & Company (September 1993),

Text derived from https://en.wikipedia.org/w/index.php?oldid=1239326243 under CC BY-SA 4.0; modifications have been made.