Centered triangular number explained

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 pointis 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 (n(n +1 ))/(2) = (3 n ^{2} +3 n +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} = (3 C _{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 < 3000) 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, … .

The first simultaneously triangular and centered triangular numbers (C_3,n = T_N < 10⁹) 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| < 1 , in which case it can be expressed as the meromorphic generating function1 + 4 x + 10 x ^{2} + 19 x ^{3} + 31 x ^{4} + ~... = (1- x ^{3})/(( 1- x) ^{4}) = (x ^{2+x+1})/(( 1- x) ^{3}) ~.

References

• Lancelot Hogben

Mathematics for the Million (1936), republished by W. W. Norton & Company (September 1993),