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Centered triangular number - Wikipedia
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Centered figurate number that represents a triangle with a dot in the center
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 {\displaystyle 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).
The first eight centered triangular numbers on a hex grid
- The gnomon of the n-th centered triangular number, corresponding to the (n + 1)-th triangular layer, is:
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C 3 , n + 1 − C 3 , n = 3 ( n + 1 ) . {\displaystyle 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:
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C 3 , n = 1 + 3 n ( n + 1 ) 2 = 3 n 2 + 3 n + 2 2 . {\displaystyle 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.
Relationship with centered square numbers[edit]
The centered triangular numbers can be expressed in terms of the centered square numbers:
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C 3 , n = 3 C 4 , n + 1 4 , {\displaystyle C_{3,n}={\frac {3C_{4,n}+1}{4}},}
where
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C 4 , n = n 2 + ( n + 1 ) 2 . {\displaystyle C_{4,n}=n^{2}+(n+1)^{2}.}
Lists of centered triangular numbers[edit]
The first centered triangular numbers (C3,n < 3000) are:
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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, … (sequence A005448 in the OEIS).
The first simultaneously triangular and centered triangular numbers (C3,n = TN < 109) are:
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1, 10, 136, 1 891, 26 335, 366 796, 5 108 806, 71 156 485, 991 081 981, … (sequence A128862 in the OEIS).
The generating function[edit]
If the centered triangular numbers are treated as the coefficients of the McLaurin series of a function, that function converges for all | x | < 1 {\displaystyle |x|<1} , in which case it can be expressed as the meromorphic generating function
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1 + 4 x + 10 x 2 + 19 x 3 + 31 x 4 + . . . = 1 − x 3 ( 1 − x ) 4 = x 2 + x + 1 ( 1 − x ) 3 . {\displaystyle 1+4x+10x^{2}+19x^{3}+31x^{4}+~...={\frac {1-x^{3}}{(1-x)^{4}}}={\frac {x^{2}+x+1}{(1-x)^{3}}}~.}
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