A centered (or centred) 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 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 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, … (sequence A005448 in the OEIS).
Each centered triangular number from 10 onwards is the sum of three consecutive regular triangular numbers. Also each centered 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.
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Tetrahedral and Octahedral Sites

Calculating the limiting radius ratio

Pentagonal Numbers
Transcription
here will describe where the tetrahedral sites and the octahedral sites are located on aface centered cubic or FCC crystal structure and a question first would be why would be interested in this so we could look at structure of something like aluminum oxide it turns out this structure consist of the oxygen in the face centered cubic structure and the aluminum then In these interstitial sites and depending on the type of aluminum oxide crystal structure determines exactly where they're located but for gamma alumina which just one type of aluminum structure they are both in tetrahedral and octahedral sites and the reason this happens is aluminum if we look at the iconic radius it's much smaller than the ionic radius of oxygen. So we have the oxygen's in this crystal structure and then the spaces between those ions we can fit the much smaller aluminum. That's a reason we are interested so let's look and see exactly where the sites are So what I have drawn here is just the start of the first couple layers of a FCC structure so FCC structure and what we're looking at is the Miller indices plane labeled 111 so that has a hexagonal symmetry and if you remember an FCC structure can be envisioned as an ABCABC tight packing where we have a hexagonal layer. We put now the second layer above the openings not above the atoms themselves and we put the third layer above the openings that are not directly above the first layer. We do not put the third layer here because that would be directly above the "A" layer but we put it here here ext. and so if we look at just a few atoms in the second structure again these could extend out in the same type the packing as the first layer then we can look at if we put just one atom in that third layer so let's say we put one atom this orange atom what we've done is say we're going to put an atom here well. This space then underneath that atom that's under here so let's go back and say this then that space they're going to create is a tetrahedral sites is smaller certainly then the atoms of the FCC structure as we create a tetrahedral site and here it just shown with the third atom much lighter so notice the third atom is not above an atom in the first layer and we can see the tetrahedral structure These four atoms form a tetrahedron and we've identified the location then of this tetrahedral site. So turns out we can identify the locations now in a unit cell so what I've done is drawn just the four corners of the unit cell FCC structuring of courses there is going to be an atom here. Atom on this face this face this face the one in the back the green locations are tetrahedral sites. One quarter in one quarter up onequarter over and if we look at the structure we can see there are eight sites. These are interstitial sites and the unit cell of the FCC structure has four atoms in the unit cell so 8 tetrahedral sites unit cell four atoms. The four atoms correspond to we have six faces. Atoms on each face it shared with the unit cell next to it so it's six times 1/2 so we have three atoms from the faces and then the corner we have 8 atoms but they are shared each of them with seven others so there's only 1/8 on the corners. You can see now we have a total of four atoms in the unit cell 8 interstitial sites we have two interstitial tetrahedral sites two interstitial tetrahedral sites for every atom of the FCC structure what we ant to now look at is octahedral sites again I'm showing the FCC the 111 plane. Miller indices 111 the hexagonal structure here's to "B" layer. Now the "C" layer remember is not located above the "A" layer so I've indicated now three atoms and "C" layer so we can visualized the octahedral sites so these three items here shown lighter. See that this atom in layer "C" is not above the atom right this is layer "A" this is layer "B" and this atom is layer "C" and since "C" are not above the atoms in layer "A" this site here then is the octahedral site and it has sixfold coordination and it's surrounded by six atoms of the FCC structure but we call an octahedral site because these six atoms form if we connect the centers of the atoms a structured that has eight sides an let's try and visualize that and the easiest way is to look at this atom draw a line to the center this atom in the"C" layer this atom in the "C" layer and then this atom in the "B" layer and then connect them so we have a square and we have this atom above this atom below and maybe it's easy so what we're looking at and why they call octahedral site so what we have is four atoms in a square atom above so if I connect similar atom below again connecting what we show here is that eight sided structure at four triangles and top four triangles on bottom. 6 coordination so we call an octahedral site because of this structure. This eight sided structure that forms by connecting atoms and then the octahedral sites reproduce the FCC structure and now drawn the octahedral sites in The octahedral sites happen to be located along each side. So there is one octahedral site on each side and there's one in the center so if we wanted to add up in determining number of octahedral sites So one site is in the center and then we have 4, 8 12 sites but these are each shared with three other so only 1/4 three other unit cells only 1/4 are there so so this is three from the edges. So total the of four octahedral sites So four octahedral sites per cell. Remember we showed early we had four atoms per cells so we have one octahedral site for each atom and as we showed previously two tetrahedral sites. So a total of three interstitial sites for each atom in the FCC structure
References
 Lancelot Hogben: Mathematics for the Million.(1936), republished by W. W. Norton & Company (September 1993), ISBN 9780393310719
 Weisstein, Eric W. "Centered Triangular Number". MathWorld.