Tritriangular Number http://mathworld.wolfram.com/TritriangularNumber.html

Determine

I might be reading it wrong but a triangular number with a triangular index is different to what you get, I seem to get 1/8 n (n + 1)(n^2 + n + 2)

Expanding mine gives n/4 + (3 n^2)/8 + n^3/4 + n^4/8 and expanding yours gives (3 n)/4 + (11 n^2)/8 + (3 n^3)/4 + n^4/8

Anyway here’s what I get forT (tn+1)

1/8 (2 + n + n^2) (4 + n + n^2)

and one minus the other is

1/8 (2 + n + n^2) (4 + n + n^2) – 1/8 n (n + 1)(n^2 + n + 2) = 1/2(n^2 + n + 2)

Answer = 1/2(n^2 + n + 2)

First 16 Tn’s are {1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136}

for n= {2, 3, 4, 5} the T tn+1 is {10, 28, 66, 136} the T tn is {6, 21, 55, 120} difference is {4, 7, 11, 16}

Paul.

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I might be reading it wrong but a triangular number with a triangular index is different to what you get, I seem to get 1/8 n (n + 1)(n^2 + n + 2)

Expanding mine gives

n/4 + (3 n^2)/8 + n^3/4 + n^4/8

and expanding yours gives

(3 n)/4 + (11 n^2)/8 + (3 n^3)/4 + n^4/8

Anyway here’s what I get forT (tn+1)

1/8 (2 + n + n^2) (4 + n + n^2)

and one minus the other is

1/8 (2 + n + n^2) (4 + n + n^2) – 1/8 n (n + 1)(n^2 + n + 2) = 1/2(n^2 + n + 2)

Answer = 1/2(n^2 + n + 2)

First 16 Tn’s are

{1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136}

for n= {2, 3, 4, 5}

the T tn+1 is

{10, 28, 66, 136}

the T tn is

{6, 21, 55, 120}

difference is

{4, 7, 11, 16}

Paul.