Pentagonal Triangular Number

Author

Eric W. Weisstein
September 3, 2003

This notebook downloaded from http://mathworld.wolfram.com/notebooks/IntegerSequences/PentagonalTriangularNumber.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/PentagonalTriangularNumber.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Initialization

<<Figurate`

Computation

Reduce[PentagonalNumber[n]==TriangularNumber[m]&&(m|n)∈Integers&&m>0&&n>0,{n,m},Integers]/.C[1]->k

soln=Table[
{(1/12) (-6+3 (((7-4 (Sqrt[3])))^k)+(Sqrt[3]) (((7-4 (Sqrt[3])))^k)+3 (((7+4 (Sqrt[3])))^k)-(Sqrt[3]) (((7+4 (Sqrt[3])))^k)),
(1/6) (1+(1/2) (-(((7-4 (Sqrt[3])))^k)-(Sqrt[3]) (((7-4 (Sqrt[3])))^k)-(((7+4 (Sqrt[3])))^k)+(Sqrt[3]) (((7+4 (Sqrt[3])))^k)))},{k,10}]//Expand

TriangularNumber/@First/@soln