Pythagoras sequence

3^2+4^2=5^2

3 4 5, 5 12 13, 13 84 85, 85 132 157, 157 12324 12325, 12325 1836 12461,
12461 105552 106285,.....

The form:  a^2+b^2=(b+1)^2 appear many time
But            a^2+(a+1)^2=b^2 found only one time i.e. 20^2+21^2=29^2

1 Are there other set of a^2+(a+1)^2=b^2 ?
2 Can we plot the triangle relation in the whole sequence?
   The above is main pythagoras sequence, I found some sub sequence e.g.
    40 52 58, 58 840 842
     9  40 41, 41 840 841