Not sure but this is what I have...
1≤a
b
a2<100a2<100 so a<10a<10.
ab<100ab<100 so ab≤99ab≤99 so b≤99ab≤99a so b≤⌊99a⌋b≤⌊99a⌋.
b+1≤c≤100b+1≤c≤100.
∑9a=1(∑⌊99a⌋b=a+1(∑100c=ab+11))=∑a=19(∑b=a+1⌊99a⌋(∑c=ab+11001))=.
∑9a=1(∑⌊99a⌋b=a+1[100−ab])∑a=19(∑b=a+1⌊99a⌋[100−ab])
∑99b=2(100−b)+∑49b=3(100−2b)+∑33b=4(100−3b)+∑24b=5(100−4b)+∑19b=6(100−5b)+∑16b=7(100−6b)+∑14b=8(100−7b)+∑12b=9(100−8b)+∑11b=10(100−9b)=∑b=299(100−b)+∑b=349(100−2b)+∑b=433(100−3b)+∑b=524(100−4b)+∑b=619(100−5b)+∑b=716(100−6b)+∑b=814(100−7b)+∑b=912(100−8b)+∑b=1011(100−9b)=
∑98k=1k+∑94k=2;+2k+∑88k=1;+3k+∑80k=4;+4k+∑70k=5;+5k+∑58k=4;+6k+∑44k=2;+7k+∑28k=4;+8k+∑10k=1;+9k=∑k=198k+∑k=2;+294k+∑k=1;+388k+∑k=4;+480k+∑k=5;+570k+∑k=4;+658k+∑k=2;+744k+∑k=4;+828k+∑k=1;+910k=
∑98k=1k+∑47k=12k+∑29k=0;(3k+1)+∑20k=14k+∑14k=15k+∑9k=06k+4+∑7k=07k+2+∑308k+4+∑1k=09k+1=∑k=198k+∑k=1472k+∑k=0;29(3k+1)+∑k=1204k+∑k=1145k+∑k=096k+4+∑k=077k+2+∑038k+4+∑k=019k+1=
98∗992+2∗47∗482+329∗302+30+4∗20∗212+514∗152+69∗102+4∗10+7∗7∗82+2∗8+83∗42+4∗4+9∗1∗22+2∗1=98∗992+2∗47∗482+329∗302+30+4∗20∗212+514∗152+69∗102+4∗10+7∗7∗82+2∗8+83∗42+4∗4+9∗1∗22+2∗1=
49∗99+47∗48+3∗29∗15+30+2∗20∗21+5∗7∗15+3∗9∗5+40+7∗7∗4+16+4∗3∗4+16+9+2=49∗99+47∗48+3∗29∗15+30+2∗20∗21+5∗7∗15+3∗9∗5+40+7∗7∗4+16+4∗3∗4+16+9+2=
49∗99+47∗48+29∗45+30+20∗42+35∗15+27∗5+40+49∗4+32+48+11=10269
(Okay, I don't know how to simplify)
+240 
+240