NMAC DURING A VISUAL APCH TO HPN IN A NIGHT OP.

Narrative

OVER BRIDGEPORT VOR; NEW YORK APCH CTLR INSTRUCTED US TO FOLLOW THE CONNECTICUT SHORELINE AND DSND TO 3000 FT FROM 5000 FT. WE WERE THEN HANDED OFF TO THE FINAL CTLR WHO IDENTED US AND LATER CLRED US FOR THE SOUND VISUAL TO RWY 34 AT HPN. THE ZNY CTLR WAS BUSY SEQUENCING ACFT THAT WERE TO FOLLOW US BUT NEVER CALLED OUT ANY TFC TO US. AS WE CROSSED THE SHORELINE; I JOINED THE ILS AT 2000 FT AND CONFIGURED THE ACFT; CALLING FOR THE LNDG CHKLIST. NEW YORK APCH (126.4) NEVER HANDED US OVER TO TWR; SO AT ABOUT 4 MI OUT; WE SWITCHED OVER. BEFORE WE EVEN CALLED THE TWR WE GOT A TA ON THE TCASII WHICH QUICKLY WENT TO AN RA AND INSTRUCTIONS TO CLB. I IMMEDIATELY ADDED PWR AND BEGAN TO CLB CALLING FOR FLAPS 7 DEGS AND THEN GEAR UP. HPN TWR ASKED IF WE HAD TFC IN SIGHT THEN CLRED US TO LAND. TWR APPARENTLY DIDN'T KNOW HOW CLOSE WE HAD MISSED; ONLY INSTRUCTING THE TFC TO FLY A 90 DEG OFF THE FINAL APCH COURSE. WE SUBSEQUENTLY WENT AROUND AND LANDED UNEVENTFULLY. NEITHER APCH NOR TWR (FOR A LATE HDOF) EVER ADVISED US OF TFC WE WERE FOLLOWING. WE ONLY HAD A VISUAL ON THE TFC AFTER THE RA AND GAR. OUR VISUAL APCH WITH NO SEQUENCE INFO LED US TO BELIEVE WE WERE #1 TO THE RWY AND A LATE HDOF TO THE TWR; INCLUDING RUNNING OUR LNDG CHKLIST; ALL PLAYED A PART IN OUR LACK OF VISUAL SCAN FOR TFC.