A PA28 PLT LANDED GEAR UP FOLLOWING A SHORT APCH FOR TFC AND AN INOP GEAR WARNING HORN. MENTAL VERSUS PAPER CHKLIST USED.

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DEPARTED ZZZ1 VFR; CLBED TO 4500 FT MSL. CONTACTED APCH AND SET UP FLT FOLLOWING. WHEN HANDED OFF; I CHKED ATIS AT ZZZ2. WHEN 15 NM FROM ZZZ2; ASKED APCH FOR LCL IFR CLRNC TO DSND THROUGH SCATTERED CLOUDS AT 3000 FT MSL. ONCE ZZZ2 WAS IN SIGHT; WAS CLRED FOR VISUAL APCH AND OVER TO ZZZ2 TWR. ZZZ2 CLRED TO LAND VISUAL RWY 32. RAN MENTAL CHKLIST (LC GUMP). AT THE SAME TIME; TWR NOTED OTHER TFC IN PATTERN. I SET UP A SHORT APCH TO STAY CLR OF UNSEEN TFC DOING A '360 DEG TURN.' ALL SYS SEEMED FINE. NO WARNING BUZZERS; NO WARNING LIGHTS -- UNTIL THE SOUND OF METAL. I HAD LANDED 'GEAR UP!' THOUGHTS: THOUGH I HAD THOUGHT I HAD PLACED THE GEAR DOWN; TWR'S CALLS ABOUT OTHER TFC MADE ME SKIP LOWERING GEAR; THOUGH I DID SET PROP; MIXTURE; FUEL PUMP ON; ETC. SHORT APCH CAUSED ME TO SKIP FINAL 3 GREEN CHK. AND FINALLY AUTO GEAR DROP AND WARNING LIGHTS/BUZZER DID NOT FUNCTION. WHAT I HAVE LEARNED: I KNOW SOME MAY SAY BETTER USE OF CHKLIST WOULD HAVE RESOLVED THIS PROB. BUT THE BOTTOM LINE IS; I AM GOOD ABOUT CHKLISTS -- BOTH WRITTEN AND VERBAL QUEUES. THERE WAS SOMETHING ABOUT THIS APCH THAT INTERRUPTED MY PROCESS AT A KEY MOMENT. YES; I NEED TO CREATE A 'FAIL SAFE' DOUBLE/TRIPLE CHKLIST PROCESS; AND IT IS HARD TO IMAGINE A MORE LIFE CHANGING EVENT THAN A GEAR-UP LNDG. BUT WHERE I REALLY FAILED WAS NOT PERIODICALLY CHKING THE GEAR DOWN WARNING AND FAIL SAFE SYS IN THE PLANE. THE PLANE WAS JUST OUT OF AN ANNUAL; BUT I DID NOT DO AN INFLT SYS CHK. THIS IS THE NEW AREA/PROCESS I WILL ADD. A WORKING WARNING SYS WOULD HAVE 'SAVED' ME WHEN ALL OTHER BEST INTENTIONS FAILED.