A lead-free neutron monitor operating at High Altitude Research Laboratory (HARL), Gulmarg optimized for detecting 2.45MeV neutron bursts produced during the atmospheric lightning discharges is also concurrently used for studying background neutron component present in the atmosphere. These background neutrons are produced due to the interaction of primary cosmic rays with the atmospheric constituents. In order to study and extract the information about the yield of the neutron production during transient atmospheric lightning discharges, the system is continuously operated to monitor and record the cosmic ray produced background secondary neutrons in the atmosphere. The data analysis of the background neutrons recorded by Lead-Free Gulmarg Neutron Monitor (LFGNM) has convincingly established that the modulation effects due to solar activity phenomena compare very well with those monitored by the worldwide IGY or NM64 type neutron monitors which have optimum energy response relatively towards the higher energy regime of the cosmic rays. The data has revealed various types of modulation phenomena like diurnal variation, Forbush decrease etc. during its entire operational period. However, a new kind of a periodic/seasonal variation pattern is also revealed in the data from September 2007 to September 2012, which is seen to be significantly consistent with the data recorded by Emilio Segre observatory, Israel (ESOI) Neutron Monitor. Interestingly, both these neutron monitors have comparable latitude and altitude. However, the same type of consistency is not observed in the data recorded by the other conventional neutron monitors operating across the globe.