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infinite square board with an integer writter in each square, <=N zeroes

Source: 2003 Swedish Mathematical Competition p6

March 21, 2021
combinatorics

Problem Statement

Consider an infinite square board with an integer written in each square. Assume that for each square the integer in it is equal to the sum of its neighbor to the left and its neighbor above. Assume also that there exists a row R0R_0 in the board such that all numbers in R0R_0 are positive. Denote by R1R_1 the row below R0R_0 , by R2R_2 the row below R1R_1 etc. Show that for each N≥1N \ge 1 the row RNR_N cannot contain more than NN zeroes.
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