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For two given integers    k   ,    m   , we introduce the    k   -step sumand    m   -step gap Fibonacci sequence by presenting a recurrence formula that generates the    n   th term as the sum of    k    successive previous terms starting the sum at the    m   th previous term. Known sequences, like Fibonacci, tribonacci, tetranacci, and Padovan sequences, are derived for specific values of    k   ,    m   . Two limiting properties concerning the terms of the sequence are presented. The limits are related to the spectral radius of the associated   {  0  ,  1  }  -matrix.

isrn discrete mathematics

<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:math>-Step Sum and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:mrow><mml:mi>m</mml:mi></mml:mrow></mml:math>-Step Gap Fibonacci Sequence

k-Step Sum and m-Step Gap Fibonacci Sequence