2 k + 1 ). ▁If ▁k ▁were ▁to ▁appear ▁in ▁the ▁sequence ▁( ai ), ▁this ▁would ▁cause ▁the ▁sequence ▁( qi ) ▁to ▁increase ▁by ▁ 2 − k - 1 , ▁but ▁this ▁cannot ▁happen ▁once ▁all ▁the ▁elements ▁of ▁( qi ) ▁are ▁within ▁ 2 − k - 1 ▁of ▁each ▁other . ▁Thus , ▁if ▁k ▁is ▁going ▁to ▁be ▁enumer ated ▁into ▁ai , ▁it ▁must ▁be ▁enumer ated ▁for ▁a ▁value ▁of ▁i ▁less ▁than ▁r ( 2 k + 1 ). ▁This ▁leaves ▁a ▁finite ▁number ▁of ▁possible ▁places ▁where ▁k ▁could ▁be ▁enumer ated . ▁To ▁complete ▁the ▁decision ▁procedure , ▁check ▁these ▁in ▁an ▁effective ▁manner ▁and ▁the ▁return ▁ 0 ▁or ▁ 1 ▁depending ▁on ▁whether ▁k ▁is ▁found . ▁ ▁See ▁also ▁ ▁Comput ation ▁in ▁the ▁limit ▁ ▁References ▁▁ ▁Douglas ▁Br id ges ▁and ▁Fred ▁Rich man . ▁Vari eties ▁of ▁Con struct ive ▁Mathemat ics , ▁Oxford , ▁ 1 9 8 7 . ▁ ▁B . A . ▁K ush ner ▁( 1 9 8 4 ), ▁L ect ures ▁on ▁construct ive ▁mathematical ▁analysis , ▁American ▁Mathemat ical ▁Society , ▁Trans l ations ▁of ▁Mathemat ical ▁Mon ograph s ▁v . ▁ 6 0 . ▁▁ ▁Jakob ▁G . ▁Sim ons en ▁( 2 0 0 5 ), ▁" Spe cker ▁sequences ▁revis ited ", ▁Mathemat ical ▁Log ic ▁Qu arter ly , ▁v . ▁ 5 1 , ▁pp . ▁ 5 3 2