%I #26 Dec 07 2015 22:54:25
%S 1,1,1,2,1,2,1,1,1,2,1,3,1,1,1,2,3,1,3,1,1,1,1,1,2,2,3,2,1,1,3,1,4,1,
%T 1,1,1,2,1,2,1,2,2,3,2,5,1,1,1,3,4,1,4,1,1,1,1,1,2,1,2,1,1,1,2,1,3,2,
%U 2,2,3,5,2,5,2,1,1,1,1,2,3,4,3,1,1,4,1,5
%N Number of distinct sum representations of n by Fibonacci numbers with minimal digit sum.
%C The digits are any nonnegative integers. The value of the minimal sum of digits is given by A007895. The sequence of those numbers where this sequence has value 1 is A256133.
%H M. Drmota and M. Gajdosik, <a href="http://www.fq.math.ca/Scanned/36-1/drmota.pdf">The parity of the sum of digits function of generalized Zeckendorf expansions</a>, The Fibonacci Quarterly, 36:1 (1988), pp. 3-19.
%e a(12) = 3 because 12 = 8 + 3 + 1 = 8 + 2 + 2 = 5 + 5 + 2 has three distinct representations.
%p L:=[1,2,3,5,8,13,21,34,55]; LC:=[1,1,1,2,1,2,1,1,1]:LS:=[1,1,1,2,1,2,2,1,2]: for n from 10 to 88 do: ct:=0: ss:=n: sm:=n: b0:=1: b1:=2: b2:=3: b3:=4: b4:=trunc(n/L[5]): b5:=trunc(n/L[6]): b6:=trunc(n/L[7]):b7:=trunc(n/L[8]):b8:=trunc(n/L[9]):
%p > for n0 from 0 to b0 do:for n1 from 0 to b1 do: for n2 from 0 to b2 do:for n3 from 0 to b3 do: for n4 from 0 to b4 do: for n5 from 0 to b5 do: for n6 from 0 to b6 do:
%p > for n7 from 0 to b7 do:for n8 from 0 to b8 do: if n=n0*L[1]+n1*L[2]+n2*L[3]+n3*L[4]+n4*L[5]+n5*L[6]+n6*L[7]+n7*L[8]+n8*L[9] then ss:=n0+n1+n2+n3+n4+n5+n6+n7+n8:fi:
%p > if sm>ss then sm:=ss: fi: od:od:od:od:od:od:od:od:od:for n0 from 0 to b0 do:for n1 from 0 to b1 do: for n2 from 0 to b2 do:for n3 from 0 to b3 do:
%p > for n4 from 0 to b4 do:for n5 from 0 to b5 do:for n6 from 0 to b6 do:
%p > for n7 from 0 to b7 do:for n8 from 0 to b8 do:
%p > if n=n0*L[1]+n1*L[2]+n2*L[3]+n3*L[4]+n4*L[5]+n5*L[6]+n6*L[7]+n7*L[8]+n8*L[9] then st:=n0+n1+n2+n3+n4+n5+n6+n7+n8: if st=sm then ct:=ct+1: fi:fi: od;od:od:od:od:od:od:od:od: LS:=[op(LS),sm]: LC:=[op(LC),ct]: od: print(LC):
%Y Cf. A000045, A007895, A256133.
%K nonn
%O 1,4
%A _Patrick Okolo Edeogu_, Oct 20 2015