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A144116
Number of non-Fibonacci parts in all partitions of n.
7
0, 0, 0, 1, 1, 3, 5, 9, 14, 23, 34, 54, 76, 113, 158, 226, 309, 431, 580, 790, 1049, 1402, 1838, 2423, 3140, 4081, 5242, 6739, 8574, 10918, 13780, 17392, 21795, 27291, 33969, 42248, 52258, 64572, 79429, 97577, 119388, 145891, 177630, 215986, 261789
OFFSET
1,6
LINKS
FORMULA
a(n) = A006128(n)-A144115(n).
MAPLE
b:= proc(n) option remember; true end: l:= [0, 1]: for k to 100 do b(l[1]):= false; l:= [l[2], l[1]+l[2]] od: aa:= proc(n, i) option remember; local g, h; if n=0 then [1, 0] elif i=0 or n<0 then [0, 0] else g:= aa(n, i-1); h:= aa(n-i, i); [g[1]+h[1], g[2]+h[2] +`if`(b(i), h[1], 0)] fi end: a:= n-> aa(n, n)[2]: seq(a(n), n=1..60); # Alois P. Heinz, Jul 28 2009
MATHEMATICA
Clear[b]; b[_] = True; l = {0, 1}; For[k=1, k <= 100, k++, b[l[[1]]] = False; l = {l[[2]], l[[1]] + l[[2]]}]; aa[n_, i_] := aa[n, i] = Module[{g, h}, If[n==0, {1, 0}, If[i==0 || n<0, {0, 0}, g = aa[n, i-1]; h = aa[n-i, i]; {g[[1]] + h[[1]], g[[2]] + h[[2]] + If[b[i], h[[1]], 0]}] ]]; a[n_] := aa[n, n][[2]]; Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Jul 30 2015, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Omar E. Pol, Sep 11 2008
EXTENSIONS
More terms from Alois P. Heinz, Jul 28 2009
STATUS
approved