A278765 Engel expansion of natural logarithm of golden ratio.
3, 3, 4, 4, 4, 6, 15, 48, 118, 147, 671, 1026, 3075, 44641, 48364, 1868380, 75080506, 96848501, 911582093, 2511879981, 8700005050, 15888441652, 108526838262, 446779835336, 632466801279, 1084794852728, 1184722346307, 1657692322844, 12376968750845, 17341469111712, 27996895637798, 38935285631573
Offset: 1
Examples
log(phi) = 0.4812118250596... = 1/3 + 1/(3*3) + 1/(3*3*4) + 1/(3*3*4*4) + 1/(3*3*4*4*4) + 1/(3*3*4*4*4*6) + ...
Links
- Eric Weisstein's World of Mathematics, Engel Expansion
- Eric Weisstein's World of Mathematics, Golden Ratio
- Index entries for sequences related to Engel expansions
Programs
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Mathematica
EngelExp[A_, n_]:=Join[Array[1&, Floor[A]], First@Transpose@NestList[{Ceiling[1/Expand[ #[[1]]#[[2]]-1]], Expand[ #[[1]]#[[2]]-1]}&, {Ceiling[1/(A-Floor[A])], A-Floor[A]}, n-1]]; EngelExp[N[Log[GoldenRatio], 7! ], 40]