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 A055778 Number of 1's in base phi representation of n. 15
 0, 1, 2, 2, 3, 3, 3, 2, 3, 4, 4, 5, 4, 4, 4, 5, 4, 4, 2, 3, 4, 4, 5, 5, 5, 4, 5, 6, 6, 7, 5, 5, 5, 6, 5, 5, 4, 5, 6, 6, 7, 5, 5, 5, 6, 5, 5, 2, 3, 4, 4, 5, 5, 5, 4, 5, 6, 6, 7, 6, 6, 6, 7, 6, 6, 4, 5, 6, 6, 7, 7, 7, 6, 7, 8, 8, 9, 6, 6, 6, 7, 6, 6, 5, 6, 7, 7, 8, 6, 6, 6, 7, 6, 6, 4, 5, 6, 6, 7, 7, 7, 6, 7, 8, 8 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Uses greedy algorithm (start with largest possible power of phi, then work downward) - see pseudo-code below. Conjecture: For all n, A007895(n) <= A055778(n). There is equality at 1, 7, 18, 19, 47, 48, 54, 123, 124, 130, 141, 142, 322, 323, 329, 340, 341, 369, 370, 376, 843, 844, 850, 861, 862, 890, 891, 897, 966, 967, 973, 984, 985, 2207, 2208, 2214, 2225, 2226, 2254, 2255, 2261, 2330, 2331, 2337, 2348, 2349, 2529, 2530, 2536, 2547, 2548, 2576, 2577, 2583, ... - Dale Gerdemann at Sun Apr 01 17:09:19 EDT 2012 LINKS Carmine Suriano, Table of n, a(n) for n = 0..5000 Ron Knott, Using Powers of Phi to represent Integers (Base Phi) (inspiration for this sequence). Eric Weisstein's World of Mathematics, Phi Number System EXAMPLE The phi-expansions for n<=15 are:    n   phi-rep(n)     a(n)    0       0.           0    1       1.           1    2      10.01         2    3     100.01         2    4     101.01         3    5    1000.1001       3    6    1010.0001       3    7   10000.0001       2    8   10001.0001       3    9   10010.0101       4   10   10100.0101       4   11   10101.0101       5   12  100000.101001     4   13  100010.001001     4   14  100100.001001     4   15  100101.001001     5 [Joerg Arndt, Jan 30 2012] MATHEMATICA nn = 100; len = 2*Ceiling[Log[GoldenRatio, nn]]; Table[d = RealDigits[n, GoldenRatio, len]; Total[d[[1]]], {n, 0, nn}] (* T. D. Noe, May 20 2011 *) PROG (Pseudo-code from Henry Bottomley): constant (float): phi=(sqrt(5)+1)/2; function: lphi(x)=log(x)/log(phi); variable (float): rem=n; variable (integer): count=0; loop: while rem>0 {rem=rem-phi^floor[lphi(x)]; count++; } result: return count; CROSSREFS Sequence in context: A194883 A175453 A014499 * A106482 A260236 A122462 Adjacent sequences:  A055775 A055776 A055777 * A055779 A055780 A055781 KEYWORD base,easy,nonn AUTHOR Robert Lozyniak (11(AT)onna.com), Jul 12 2000 EXTENSIONS More terms and algorithm from Henry Bottomley, Aug 04 2000 STATUS approved

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