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A075253
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Trajectory of 77 under the Reverse and Add! operation carried out in base 2.
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12
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77, 166, 267, 684, 897, 1416, 1557, 2904, 3333, 5904, 6189, 11952, 12813, 24096, 24669, 48480, 50205, 97344, 98493, 195264, 198717, 391296, 393597, 783744, 790653, 1569024, 1573629, 3140352, 3154173, 6283776, 6292989, 12572160
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,1
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COMMENTS
| 22 is the smallest number whose base 2 trajectory (A061561) provably does not contain a palindrome. 77 is the next number (cf. A075252) with a completely different trajectory which has this property. A proof along the lines of Klaus Brockhaus, On the 'Reverse and Add!' algorithm in base 2, can be based on the formula given below.
lim_{n -> infinity} a(n)/a(n-1) = 2 for n mod 2 = 1.
lim_{n -> infinity} a(n)/a(n-1) = 1 for n mod 2 = 0.
Interleaving of A176632, 2*A176633, 3*A176634, 12*A176635.
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LINKS
| Klaus Brockhaus, On the 'Reverse and Add!' algorithm in base 2
Index entries for sequences related to Reverse and Add!
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FORMULA
| a(0) = 77; a(1) = 166; a(2) = 267; for n > 2 and
n = 3 (mod 4): a(n) = 48*2^(2*k)-21*2^k where k = (n+5)/4;
n = 0 (mod 4): a(n) = 48*2^(2*k)+33*2^k-3 where k = (n+4)/4;
n = 1 (mod 4): a(n) = 96*2^(2*k)-30*2^k where k = (n+3)/4;
n = 2 (mod 4): a(n) = 96*2^(2*k)+6*2^k-3 where k = (n+2)/4.
G.f.: (77+166*x+36*x^2+186*x^3+96*x^4-636*x^5-672*x^6-348*x^7-44*x^8 +632*x^9+504*x^10) / ((1-x)*(1+x)*(1-2*x^2)*(1-2*x^4)).
G.f. for the sequence starting at a(3): 3*x^3*(228+299*x-212*x^2 -378*x^3-448*x^4-446*x^5+432*x^6+524*x^7) / ((1-x)*(1+x)*(1-2*x^2)*(1-2*x^4)).
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EXAMPLE
| 267 (decimal) = 100001011 -> 100001011 + 110100001 = 1010101100 = 684 (decimal).
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PROG
| (PARI) {m=77; stop=34; c=0; while(c<stop, print1(k=m, ", "); rev=0; while(k>0, d=divrem(k, 2); k=d[1]; rev=2*rev+d[2]); c++; m=m+rev)}
(MAGMA) trajectory:=function(init, steps, base) S:=[init]; a:=S[1]; for n in [1..steps] do a+:=Seqint(Reverse(Intseq(a, base)), base); Append(~S, a); end for; return S; end function; trajectory(77, 31, 2);
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CROSSREFS
| Cf. A061561 (trajectory of 22 in base 2), A075268 (trajectory of 442 in base 2), A077076 (trajectory of 537 in base 2), A077077 (trajectory of 775 in base 2), A066059 (trajectory of n in base 2 presumably does not reach a palindrome), A075252 (trajectory of n in base 2 does not reach a palindrome and presumably does not join the trajectory of any term m < n), A092210 (trajectory of n in base 2 presumably does not join the trajectory of any m < n).
Cf. A176632 (a(4*n)), A176633 (a(4*n+1)/2), A176634 (a(4*n+2)/3), A176635 (a(4*n+3)/12).
Sequence in context: A113945 A044328 A044709 * A046513 A199994 A043518
Adjacent sequences: A075250 A075251 A075252 * A075254 A075255 A075256
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KEYWORD
| base,nonn
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AUTHOR
| Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Sep 10 2002
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EXTENSIONS
| Three comments added, g.f. edited, MAGMA program and crossrefs added by Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Apr 25 2010
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