Search: a176633 -id:a176633
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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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+10
13
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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
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OFFSET
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0,1
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COMMENTS
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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.
Pattern with cycle length 4 in binary representation, represented by contextfree grammars with production rules:
S_a -> 10 T_a 00, T_a -> 1 T_a 0 | 1100010;
S_b -> 11 T_b 01, T_b -> 0 T_b 1 | 0000101;
S_c -> 10 T_c 000, T_c -> 1 T_c 0 | 1101011;
S_d -> 11 T_d 101, T_d -> 0 T_d 1 | 0100000;
the trajectory is similar to that of 22 (see A058042) except for the stopping strings in T_a, T_b, T_c and T_d. (End)
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LINKS
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FORMULA
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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
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267 (decimal) = 100001011 -> 100001011 + 110100001 = 1010101100 = 684 (decimal).
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MAPLE
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seq(coeff(series((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)), x, n+1), x, n), n = 0 .. 40); # Muniru A Asiru, Feb 12 2019
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MATHEMATICA
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CoefficientList[Series[(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)), {x, 0, 40}], x] (* G. C. Greubel, Feb 11 2019 *)
NestWhileList[# + IntegerReverse[#, 2] &, 77, # !=
IntegerReverse[#, 2] &, 1, 31] (* Robert Price, Oct 18 2019 *)
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PROG
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(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);
(Haskell)
a075253 n = a075253_list !! n
(Sage) ((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))).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Feb 11 2019
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CROSSREFS
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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).
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KEYWORD
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base,nonn
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AUTHOR
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EXTENSIONS
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Three comments added, g.f. edited, MAGMA program and crossrefs added by Klaus Brockhaus, Apr 25 2010
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STATUS
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approved
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A176635
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a(n) = 6*a(n-1)-8*a(n-2) for n > 1; a(0) = 57, a(1) = 242.
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+10
6
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57, 242, 996, 4040, 16272, 65312, 261696, 1047680, 4192512, 16773632, 67101696, 268421120, 1073713152, 4294909952, 17179754496, 68719247360, 274877448192, 1099510710272, 4398044676096, 17592182374400, 70368736837632
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OFFSET
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0,1
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COMMENTS
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Related to Reverse and Add trajectory of 77 in base 2: a(n) = A075253(4*n+3)/12, i.e., one twelfth of fourth quadrisection of A075253.
Second binomial transform of 57 followed by 128*A000079.
Third binomial transform of A176636.
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LINKS
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FORMULA
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a(n) = 64*4^n-7*2^n.
G.f.: (57-100*x)/((1-2*x)*(1-4*x)).
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MATHEMATICA
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CoefficientList[Series[(57 - 100 x)/((1 - 2 x) (1 - 4 x)), {x, 0, 40}], x] (* Vincenzo Librandi, Sep 24 2013 *)
LinearRecurrence[{6, -8}, {57, 242}, 30] (* Harvey P. Dale, Jun 08 2016 *)
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PROG
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(PARI) {m=21; v=concat([57, 242], vector(m-2)); for(n=3, m, v[n]=6*v[n-1]-8*v[n-2]); v}
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A176632
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a(n) = 6*a(n-1)-8*a(n-2)-9 for n > 2; a(0) = 77, a(1) = 897, a(2) = 3333.
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+10
5
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77, 897, 3333, 12813, 50205, 198717, 790653, 3154173, 12599805, 50365437, 201394173, 805441533, 3221495805, 12885442557, 51540688893, 206160592893, 824638046205, 3298543534077, 13194156834813, 52776592736253
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OFFSET
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0,1
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COMMENTS
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Related to Reverse and Add trajectory of 77 in base 2: a(n) = A075253(4*n), i.e., first quadrisection of A075253.
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LINKS
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FORMULA
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a(n) = 3*(64*4^n+22*2^n-1) for n > 0, a(0) = 77.
G.f.: (77+358*x-1868*x^2+1424*x^3)/((1-x)*(1-2*x)*(1-4*x)).
G.f. for the sequence starting at a(1): 3*x*(299-982*x+680*x^2)/((1-x)* (1-2*x)*(1-4*x)).
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MATHEMATICA
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CoefficientList[Series[(77 + 358 x - 1868 x^2 + 1424 x^3)/((1 - x) (1 - 2 x) (1 - 4 x)), {x, 0, 40}], x] (* Vincenzo Librandi, Sep 24 2013 *)
Join[{77}, RecurrenceTable[{a[1]==897, a[2]==3333, a[n]==6a[n-1]-8a[n-2]- 9}, a[n], {n, 20}]] (* Harvey P. Dale, May 21 2019 *)
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PROG
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(PARI) {m=20; v=concat([77, 897, 3333], vector(m-3)); for(n=4, m, v[n]=6*v[n-1]-8*v[n-2]-9); v}
(Magma) [77] cat [3*(64*4^n+22*2^n-1): n in [1..25]]; // Vincenzo Librandi, Sep 24 2013
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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A176634
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a(n) = 6*a(n-1)-8*a(n-2)-3 for n > 2; a(0) = 89, a(1) = 519, a(2) = 2063.
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+10
5
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89, 519, 2063, 8223, 32831, 131199, 524543, 2097663, 8389631, 33556479, 134221823, 536879103, 2147500031, 8589967359, 34359803903, 137439084543, 549756076031, 2199023779839, 8796094070783, 35184374185983
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,1
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COMMENTS
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Related to Reverse and Add trajectory of 77 in base 2: a(n)= A075253(4*n+2)/3, i.e., one third of third quadrisection of A075253.
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LINKS
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FORMULA
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a(n) = 128*4^n+4*2^n-1 for n > 0, a(1) = 89.
G.f.: (89-104*x-324*x^2+336*x^3)/((1-x)*(1-2*x)*(1-4*x)).
G.f. for the sequence starting at a(1): x*(519-1570*x+1048*x^2)/((1-x)* (1-2*x)*(1-4*x)).
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MATHEMATICA
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CoefficientList[Series[(89 - 104 x - 324 x^2 + 336 x^3)/((1 - x) (1 - 2 x) (1 - 4 x)), {x, 0, 40}], x] (* Vincenzo Librandi, Sep 24 2013 *)
LinearRecurrence[{7, -14, 8}, {89, 519, 2063, 8223}, 20] (* Harvey P. Dale, Jun 20 2023 *)
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PROG
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(PARI) {m=20; v=concat([89, 519, 2063], vector(m-3)); for(n=4, m, v[n]=6*v[n-1]-8*v[n-2]-3); v}
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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