A133048
Powerback(n): reverse the decimal expansion of n, drop any leading zeros, then apply the powertrain map of A133500 to the resulting number.
Original entry on oeis.org
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 2, 1, 4, 9, 16, 25, 36, 49, 64, 81, 3, 1, 8, 27, 64, 125, 216, 343, 512, 729, 4, 1, 16, 81, 256, 625, 1296, 2401, 4096, 6561, 5, 1, 32, 243, 1024, 3125, 7776, 16807, 32768, 59049, 6, 1, 64, 729, 4096, 15625, 46656, 117649
Offset: 0
E.g. 240 -> (0)42 -> 4^2 = 16; 12345 -> 54321 -> 5^4*3^2*1 = 5625.
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a133048 0 = 0
a133048 n = train $ dropWhile (== 0) $ a031298_row n where
train [] = 1
train [x] = x
train (u:v:ws) = u ^ v * (train ws)
-- Reinhard Zumkeller, May 27 2013
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powerback:=proc(n) local a,i,j,t1,t2,t3;
if n = 0 then RETURN(0); fi;
t1:=convert(n, base, 10); t2:=nops(t1);
for i from 1 to t2 do if t1[i] > 0 then break; fi; od:
a:=1; t3:=t2-i+1;
for j from 0 to floor(t3/2)-1 do a := a*t1[i+2*j]^t1[i+2*j+1]; od:
if t3 mod 2 = 1 then a:=a*t1[t2]; fi;
RETURN(a); end;
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ptm[n_]:=Module[{idn=IntegerDigits[IntegerReverse[n]]},If[ EvenQ[ Length[idn]],Times@@ (#[[1]]^#[[2]]&/@Partition[idn,2]),(Times@@(#[[1]]^#[[2]]&/@Partition[ Most[ idn],2]))Last[idn]]];Array[ptm,70,0] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jul 05 2020 *)
A133474
Inverse binomial transform of (A113405 preceded by 0).
Original entry on oeis.org
0, 0, 0, 1, 6, 24, 81, 252, 756, 2241, 6642, 19764, 59049, 176904, 530712, 1592865, 4780782, 14346720, 43046721, 129146724, 387440172, 1162300833, 3486843450, 10460412252, 31381059609, 94143001680, 282429005040, 847287546561
Offset: 0
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a:=[0,0,1];; for n in [4..30] do a[n]:=6*a[n-1]-12*a[n-2]+9*a[n-3]; od; a; # G. C. Greubel, Nov 21 2019
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R:=PowerSeriesRing(Integers(), 30); [0,0,0] cat Coefficients(R!( x^3/((1-3*x)*(1-3*x+3*x^2)) )); // G. C. Greubel, Nov 21 2019
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seq(coeff(series(x^3/((1-3*x)(1-3*x+3*x^2)), x, n+1), x, n), n = 0 .. 40); # G. C. Greubel, Nov 21 2019
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LinearRecurrence[{6,-12,9}, {0,0,0,1}, 30] (* G. C. Greubel, Nov 21 2019 *)
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my(x='x+O('x^30)); concat([0,0,0], Vec(x^3/((1-3*x)*(1-3*x+3*x^2)))) \\ G. C. Greubel, Nov 21 2019
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def A133474_list(prec):
P. = PowerSeriesRing(ZZ, prec)
return P(x^3/((1-3*x)*(1-3*x+3*x^2))).list()
A133474_list(30) # G. C. Greubel, Nov 21 2019
A133144
Start with n and repeatedly apply the powerback map of A133048. Sequence gives number of steps to the point where the next number would be one that has appeared before.
Original entry on oeis.org
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 0, 4, 6, 3, 2, 1, 1, 1, 7, 3, 2, 3, 6, 3, 7, 1, 1, 2, 2, 4, 3, 3, 3, 2, 5, 1, 1, 2, 5, 2, 2, 6, 5, 4, 2, 1, 1, 3, 7, 2, 4, 4, 5, 4, 5, 1, 1, 4, 4, 5, 2, 3, 6, 4, 3, 1, 1, 4, 5, 3, 5, 5, 3, 4, 5, 1, 1, 3, 9, 4, 6, 2, 2, 5, 3
Offset: 0
n, a(n), trajectory
22, 1, [22, 4]
23, 1, [23, 9]
24, 2, [24, 16, 6]
25, 0, [25]
26, 4, [26, 36, 216, 12, 2]
27, 6, [27, 49, 6561, 15625, 194400, 2304, 9]
28, 3, [28, 64, 4096, 0]
29, 2, [29, 81, 1]
30, 1, [30, 3]
31, 1, [31, 1]
32, 1, [32, 8]
33, 7, [33, 27, 49, 6561, 15625, 194400, 2304, 9]
34, 3, [34, 64, 4096, 0]
35, 2, [35, 125, 25]
36, 3, [36, 216, 12, 2]
37, 6, [37, 343, 243, 162, 64, 4096, 0]
38, 3, [38, 512, 10, 1]
39, 7, [39, 729, 567, 588245, 5242880000, 8589934592, 105911076180375000000000, 0]
A145608
Numbers a(n)=k such that (1/3)*(5*(2k+1)^2-2) is A057080(n)^2.
Original entry on oeis.org
0, 3, 27, 216, 1704, 13419, 105651, 831792, 6548688, 51557715, 405913035, 3195746568, 25160059512, 198084729531, 1559517776739, 12278057484384, 96664942098336, 761041479302307, 5991666892320123, 47172293659258680, 371386682381749320, 2923921165394735883, 23019982640776137747
Offset: 0
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RecurrenceTable[{a[0]==0,a[1]==3,a[n]==8a[n-1]-a[n-2]+3},a,{n,30}] (* or *) LinearRecurrence[{9,-9,1},{0,3,27},30] (* Harvey P. Dale, May 06 2013 *)
Made definition and sequence consistent. Changed offset to 0. -
R. J. Mathar, Oct 16 2008
Showing 1-4 of 4 results.
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