A168607
a(n) = 3^n + 2.
Original entry on oeis.org
3, 5, 11, 29, 83, 245, 731, 2189, 6563, 19685, 59051, 177149, 531443, 1594325, 4782971, 14348909, 43046723, 129140165, 387420491, 1162261469, 3486784403, 10460353205, 31381059611, 94143178829, 282429536483, 847288609445
Offset: 0
- Vincenzo Librandi, Table of n, a(n) for n = 0..1000
- Gennady Eremin, Arithmetization of well-formed parenthesis strings. Motzkin Numbers of the Second Kind, arXiv:2012.12675 [math.CO], 2020.
- Kurt Mahler, The representation of squares to the base 3, Acta Arith. Vol. 53, Issue 1 (1989), p. 99-106.
- Index entries for linear recurrences with constant coefficients, signature (4,-3).
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[3^n+2: n in [0..30]];
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A168607:=n->3^n + 2; seq(A168607(n), n=0..30); # Wesley Ivan Hurt, Mar 21 2014
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CoefficientList[Series[(3 - 7 x)/((1-x) (1-3 x)), {x, 0, 30}], x] (* Vincenzo Librandi, Feb 06 2013 *)
NestList[3 # - 4 & , 3, 25] (* Bruno Berselli, Feb 06 2013 *)
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a(n)=3^n+2 \\ Charles R Greathouse IV, Oct 07 2015
A340131
Numbers whose ternary expansions have the same number of 1's and 2's and, in each prefix (initial fragment), at least as many 1's as 2's.
Original entry on oeis.org
0, 5, 11, 15, 29, 33, 44, 45, 50, 83, 87, 98, 99, 104, 116, 128, 132, 135, 140, 146, 150, 245, 249, 260, 261, 266, 278, 290, 294, 297, 302, 308, 312, 332, 344, 348, 377, 380, 384, 395, 396, 401, 405, 410, 416, 420, 434, 438, 449, 450, 455, 731, 735, 746, 747
Offset: 1
The first terms 0 and 5 are obvious, because the four intermediate ternary codes 1, 2, 10[3], and 11[4] are rejected due to a violation of the balance of 1's and 2's. Next, the successor function S works: for any term x, the next term is S(x).
Iterating over numbers is inefficient; code suffixes (final digits) can be processed faster. The transition from 0 to 12[5] is generalized for terms that are multiples of 9. For example,
S(10200[99]) = 10212[104], S(1122000[1188]) = 1122012[1193], etc.
In this case, the calculation of the subsequent term is reduced to simply replacing the suffix s = 00 with the subsequent suffix s'= 12.
Another common suffix is s = 02..2 = 02^k (twos are repeated at the end of the ternary code). Then the subsequent suffix is s'= 202..2 = 202^(k-1), i.e., within such a suffix, the first two digits are reversed. Here are some examples:
k = 1, S(1002[29]) = 1020[33], the increment is 4*3^0 = 4;
k = 2, S(110022[332]) = 110202[344], the increment is 4*3^1 = 12;
k = 3, S(10110222[2537]) = 10112022[2573], the increment is 4*3^2 = 36;
k = 4, S(111102222[9800]) = 111120222[9908], the increment is 4*3^3 = 108.
There are 5 such group suffixes.
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is(n) = {my(d = digits(n, 3), v = [0, 0]); for(i = 1, #d, if(d[i] > 0, v[d[i]]++); if(v[1] < v[2], return(0))); v[1] == v[2] } \\ David A. Corneth, Dec 29 2020
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def digits(n, b):
out = []
while n >= b:
out.append(n % b)
n //= b
return [n] + out[::-1]
def ok(n):
t = digits(n, 3)
if t.count(1) != t.count(2): return False
return all(t[:i].count(1) >= t[:i].count(2) for i in range(1, len(t)))
print([n for n in range(750) if ok(n)]) # Michael S. Branicky, Dec 29 2020
A134753
Numbers k such that 3^(2*k-1) + 2 is prime.
Original entry on oeis.org
1, 2, 8, 32, 62, 70, 118, 122, 158, 182, 196, 566, 752, 3602, 21896, 22768, 53072
Offset: 1
A340544
Numbers from A340131 that are not multiples of 3.
Original entry on oeis.org
5, 11, 29, 44, 50, 83, 98, 104, 116, 128, 140, 146, 245, 260, 266, 278, 290, 302, 308, 332, 344, 377, 380, 395, 401, 410, 416, 434, 449, 455, 731, 746, 752, 764, 776, 788, 794, 818, 830, 863, 866, 881, 887, 896, 902, 920, 935, 941, 980, 992, 1025, 1028, 1043
Offset: 1
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def digits(n, b):
out = []
while n >= b:
out.append(n % b)
n //= b
return [n] + out[::-1]
def ok(n):
if n%3 == 0: return False
t = digits(n, 3)
if t.count(1) != t.count(2): return False
return all(t[:i].count(1) >= t[:i].count(2) for i in range(1, len(t)))
print([n for n in range(750) if ok(n)]) # after Michael S. Branicky (A340131)
Showing 1-4 of 4 results.
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