keyword:hear - cp's OEIS frontend

A243658 a(0)=0; thereafter a(n) = noz(n+a(n-1)), where noz(n) = A004719(n).

0, 1, 3, 6, 1, 6, 12, 19, 27, 36, 46, 57, 69, 82, 96, 111, 127, 144, 162, 181, 21, 42, 64, 87, 111, 136, 162, 189, 217, 246, 276, 37, 69, 12, 46, 81, 117, 154, 192, 231, 271, 312, 354, 397, 441, 486, 532, 579, 627, 676, 726, 777, 829, 882, 936, 991, 147, 24, 82, 141, 21, 82, 144, 27, 91, 156, 222, 289

Offset: 0

N. J. A. Sloane, Jun 11 2014

Zeroless analog of triangular numbers.

N. J. A. Sloane, Table of n, a(n) for n = 0..5000

Cf. A004719, A242350, A243063, A243657.

Row n = 3 of A373169.

noz:=proc(n) local a,t1,i,j; a:=0; t1:=convert(n,base,10); for i from 1 to nops(t1) do j:=t1[nops(t1)+1-i]; if j <> 0 then a := 10*a+j; fi; od: a; end;
t1:=[0]; for n from 1 to 50 do t1:=[op(t1),noz(n+t1[n])]; od: t1;

noz[n_] := FromDigits[DeleteCases[IntegerDigits[n], 0]];
Block[{n = 0}, NestList[noz[++n+#] &, 0, 100]] (* Paolo Xausa, Apr 17 2024 *)

from itertools import count, islice
def noz(n): return int(str(n).replace("0", ""))
def agen(): # generator of terms
    yield (an:=0)
    yield from (an:=noz(n+an) for n in count(1))
print(list(islice(agen(), 68))) # Michael S. Branicky, Jul 02 2024

A067992 a(0)=1 and, for n > 0, a(n) is the smallest positive integer such that the ratios min(a(k)/a(k-1), a(k-1)/a(k)) for 0 < k <= n are all distinct.

Original entry on oeis.org

1, 1, 2, 3, 1, 4, 3, 5, 1, 6, 5, 2, 7, 1, 8, 3, 7, 4, 5, 7, 6, 11, 1, 9, 2, 11, 3, 10, 1, 12, 5, 8, 7, 9, 4, 11, 5, 9, 8, 11, 7, 10, 9, 11, 10, 13, 1, 14, 3, 13, 2, 15, 1, 16, 3, 17, 1, 18, 5, 13, 4, 15, 7, 12, 11, 13, 6, 17, 2, 19, 1, 20, 3, 19, 4, 17, 5, 14, 9, 13, 7, 16, 5, 19, 6, 23, 1, 21, 2

Offset: 0

John W. Layman, Feb 06 2002

Every positive rational number appears exactly once as the ratio of adjacent terms (in either order). Conjecture: adjacent terms are always relatively prime. - Franklin T. Adams-Watters, Sep 13 2006

			The sequence of all rational numbers between 0 and 1 obtained by taking ratios of sorted consecutive terms begins: 1/2, 2/3, 1/3, 1/4, 3/4, 3/5, 1/5, 1/6, 5/6, 2/5, 2/7, 1/7, 1/8, 3/8, 3/7, 4/7, 4/5, 5/7, 6/7. - _Gus Wiseman_, Aug 30 2018

Rémy Sigrist, Table of n, a(n) for n = 0..10000
Neil Calkin and Herbert S. Wilf, Recounting the rationals, The American Mathematical Monthly, Vol. 107, No. 4 (2000), 360-363.
Neil Calkin and Herbert S. Wilf, Recounting the rationals, Fermat's Library (2008).

See A066720 for a somewhat similar sequence.

Cf. A002487, A025582, A050376, A057979.

Nest[Function[seq,Append[seq,NestWhile[#+1&,1,MemberQ[Divide@@@Sort/@Partition[seq,2,1],Min[Last[seq]/#,#/Last[seq]]]&]]],{1},100] (* Gus Wiseman, Aug 30 2018 *)

seen = Set([]); other(p) = for (v=1, oo, my (r = min(v,p)/max(v,p)); if (!set search(seen, r), seen = set union(seen, Set([r])); return (v)))
for (n=0, 88, v = if (n==0, 1, other(v)); print1 (v ", ")) \\ Rémy Sigrist, Aug 07 2017

a(6)=3, since 1/4 and 2/4 = 1/2 have already occurred as ratios of adjacent terms.

A080099 Triangle T(n,k) = n AND k, 0<=k<=n, bitwise logical AND, read by rows.

Original entry on oeis.org

0, 0, 1, 0, 0, 2, 0, 1, 2, 3, 0, 0, 0, 0, 4, 0, 1, 0, 1, 4, 5, 0, 0, 2, 2, 4, 4, 6, 0, 1, 2, 3, 4, 5, 6, 7, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 1, 0, 1, 0, 1, 0, 1, 8, 9, 0, 0, 2, 2, 0, 0, 2, 2, 8, 8, 10, 0, 1, 2, 3, 0, 1, 2, 3, 8, 9, 10, 11, 0, 0, 0, 0, 4, 4, 4, 4, 8, 8, 8, 8, 12, 0, 1, 0, 1, 4, 5, 4, 5, 8, 9, 8, 9

Offset: 0

Reinhard Zumkeller, Jan 28 2003

A080100(n) = number of numbers k such that n AND k = 0 in n-th row of the triangular array.

			Triangle starts:
0
0 1
0 0 2
0 1 2 3
0 0 0 0 4
0 1 0 1 4 5
0 0 2 2 4 4 6
0 1 2 3 4 5 6 7
...

Rick L. Shepherd, Rows n=0..500 of triangle, flattened
Eric Weisstein's World of Mathematics, AND.

Cf. A080100, A222423 (row sums), A004198 (array).

Other triangles: A080098 (OR), A051933 (XOR), A265705 (IMPL), A102037 (CNIMPL).

import Data.Bits ((.&.))
a080099 n k = n .&. k :: Int
a080099_row n = map (a080099 n) [0..n]
a080099_tabl = map a080099_row [0..]
-- Reinhard Zumkeller, Aug 03 2014, Jul 05 2012

Column[Table[BitAnd[n, k], {n, 0, 15}, {k, 0, n}], Center] (* Alonso del Arte, Jun 19 2012 *)

T(n,k)=bitand(n,k) \\ Charles R Greathouse IV, Jan 26 2013

def T(n, k): return n & k
print([T(n, k) for n in range(14) for k in range(n+1)]) # Michael S. Branicky, Dec 16 2021

A082850 Let S(0) = {}, S(n) = {S(n-1), S(n-1), n}; sequence gives S(infinity).

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 1, 1, 2, 3, 4, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 1, 1, 2, 3, 4, 5, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 1, 1, 2, 3, 4, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 1, 1, 2, 3, 4, 5, 6, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 1, 1, 2, 3, 4, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 1, 1, 2, 3, 4, 5, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 1

Offset: 1

Benoit Cloitre, Apr 14 2003

Sequence counts up to successive values of A001511; i.e., apply the morphism k -> 1,2,...,k to A001511. If all 1's are removed from the sequence, the resulting sequence b has b(n) = a(n)+1. A101925 lists the positions of 1's in this sequence.

The geometric mean of this sequence approaches the Somos constant (A112302). - Jwalin Bhatt, Jan 30 2025

			S(1) = {1}, S(2) = {1,1,2}, S(3) = {1,1,2,1,1,2,3}, etc.

Alois P. Heinz, Table of n, a(n) for n = 1..65535

Cf. A082851 (partial sums).
Cf. A001511, A101925, A112302.
Cf. A215020.

Fold[Flatten[{#1, #1, #2}] &, {}, Range[5]] (* Birkas Gyorgy, Apr 13 2011 *)
Flatten[Table[Length@Last@Split@IntegerDigits[2 n, 2], {n, 20}] /. {n_ ->Range[n]}] (* Birkas Gyorgy, Apr 13 2011 *)

S = []; [S.extend(S + [n]) for n in range(1, 8)]
print(S) # Michael S. Branicky, Jul 02 2022

from itertools import count, islice
def A082850_gen(): # generator of terms
    S = []
    for n in count(1):
        yield from (m:=S+[n])
        S += m #
A082850_list = list(islice(A082850_gen(),20)) # Chai Wah Wu, Mar 06 2023

a(2^m - 1) = m.
If n = 2^m - 1 + k with 0 < k < 2^m, then a(n) = a(k). - Franklin T. Adams-Watters, Aug 16 2006
a(n) = log_2(A182105(n)) + 1. - Laurent Orseau, Jun 18 2019
a(n) = 1 + A215020(n). - Joerg Arndt, Mar 04 2025

A247074 a(n) = phi(n)/(Product_{primes p dividing n } gcd(p - 1, n - 1)).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 4, 3, 4, 1, 4, 1, 6, 2, 8, 1, 6, 1, 8, 3, 10, 1, 8, 5, 12, 9, 4, 1, 8, 1, 16, 5, 16, 6, 12, 1, 18, 6, 16, 1, 12, 1, 20, 3, 22, 1, 16, 7, 20, 8, 8, 1, 18, 10, 24, 9, 28, 1, 16, 1, 30, 9, 32, 3, 4, 1, 32, 11, 8, 1, 24, 1, 36, 10, 12, 15, 24, 1, 32, 27, 40, 1, 24, 4, 42, 14, 40, 1, 24, 2, 44, 15, 46

Offset: 1

Eric Chen, Nov 16 2014

a(n) = A000010(n)/A063994(n). - Eric Chen, Nov 29 2014
Does every natural number appear in this sequence? If so, do they appear infinitely many times? - Eric Chen, Nov 26 2014
A063994(n) must be a factor of EulerPhi(n). - Eric Chen, Nov 26 2014
Number n is (Fermat) pseudoprimes (or prime) to one in a(n) of its coprime bases. That is, b^(n-1) = 1 (mod n) for one in a(n) of numbers b coprime to n. - Eric Chen, Nov 26 2014
a(n) = 1 if and only if n is 1, prime (A000040), or Carmichael number (A002997). - Eric Chen, Nov 26 2014
a(A191311(n)) = 2. - Eric Chen, Nov 26 2014
a(p^n) = p^(n-1), where p is a prime. - Eric Chen, Nov 26 2014
a(A209211(n)) = EulerPhi(A209211(n)), because A063994(A209211(n)) = 1. - Eric Chen, Nov 26 2014

			EulerPhi(15) = 8, and that 15 is a Fermat pseudoprime in base 1, 4, 11, and 14, the total is 4 bases, so a(15) = 8/4 = 2.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Gérard P. Michon, Pseudoprimes

Cf. A063994, A182816, A002997, A191311, A064234, A000010, A063994, A003557, A209211.

a063994[n_] := Times @@ GCD[n - 1, First /@ FactorInteger@ n - 1]; a063994[1] = 1; a247074[n_] := EulerPhi[n]/a063994[n]; Array[a247074, 150]

a(n)=my(f=factor(n));eulerphi(f)/prod(i=1,#f~,gcd(f[i,1]-1,n-1)) \\ Charles R Greathouse IV, Nov 17 2014

A003557(n) <= a(n) <= n, and a(n) is a multiple of A003557(n). - Charles R Greathouse IV, Nov 17 2014

A249814 "Mountains of Eratosthenes" permutation: a(1) = 1, a(n) = A249741(A001511(n), a(A003602(n))).

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 7, 6, 9, 14, 15, 24, 13, 20, 11, 10, 17, 26, 27, 34, 29, 44, 47, 48, 25, 38, 39, 54, 21, 32, 19, 12, 33, 50, 51, 64, 53, 80, 67, 76, 57, 86, 87, 114, 93, 140, 95, 120, 49, 74, 75, 94, 77, 116, 107, 90, 41, 62, 63, 84, 37, 56, 23, 16, 65, 98, 99, 124, 101, 152, 127, 118, 105, 158, 159, 204, 133, 200, 151, 142

Offset: 1

Antti Karttunen, Nov 06 2014

This sequence is a "recursed variant" of A249811.
From Antti Karttunen, Jan 18 2015: (Start)
This can be viewed as an entanglement or encoding permutation where the complementary pairs of sequences to be interwoven together are even and odd numbers (A005843/A005408) which are entangled with another complementary pair: even numbers in the order they appear in A253886 and odd numbers in their usual order: (A253886/A005408).
From the above follows also that this sequence can be represented as a binary tree. Each child to the left is obtained by doubling the parent and subtracting one, and each child to the right is obtained by applying A253886 to the parent:
                                    1
                                    |
                 ...................2...................
                3                                       4
      5......../ \........8                   7......../ \........6
     / \                 / \                 / \                 / \
    /   \               /   \               /   \               /   \
   /     \             /     \             /     \             /     \
  9       14         15       24         13       20         11       10
17 26   27  34     29  44   47  48     25  38   39  54     21  32   19  12
(End)
For listening I recommend some (mostly) percussive MIDI-instrument and the pitch offset set to at least 29 and the tempo (rate) to about 60. - Antti Karttunen, Feb 17 2015

Antti Karttunen, Table of n, a(n) for n = 1..8191
Index entries for sequences that are permutations of the natural numbers

Inverse: A249813.
Similar or related permutations: A246684, A249811, A250244, A252755.
Cf. A000079, A000265, A001511, A003602, A006093, A083140, A083221, A249741, A250469, A253886.
Compare also the scatterplot of this sequence to the graphs of A252755 and A246684.
Differs from A246684 for the first time at n=14, where a(14) = 20, while A246684(14) = 26.

In the following formulas, A083221 and A249741 are interpreted as bivariate functions:
a(1) = 1, for n>1: a(n) = A083221(A001511(n), a(A003602(n))) - 1 = A249741(A001511(n), a(A003602(n))).
a(1) = 1, a(2n) = A253886(a(n)), a(2n+1) = (2*a(n+1))-1. - Antti Karttunen, Jan 18 2015
As a composition of other permutations:
a(n) = A250244(A246684(n)).
Other identities. For all n >= 1, the following holds:
a(n) = (1+a((2*n)-1))/2. [The odd bisection from a(1) onward with one added and then halved gives the sequence back.]
a(A000079(n-1)) = A006093(n).

A299918 Motzkin numbers (A001006) mod 8.

Original entry on oeis.org

1, 1, 2, 4, 1, 5, 3, 7, 3, 3, 4, 6, 7, 3, 2, 4, 3, 3, 6, 4, 3, 7, 7, 7, 5, 5, 4, 2, 1, 5, 3, 7, 3, 3, 6, 4, 3, 7, 1, 5, 1, 1, 4, 2, 5, 1, 4, 6, 5, 5, 2, 4, 5, 1, 1, 1, 3, 3, 4, 6, 7, 3, 2, 4, 3, 3, 6, 4, 3, 7, 1, 5, 1, 1, 4, 2, 5, 1, 6, 4, 1, 1, 2, 4, 1

Offset: 0

N. J. A. Sloane, Mar 16 2018

Robert Israel, Table of n, a(n) for n = 0..10000
S.-P. Eu, S.-C. Liu, and Y.-N. Yeh, Catalan and Motzkin numbers modulo 4 and 8, Europ. J. Combin. 29 (2008), 1449-1466.
Christian Krattenthaler and Thomas W. Müller, Motzkin numbers and related sequences modulo powers of 2, arXiv:1608.05657 [math.CO], 2016-2018.
E. Rowland and R. Yassawi, Automatic congruences for diagonals of rational functions, J. Théorie Nombres Bordeaux 27 (2015), 245-288.
Ying Wang and Guoce Xin, A Classification of Motzkin Numbers Modulo 8, Electron. J. Combin., 25(1) (2018), #P1.54.

Motzkin numbers A001006 read mod 2,3,4,5,6,7,8,11: A039963, A039964, A299919, A258712, A299920, A258711, A299918, A258710.

f:= rectoproc({(3+3*n)*a(n)+(5+2*n)*a(1+n)+(-4-n)*a(n+2), a(0) = 1, a(1) = 1}, a(n), remember): seq(f(n) mod 8, n=0..200); # Robert Israel, Mar 16 2018

Table[Mod[GegenbauerC[n, -n - 1, -1/2] / (n + 1), 8], {n, 0, 100}] (* Vincenzo Librandi, Sep 08 2018 *)

catalan(n) = binomial(2*n, n)/(n+1);
a(n) = sum(k=0, n, (-1)^(n-k)*binomial(n, k)*catalan(k+1)) % 8; \\ Michel Marcus, May 23 2022

A309890 Lexicographically earliest sequence of positive integers without triples in weakly increasing arithmetic progression.

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 2, 4, 4, 1, 1, 2, 1, 1, 2, 2, 4, 4, 2, 4, 4, 5, 5, 8, 5, 5, 9, 1, 1, 2, 1, 1, 2, 2, 4, 4, 1, 1, 2, 1, 1, 2, 2, 4, 4, 2, 4, 4, 5, 5, 8, 5, 5, 9, 2, 4, 4, 5, 5, 10, 5, 5, 10, 10, 11, 13, 10, 11, 10, 11, 13, 10, 10, 12, 13, 10, 13, 11, 12, 20, 11, 1, 1, 2, 1, 1, 2, 2, 4, 4, 1, 1, 2, 1, 1, 2, 2, 4, 4, 2

Offset: 1

Sébastien Palcoux, Aug 21 2019

Formal definition: lexicographically earliest sequence of positive integers a(n) such that for any i > 0, there is no n > 0 such that 2a(n+i) = a(n) + a(n+2i) AND a(n) <= a(n+i) <= a(n+2i).
Sequence suggested by Richard Stanley as a variant of A229037. They differ from the 55th term. The numbers n for which a(n) = 1 are given by A003278, or equally by A005836 (Richard Stanley).
The sequence defined by c(n) = 1 if a(n) = 1 and otherwise c(n) = 0 is A039966 (although with a different offset). - N. J. A. Sloane, Dec 01 2019
Pleasant to listen to (button above) with Instrument no. 13: Marimba (and for better listening, save and convert to MP3).

Sébastien Palcoux, Table of n, a(n) for n = 1..100000
Sébastien Palcoux, On the first sequence without triple in arithmetic progression (version: 2019-08-21), second part, MathOverflow
Sébastien Palcoux, Table of n, a(n) for n = 1..1000000
Sébastien Palcoux, Density plot of the first 1000000 terms

Cf. A003278, A005836, A039966, A229037.

from itertools import count, islice
def A309890_gen(): # generator of terms
    blist = []
    for n in count(0):
        i, j, b = 1, 1, set()
        while n-(i<<1) >= 0:
            x, y = blist[n-2*i], blist[n-i]
            z = (y<<1)-x
            if x<=y<=z:
                b.add(z)
                while j in b:
                    j += 1
            i += 1
        blist.append(j)
        yield j
A309890_list = list(islice(A309890_gen(),30)) # Chai Wah Wu, Sep 12 2023

# %attach SAGE/ThreeFree.spyx
from sage.all import *
cpdef ThreeFree(int n):
     cdef int i,j,k,s,Li,Lj
     cdef list L,Lb
     cdef set b
     L=[1,1]
     for k in range(2,n):
          b=set()
          for i in range(k):
               if 2*((i+k)/2)==i+k:
                    j=(i+k)/2
                    Li,Lj=L[i],L[j]
                    s=2*Lj-Li
                    if s>0 and Li<=Lj:
                         b.add(s)
          if 1 not in b:
               L.append(1)
          else:
               Lb=list(b)
               Lb.sort()
               for t in Lb:
                    if t+1 not in b:
                         L.append(t+1)
                         break
     return L

A001612 a(n) = a(n-1) + a(n-2) - 1 for n > 1, a(0)=3, a(1)=2.

Original entry on oeis.org

3, 2, 4, 5, 8, 12, 19, 30, 48, 77, 124, 200, 323, 522, 844, 1365, 2208, 3572, 5779, 9350, 15128, 24477, 39604, 64080, 103683, 167762, 271444, 439205, 710648, 1149852, 1860499, 3010350, 4870848, 7881197, 12752044, 20633240, 33385283, 54018522

Offset: 0

N. J. A. Sloane

a(n+3) = A^(n)B^(2)(1), n >= 0, with compositions of Wythoff's complementary A(n):=A000201(n) and B(n)=A001950(n) sequences. See the W. Lang link under A135817 for the Wythoff representation of numbers (with A as 1 and B as 0 and the argument 1 omitted). E.g., 5=`00`, 8=`100`, 12=`1100`, ..., in Wythoff code.
From Petros Hadjicostas, Jan 11 2017: (Start)
a(n) is the number of cyclic sequences consisting of zeros and ones that avoid the pattern 001 (or equivalently, the pattern 110) provided the positions of zeros and ones on a circle are fixed. This can easily be proved by considering that sequence A000071(n+3) is the number of binary zero-one words of length n that avoid the pattern 001 and that a(n) = A000071(n+3) - 2*A000071(n). (From the collection of all zero-one binary sequences that avoid 001 subtract those that start with 1 and end with 00 and those that start with 01 and end with 0.)
For n = 1,2, the number a(n) still gives the number of cyclic sequences consisting of zeros and ones that avoid the pattern 001 (provided the positions of zeros and ones on a circle are fixed) even if we assume that the sequence wraps around itself on the circle. For example, when 01 wraps around itself, it becomes 01010..., and it never contains the pattern 001. (End)
For n >= 3, a(n) is also the number of independent vertex sets and vertex covers in the wheel graph on n+1 nodes. - Eric W. Weisstein, Mar 31 2017

			a(3) = 5 because the following cyclic sequences of length three avoid the pattern 001: 000, 011, 101, 110, 111. - _Petros Hadjicostas_, Jan 11 2017

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..500
Nazim Fatès, Biswanath Sethi, and Sukanta Das, On the reversibility of ECAs with fully asynchronous updating: the recurrence point of view, hal-01571847 Preprint, 2017.
Martin Griffiths, On a Matrix Arising from a Family of Iterated Self-Compositions, Journal of Integer Sequences, 18 (2015), #15.11.8.
Fumio Hazama, Spectra of graphs attached to the space of melodies, Discr. Math., 311 (2011), 2368-2383. See Table 5.2.
Martin Herschend and Peter Jorgensen, Classification of higher wide subcategories for higher Auslander algebras of type A, arXiv:2002.01778 [math.RT], 2020.
Dov Jarden, Recurring Sequences, Riveon Lematematika, Jerusalem, 1966. [Annotated scanned copy] See p. 97.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Eric Weisstein's World of Mathematics, Independent Vertex Set
Eric Weisstein's World of Mathematics, Vertex Cover
Eric Weisstein's World of Mathematics, Wheel Graph
Index entries for linear recurrences with constant coefficients, signature (2,0,-1).

Cf. A000032, A000071, A274017.

a001612 n = a001612_list !! n
a001612_list = 3 : 2 : (map (subtract 1) $
   zipWith (+) a001612_list (tail a001612_list))
-- Reinhard Zumkeller, May 26 2013

A001612:=-(-2+3*z**2)/(z-1)/(z**2+z-1); # conjectured by Simon Plouffe in his 1992 dissertation; gives sequence except for the initial 3

Join[{b=3},a=0;Table[c=a+b-1;a=b;b=c,{n,100}]] (* Vladimir Joseph Stephan Orlovsky, Mar 15 2011 *)
Table[Fibonacci[n] + Fibonacci[n - 2] + 1, {n, 20}] (* Eric W. Weisstein, Mar 31 2017 *)
LinearRecurrence[{2, 0, -1}, {3, 2, 4}, 20] (* Eric W. Weisstein, Mar 31 2017 *)
CoefficientList[Series[(3 - 4 x)/(1 - 2 x + x^3), {x, 0, 20}], x] (* Eric W. Weisstein, Sep 21 2017 *)

PARI
```
a(n)=fibonacci(n+1)+fibonacci(n-1)+1
```

G.f.: (3-4*x)/((1-x)*(1-x-x^2)).
a(n) = a(n-1) + a(n-2) - 1.
a(n) = A000032(n) + 1.
a(n) = A000071(n+3) - 2*A000071(n). - Petros Hadjicostas, Jan 11 2017

Additional comments from Michael Somos, Jun 01 2000

A245701 Permutation of natural numbers: a(1) = 1, a(A014580(n)) = 2a(n), a(A091242(n)) = 2a(n)+1, where A014580(n) = binary code for n-th irreducible polynomial over GF(2), A091242(n) = binary code for n-th reducible polynomial over GF(2).

Original entry on oeis.org

1, 2, 4, 3, 5, 9, 8, 7, 11, 19, 6, 17, 10, 15, 23, 39, 13, 35, 18, 21, 31, 47, 79, 27, 16, 71, 37, 43, 63, 95, 14, 159, 55, 33, 143, 75, 22, 87, 127, 191, 38, 29, 319, 111, 67, 287, 12, 151, 45, 175, 255, 383, 77, 59, 34, 639, 223, 135, 20, 575, 30, 25, 303, 91, 351, 511, 46, 767, 155, 119, 69, 1279, 78, 447, 271, 41, 1151, 61, 51

Offset: 1

Antti Karttunen, Aug 02 2014

Antti Karttunen, Table of n, a(n) for n = 1..10001
Antti Karttunen, Entanglement Permutations, 2016-2017
Index entries for sequences related to polynomials in ring GF(2)[X]
Index entries for sequences that are permutations of the natural numbers

Inverse: A245702.
Cf. A000035, A014580, A091242, A091225, A091226, A091245.
Similar entanglement permutations: A135141, A193231, A237427, A243287, A245703, A245704.

allocatemem(123456789);
A091226 = vector(2^22);
isA014580(n)=polisirreducible(Pol(binary(n))*Mod(1, 2)); \\ This function from Charles R Greathouse IV
n=2; while((n < 2^22), if(isA014580(n), A091226[n] = A091226[n-1]+1, A091226[n] = A091226[n-1]); n++)
A091245(n) = ((n-A091226[n])-1);
A245701(n) = if(1==n, 1, if(isA014580(n), 2*(A245701(A091226[n])), 1 + 2*(A245701(A091245(n)))));
for(n=1, 10001, write("b245701.txt", n, " ", A245701(n)));

;; With memoizing definec-macro.
(definec (A245701 n) (cond ((= 1 n) n) ((= 1 (A091225 n)) (* 2 (A245701 (A091226 n)))) (else (+ 1 (* 2 (A245701 (A091245 n)))))))

a(1) = 1, and for n > 1, if n is in A014580, a(n) = 2*a(A091226(n)), otherwise a(n) = 1 + 2*a(A091245(n)).
As a composition of related permutations:
a(n) = A135141(A245704(n)).
Other identities:
For all n >= 1, 1 - A000035(a(n)) = A091225(n). [Maps binary representations of irreducible GF(2) polynomials (= A014580) to even numbers and the corresponding representations of reducible polynomials to odd numbers].

cp's OEIS Frontend

A243658 a(0)=0; thereafter a(n) = noz(n+a(n-1)), where noz(n) = A004719(n).

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A067992 a(0)=1 and, for n > 0, a(n) is the smallest positive integer such that the ratios min(a(k)/a(k-1), a(k-1)/a(k)) for 0 < k <= n are all distinct.

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A080099 Triangle T(n,k) = n AND k, 0<=k<=n, bitwise logical AND, read by rows.

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A082850 Let S(0) = {}, S(n) = {S(n-1), S(n-1), n}; sequence gives S(infinity).

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A247074 a(n) = phi(n)/(Product_{primes p dividing n } gcd(p - 1, n - 1)).

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A249814 "Mountains of Eratosthenes" permutation: a(1) = 1, a(n) = A249741(A001511(n), a(A003602(n))).

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A299918 Motzkin numbers (A001006) mod 8.

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A245701 Permutation of natural numbers: a(1) = 1, a(A014580(n)) = 2a(n), a(A091242(n)) = 2a(n)+1, where A014580(n) = binary code for n-th irreducible polynomial over GF(2), A091242(n) = binary code for n-th reducible polynomial over GF(2).