A005352 -id:A005352 - cp's OEIS frontend

A016789 a(n) = 3*n + 2.

2, 5, 8, 11, 14, 17, 20, 23, 26, 29, 32, 35, 38, 41, 44, 47, 50, 53, 56, 59, 62, 65, 68, 71, 74, 77, 80, 83, 86, 89, 92, 95, 98, 101, 104, 107, 110, 113, 116, 119, 122, 125, 128, 131, 134, 137, 140, 143, 146, 149, 152, 155, 158, 161, 164, 167, 170, 173, 176, 179

Offset: 0

N. J. A. Sloane

Except for 1, n such that Sum_{k=1..n} (k mod 3)*binomial(n,k) is a power of 2. - Benoit Cloitre, Oct 17 2002
The sequence 0,0,2,0,0,5,0,0,8,... has a(n) = n*(1 + cos(2*Pi*n/3 + Pi/3) - sqrt(3)*sin(2*Pi*n + Pi/3))/3 and o.g.f. x^2(2+x^3)/(1-x^3)^2. - Paul Barry, Jan 28 2004 [Artur Jasinski, Dec 11 2007, remarks that this should read (3*n + 2)*(1 + cos(2*Pi*(3*n + 2)/3 + Pi/3) - sqrt(3)*sin(2*Pi*(3*n + 2)/3 + Pi/3))/3.]
Except for 2, exponents e such that x^e + x + 1 is reducible. - N. J. A. Sloane, Jul 19 2005
The trajectory of these numbers under iteration of sum of cubes of digits eventually turns out to be 371 or 407 (47 is the first of the second kind). - Avik Roy (avik_3.1416(AT)yahoo.co.in), Jan 19 2009
Union of A165334 and A165335. - Reinhard Zumkeller, Sep 17 2009
a(n) is the set of numbers congruent to {2,5,8} mod 9. - Gary Detlefs, Mar 07 2010
It appears that a(n) is the set of all values of y such that y^3 = k*n + 2 for integer k. - Gary Detlefs, Mar 08 2010
These numbers do not occur in A000217 (triangular numbers). - Arkadiusz Wesolowski, Jan 08 2012
A089911(2*a(n)) = 9. - Reinhard Zumkeller, Jul 05 2013
Also indices of even Bell numbers (A000110). - Enrique Pérez Herrero, Sep 10 2013
Central terms of the triangle A108872. - Reinhard Zumkeller, Oct 01 2014
A092942(a(n)) = 1 for n > 0. - Reinhard Zumkeller, Dec 13 2014
a(n-1), n >= 1, is also the complex dimension of the manifold E(S), the set of all second-order irreducible Fuchsian differential equations defined on P^1 = C U {oo}, having singular points at most in S = {a_1, ..., a_n, a_{n+1} = oo}, a subset of P^1. See the Iwasaki et al. reference, Proposition 2.1.3., p. 149. - Wolfdieter Lang, Apr 22 2016
Except for 2, exponents for which 1 + x^(n-1) + x^n is reducible. - Ron Knott, Sep 16 2016
The reciprocal sum of 8 distinct items from this sequence can be made equal to 1, with these terms: 2, 5, 8, 14, 20, 35, 41, 1640. - Jinyuan Wang, Nov 16 2018
There are no positive integers x, y, z such that 1/a(x) = 1/a(y) + 1/a(z). - Jinyuan Wang, Dec 31 2018
As a set of positive integers, it is the set sum S + S where S is the set of numbers in A016777. - Michael Somos, May 27 2019
Interleaving of A016933 and A016969. - Leo Tavares, Nov 16 2021
Prepended with {1}, these are the denominators of the elements of the 3x+1 semigroup, the numerators being A005408 prepended with {2}. See Applegate and Lagarias link for more information. - Paolo Xausa, Nov 20 2021
This is also the maximum number of moves starting with n + 1 dots in the game of Sprouts. - Douglas Boffey, Aug 01 2022 [See the Wikipedia link. - Wolfdieter Lang, Sep 29 2022]
a(n-2) is the maximum sum of the span (or L(2,1)-labeling number) of a graph of order n and its complement. The extremal graphs are stars and their complements. For example, K_{1,2} has span 3, and K_2 has span 2. Thus a(3-1) = 5. - Allan Bickle, Apr 20 2023

			G.f. = 2 + 5*x + 8*x^2 + 11*x^3 + 14*x^4 + 17*x^5 + 20*x^6 + ... - _Michael Somos_, May 27 2019

K. Iwasaki, H. Kimura, S. Shimomura and M. Yoshida, From Gauss to Painlevé, Vieweg, 1991. p. 149.
Konrad Knopp, Theory and Application of Infinite Series, Dover, p. 269

G. C. Greubel, Table of n, a(n) for n = 0..10000
D. Applegate and J. C. Lagarias, The 3x+1 semigroup, Journal of Number Theory, Vol. 177, Issue 1, March 2006, pp. 146-159; see also the arXiv version, arXiv:math/0411140 [math.NT], 2004-2005.
H. Balakrishnan and N. Deo, Parallel algorithm for radiocoloring a graph, Congr. Numer. 160 (2003), 193-204.
Allan Bickle, Extremal Decompositions for Nordhaus-Gaddum Theorems, Discrete Math, 346 7 (2023), 113392.
L. Euler, Observatio de summis divisorum p. 9.
L. Euler, An observation on the sums of divisors, arXiv:math/0411587 [math.HO], 2004-2009, p. 9.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 937
L. B. W. Jolley, Summation of Series, Dover, 1961, p. 16
Tanya Khovanova, Recursive Sequences
Konrad Knopp, Theorie und Anwendung der unendlichen Reihen, Berlin, J. Springer, 1922. (Original German edition of "Theory and Application of Infinite Series")
Fabian S. Reid, The Visual Pattern in the Collatz Conjecture and Proof of No Non-Trivial Cycles, arXiv:2105.07955 [math.GM], 2021.
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.
Leo Tavares, Illustration: Capped Triangular Frames
Wikipedia, Sprouts (game)
Index entries for linear recurrences with constant coefficients, signature (2,-1).

First differences of A005449.
Cf. A002939, A017041, A017485, A125202, A017233, A179896, A017617, A016957, A008544 (partial products), A032766, A016777, A124388, A005351.
Cf. A087370.
Cf. similar sequences with closed form (2*k-1)*n+k listed in A269044.
Cf. A000096, A000217.
Cf. A016933, A016969.

List([0..70],n->3*n+2); # Muniru A Asiru, Nov 02 2018

a016789 = (+ 2) . (* 3)  -- Reinhard Zumkeller, Jul 05 2013

[3*n+2: n in [0..80]]; // Vincenzo Librandi, Apr 14 2015

seq(3*n+2, n = 0 .. 50); # Matt C. Anderson, May 18 2017

Range[2, 500, 3] (* Vladimir Joseph Stephan Orlovsky, May 26 2011 *)
LinearRecurrence[{2,-1},{2,5},70] (* Harvey P. Dale, Aug 11 2021 *)

vector(100,n,3*n-1) \\ Derek Orr, Apr 13 2015

for n in range(0,100): print(3*n+2, end=', ') # Stefano Spezia, Nov 21 2018

G.f.: (2+x)/(1-x)^2.
a(n) = 3 + a(n-1).
a(n) = 1 + A016777(n).
a(n) = A124388(n)/9.
a(n) = A125199(n+1,1). - Reinhard Zumkeller, Nov 24 2006
Sum_{n>=1} (-1)^n/a(n) = (1/3)*(Pi/sqrt(3) - log(2)). - Benoit Cloitre, Apr 05 2002
1/2 - 1/5 + 1/8 - 1/11 + ... = (1/3)*(Pi/sqrt(3) - log 2). [Jolley] - Gary W. Adamson, Dec 16 2006
Sum_{n>=0} 1/(a(2*n)*a(2*n+1)) = (Pi/sqrt(3) - log 2)/9 = 0.12451569... (see A196548). [Jolley p. 48 eq (263)]
a(n) = 2*a(n-1) - a(n-2); a(0)=2, a(1)=5. - Philippe Deléham, Nov 03 2008
a(n) = 6*n - a(n-1) + 1 with a(0)=2. - Vincenzo Librandi, Aug 25 2010
Conjecture: a(n) = n XOR A005351(n+1) XOR A005352(n+1). - Gilian Breysens, Jul 21 2017
E.g.f.: (2 + 3*x)*exp(x). - G. C. Greubel, Nov 02 2018
a(n) = A005449(n+1) - A005449(n). - Jinyuan Wang, Feb 03 2019
a(n) = -A016777(-1-n) for all n in Z. - Michael Somos, May 27 2019
a(n) = A007310(n+1) + (1 - n mod 2). - Walt Rorie-Baety, Sep 13 2021
a(n) = A000096(n+1) - A000217(n-1). See Capped Triangular Frames illustration. - Leo Tavares, Oct 05 2021

A122803 Powers of -2: a(n) = (-2)^n.

Original entry on oeis.org

1, -2, 4, -8, 16, -32, 64, -128, 256, -512, 1024, -2048, 4096, -8192, 16384, -32768, 65536, -131072, 262144, -524288, 1048576, -2097152, 4194304, -8388608, 16777216, -33554432, 67108864, -134217728, 268435456, -536870912, 1073741824, -2147483648, 4294967296, -8589934592, 17179869184

Offset: 0

Franklin T. Adams-Watters, Sep 11 2006

The number -2 can be used as a base of numeration (see the Weisstein link). - Alonso del Arte, Mar 30 2014
Contribution from M. F. Hasler, Oct 21 2014: (Start)
This is the inverse binomial transform of A033999 = n->(-1)^n, and the binomial transform of A033999*A000244 = n->(-3)^n, see also A141413.
Prefixed with one 0, i.e., (0,1,-2,4,...) = -A033999*A131577, it is the binomial transform of (0, 1, -4, 13, -40, 121,...) = -A033999*A003462, and inverse binomial transform of (0,1,0,1,0,1,...) = A000035.
Prefixed with two 0's, i.e., (0,0,1,-2,4,-8,...), it is the binomial transform of (0,0,1,-5,18,-58,179,-543,...) (cf. A000340) and inverse binomial transform of (0,0,1,1,2,2,3,3,...) = A004526. (End)
Prefixed with three 0's, this is the inverse binomial difference of (0, 0, 0, 1, 2, 4, 6, 9, 12, 16,...) = concat(0, A002620), which has as successive differences (0, 0, 1, 1, 2, 2,...) = A004526, then (0, 1, 0, 1,...) = A000035, then (1, -1, 1, -1,...) = A033999, and then (-2)^k*A033999 with k=1,2,3,... - Paul Curtz, Oct 16 2014, edited by M. F. Hasler, Oct 21 2014
Stirling-Bernoulli transform of triangular numbers: 1, 3, 6, 10, 15, 21, 28, ... - Philippe Deléham, May 25 2015

Franklin T. Adams-Watters, Table of n, (-2)^n for n = 0..1000
Tanya Khovanova, Recursive Sequences
Eric Weisstein's World of Mathematics, Negabinary.
Index entries for linear recurrences with constant coefficients, signature (-2).

Cf. A000079 (powers of 2), A004526, A005351, A005352, A011782, A034008, A131577, A171449, A248800.

[(-2)^n: n in [0..60]]; // Vincenzo Librandi, Oct 22 2014

A122803:=n->(-2)^n; seq(A122803(n), n=0..50); # Wesley Ivan Hurt, Mar 30 2014

Table[(-2)^n, {n, 0, 50}] (* Vladimir Joseph Stephan Orlovsky, Feb 15 2011 *)

a(n)=(-2)^n \\ Charles R Greathouse IV, Sep 24 2015

def A122803(n): return -(1<Chai Wah Wu, Nov 18 2022

a(n) = (-2)^n = (-1)^n * 2^n.
a(n) = -2*a(n-1), n > 0; a(0) = 1. G.f.: 1/(1+2x). - Philippe Deléham, Nov 19 2008
Sum_{n >= 0} 1/a(n) = 2/3. - Jaume Oliver Lafont, Mar 01 2009
E.g.f.: 1/exp(2*x). - Arkadiusz Wesolowski, Aug 13 2012
a(n) = Sum_{k = 0..n} (-2)^(n-k)*binomial(n, k)*A030195(n+1). - R. J. Mathar, Oct 15 2012
G.f.: 1/(1+2x). A122803 = A033999 * A000079. - M. F. Hasler, Oct 21 2014
a(n) = Sum_{k = 0..n} A163626(n,k)*A000217(k+1). - Philippe Deléham, May 25 2015

A005351 Base -2 representation for n regarded as base 2, then evaluated.

Original entry on oeis.org

0, 1, 6, 7, 4, 5, 26, 27, 24, 25, 30, 31, 28, 29, 18, 19, 16, 17, 22, 23, 20, 21, 106, 107, 104, 105, 110, 111, 108, 109, 98, 99, 96, 97, 102, 103, 100, 101, 122, 123, 120, 121, 126, 127, 124, 125, 114, 115, 112, 113, 118, 119, 116, 117, 74, 75, 72, 73, 78, 79, 76

Offset: 0

N. J. A. Sloane

a(n) = n when n is a power of 4. This is because the even-indexed powers of 2 are the same as the even-indexed powers of -2. - Alonso del Arte, Feb 09 2012
a(n) = n if n is a sum of distinct powers of 4. - Michael Somos, Aug 27 2012
Write n = Sum_{i in b(n)} (-2)^(i - 1), which uniquely determines the set of positive integers b(n). Then a(n) = Sum_{i in b(n)} 2^(i - 1). For example, a(7) = 27 because 7 = (-2)^0 + (-2)^1 + (-2)^3 + (-2)^4 and 27 = 2^0 + 2^1 + 2^3 + 2^4. - Gus Wiseman, Jul 26 2019

			2 = 4+(-2)+0 = 110 => 6, 3 = 4+(-2)+1 = 111 => 7, ..., 6 = (16)+(-8)+0+(-2)+0 = 11010 => 26.

M. Gardner, Knotted Doughnuts and Other Mathematical Entertainments. Freeman, NY, 1986, p. 101.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Reinhard Zumkeller, Table of n, a(n) for n = 0..8191
Joerg Arndt, Matters Computational (The Fxtbook), p. 58-59
Eric Weisstein's World of Mathematics, Negabinary
Wikipedia, Negative base
A. Wilks, Email, May 22 1991

Cf. A039724. Complement of A005352.
Cf. A185269 (primes in this sequence).
Cf. A000120, A029931, A035327, A053985, A065359, A070939, A326032.

a005351 0 = 0
a005351 n = a005351 n' * 2 + m where
   (n', m) = if r < 0 then (q + 1, r + 2) else (q, r)
             where (q, r) = quotRem n (negate 2)
-- Reinhard Zumkeller, Jul 07 2012

a[n_] := Module[{t = 2(4^Floor[ Log[4, Abs[n] + 1] + 2] - 1)/3}, BitXor[n + t, t]]; Table[a[n], {n, 0, 60}] (* Robert G. Wilson v, Jan 24 2005 *)

a(n) = my(t=(32*4^logint(abs(n)+1,4)-2)/3); bitxor(n+t,t); \\ Ruud H.G. van Tol, Oct 18 2023

def A005351(n):
    s, q = '', n
    while q >= 2 or q < 0:
        q, r = divmod(q, -2)
        if r < 0:
            q += 1
            r += 2
        s += str(r)
    return int(str(q)+s[::-1],2) # Chai Wah Wu, Apr 10 2016

a(4n+2) = 4a(n+1)+2, a(4n+3) = 4a(n+1)+3, a(4n+4) = 4a(n+1), a(4n+5) = 4a(n+1)+1, n>-2, a(1)=1. - Ralf Stephan, Apr 06 2004

More terms from Robert G. Wilson v, Jan 24 2005

A212529 Negative numbers in base -2.

Original entry on oeis.org

11, 10, 1101, 1100, 1111, 1110, 1001, 1000, 1011, 1010, 110101, 110100, 110111, 110110, 110001, 110000, 110011, 110010, 111101, 111100, 111111, 111110, 111001, 111000, 111011, 111010, 100101, 100100, 100111, 100110, 100001, 100000, 100011, 100010, 101101, 101100, 101111, 101110, 101001, 101000, 101011, 101010, 11010101

Offset: 1

Joerg Arndt, May 20 2012

The formula a(n) = A039724(-n) is slightly misleading because sequence A039724 isn't defined for n < 0, and none of the terms a(n) is a term of A039724. It can be seen as the definition of the extension of A039724 to negative indices. Also, recursive definitions or implementations of A039724 require that function to be defined for negative arguments, and using a generic formula it will work as expected for -n, n > 0. - M. F. Hasler, Oct 18 2018

Joerg Arndt, Table of n, a(n) for n = 1..1000
Joerg Arndt, Matters Computational (The Fxtbook), pp. 58-59.

Cf. A039724 (nonnegative numbers in base -2).
Cf. A007608 (nonnegative numbers in base -4), A212526 (negative numbers in base -4).
Cf. A005352.

a212529 = a039724 . negate  -- Reinhard Zumkeller, Feb 05 2014

a:= proc(n) local d, i, l, m;
      m:= n; l:= NULL;
      for i from 0 while m>0 do
        d:= irem(m, 2, 'm');
        if d=1 and irem(i, 2)=0 then m:= m+1 fi;
        l:= d, l
      od; parse(cat(l))
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, May 20 2012

negabin[n_] := negabin[n] = If[n == 0, 0, negabin[Quotient[n - 1, -2]]*10 + Mod[n, 2]]; a[n_] := negabin[-n]; Array[a, 50] (* Amiram Eldar, Jul 23 2023 *)

A212529(n)=A039724(-n) \\ M. F. Hasler, Oct 16 2018

def A212529(n):
    s, q = '', -n
    while q >= 2 or q < 0:
        q, r = divmod(q, -2)
        if r < 0:
            q += 1
            r += 2
        s += str(r)
    return int(str(q)+s[::-1]) # Chai Wah Wu, Apr 09 2016

a(n) = A039724(-n). - Reinhard Zumkeller, Feb 05 2014

A178729 a(n) = n XOR 3n, where XOR is bitwise XOR.

Original entry on oeis.org

0, 2, 4, 10, 8, 10, 20, 18, 16, 18, 20, 42, 40, 42, 36, 34, 32, 34, 36, 42, 40, 42, 84, 82, 80, 82, 84, 74, 72, 74, 68, 66, 64, 66, 68, 74, 72, 74, 84, 82, 80, 82, 84, 170, 168, 170, 164, 162, 160, 162, 164, 170, 168, 170, 148, 146, 144, 146, 148, 138, 136, 138, 132, 130

Offset: 0

Dmitry Kamenetsky, Jun 08 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Helena A. Verrill, Note on the negative base xor sequence n⊕-b(-n), arXiv:2506.09509 [math.NT], 2025.

Cf. A048724, A048725, A178731, A178732, A178733, A178734, A178735, A178736. - Robert G. Wilson v, Jun 09 2010

Cf. A184617.

[BitwiseXor(n, 3*n): n in [0..80]]; // Vincenzo Librandi, Jul 11 2017

f[n_] := BitXor[n, 3 n]; Array[f, 70, 0] (* Robert G. Wilson v, Jun 09 2010 *)

a(n) = bitxor(n, 3*n); \\ Michel Marcus, Jul 11 2017

def A178729(n): return n^ 3*n # Chai Wah Wu, Jun 29 2022

a(n) = A005351(n) XOR A005352(n) (conjectured). Proved by Verrill link.

a(n) = 2 * A184617(n). - Alois P. Heinz, Jul 21 2017

a(30) onwards from Robert G. Wilson v, Jun 09 2010

cp's OEIS Frontend

A016789 a(n) = 3*n + 2.

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A122803 Powers of -2: a(n) = (-2)^n.

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A005351 Base -2 representation for n regarded as base 2, then evaluated.

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A212529 Negative numbers in base -2.

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Haskell

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A178729 a(n) = n XOR 3n, where XOR is bitwise XOR.

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A331819 Positive numbers k such that -k is a negative negabinary-Niven number, i.e., divisible by the sum of digits of its negabinary representation (A027615).

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