A143097 -id:A143097 - cp's OEIS frontend

A143098 First differences of A143097.

1, 2, -1, 2, 2, -1, 2, 2, -1, 2, 2, -1, 2, 2, -1, 2, 2, -1, 2, 2, -1, 2, 2, -1, 2, 2, -1, 2, 2, -1, 2, 2, -1, 2, 2, -1, 2, 2, -1, 2, 2, -1, 2, 2, -1, 2, 2, -1, 2, 2, -1, 2, 2, -1, 2, 2, -1, 2, 2, -1, 2, 2, -1, 2, 2, -1, 2, 2, -1, 2, 2, -1, 2, 2, -1, 2, 2, -1, 2

Offset: 1

Gary W. Adamson, Jul 24 2008

Binomial transform gives A143100.

1 followed by {2, -1, 2} repeated. - Amiram Eldar, Jun 02 2025

Cf. A143097, A143100.

Join[{1}, Flatten[Table[{2, -1, 2}, {33}]]] (* Amiram Eldar, Jun 02 2025 *)

G.f.: x(1+2x-x^2+x^3)/((1-x)(1+x+x^2)). - R. J. Mathar, Sep 06 2008

A143099 A007318 * A143097.

Original entry on oeis.org

1, 3, 9, 22, 50, 113, 256, 576, 1281, 2818, 6146, 13313, 28672, 61440, 131073, 278530, 589826, 1245185, 2621440, 5505024, 11534337, 24117250, 50331650, 104857601, 218103808, 452984832, 939524097, 1946157058, 4026531842, 8321499137, 17179869184, 35433480192

Offset: 1

Gary W. Adamson, Jul 24 2008

A143100 = (1, 3, 4, 6, 13, 30, 64, 129, ...).

			a(4) = 22 = (1, 3, 3, 1) dot (1, 2, 4, 3) = (1 + 6 + 12 + 3).
a(4) = 22 = 2*a(3) + A143099(3) = 2*9 + 4, where 4 = A143100(3).

Nathaniel Johnston, Table of n, a(n) for n = 1..1000

Cf. A143097, A143098, A143100.

A143097 := proc(n) if(n<=1)then return n: elif(n mod 3 <= 1)then return n+1-2*(n mod 3): else return n: fi: end: A143099 := proc(n) return add(binomial(n-1,k-1)*A143097(k),k=1..n): end: seq(A143099(n),n=1..32); # Nathaniel Johnston, Apr 30 2011

Binomial transform of A143097: (1, 2, 4, 3, 5, 7, 6, 8, 10, 9, 11, ...). a(n) = 2*a(n-1) + A143100(n-1).

G.f.: x*(5*x^4-7*x^3+5*x^2-3*x+1)/((1-x)*(x^2-x+1)*(1-2*x)^2). - Maksym Voznyy (voznyy(AT)mail.ru), Aug 14 2009; [corrected by R. J. Mathar, Sep 16 2009]

A143101 Partial sums of A143097.

Original entry on oeis.org

1, 3, 7, 10, 15, 22, 28, 36, 46, 55, 66, 79, 91, 105, 121, 136, 153, 172, 190, 210, 232, 253, 276, 301, 325, 351, 379, 406, 435, 466, 496, 528, 562, 595, 630, 667, 703, 741, 781, 820, 861, 904, 946, 990, 1036, 1081, 1128, 1177, 1225, 1275, 1327, 1378, 1431

Offset: 1

Gary W. Adamson, Jul 24 2008

			a(3) = 7 = T(3) + 1 since 3 == 0 mod 3 and T(3) = 6.
a(4) = 10 = T(4) since 4 == 1 mod 3.

Nathaniel Johnston, Table of n, a(n) for n = 1..10000

Cf. A000217, A143097.

A143097 := proc(n) if(n<=1)then return n: elif(n mod 3 <= 1)then return n+1-2*(n mod 3): else return n: fi: end: A143101 := proc(n) option remember: if(n=0)then return 0:else return procname(n-1)+A143097(n):fi: end:seq(A143101(n),n=1..60); # Nathaniel Johnston, Apr 30 2011

With[{nn=70},Accumulate[Join[{1},Riffle[Rest[Select[Range[nn], !Divisible[ #,3]&]], Range[3,nn,3],3]]]] (* Harvey P. Dale, May 06 2012 *)

a(n)=n*(n+1)/2+if(n%3==0,1,0) \\ Luc Rousseau, Jun 18 2017

G.f.: x(1+x+2x^2-2x^3+x^4)/((1-x)^3(1+x+x^2)). [R. J. Mathar, Sep 06 2008]

a(n) = n*(n+1)/2 + (3|n) = A000217(n) + A079978(n). - Luc Rousseau, Jun 18 2017

a(18) corrected by Nathaniel Johnston, Apr 30 2011

A143102 Triangle read by rows, A000012 * (A143097 * 0^(n-k)) * A000012, 1<=k<=n.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 3, 7, 9, 10, 5, 8, 12, 14, 15, 7, 12, 15, 29, 21, 22, 6, 18, 18, 21, 25, 27, 28, 8, 14, 21, 26, 29, 33, 35, 36, 10, 18, 24, 31, 36, 39, 43, 45, 46, 9, 19, 27, 33, 40, 45, 48, 52, 54, 55, 11, 20, 30, 38, 44, 51, 56, 59, 63, 65, 66, 13, 24, 33, 43, 51, 57, 64, 69, 72

Offset: 1

Gary W. Adamson, Jul 24 2008

Left border = A143097: (1, 2, 4, 3, 5, 7, 6, 8,...); right border = A143101, partial sums of A143097: (1, 3, 7, 10, 15, 22, 28,...).

Row sums = A143103: (1, 5, 17, 29, 54, 96,...).

			First few rows of the triangle are:
1;
2, 3;
4, 6, 7;
3, 7, 9, 10;
5, 8, 12, 14, 15;
7, 12, 15, 19, 21, 22;
6, 13, 18, 21, 25, 27, 28;
...

Cf. A143097, A143101, A143103.

Triangle read by rows, A000012 * (A143097 * 0(n-k)) * A000012, 1<=k<=n; where = A000012 = an infinite lower triangular matrix with all 1's and A143097 * 0^(n-k) = an infinite lower triangular matrix with A143097 (1, 2, 4, 3, 5, 7, 6,...) in the main diagonal and the rest zeros.

A064429 a(n) = floor(n / 3) * 3 + sign(n mod 3) * (3 - n mod 3).

Original entry on oeis.org

0, 2, 1, 3, 5, 4, 6, 8, 7, 9, 11, 10, 12, 14, 13, 15, 17, 16, 18, 20, 19, 21, 23, 22, 24, 26, 25, 27, 29, 28, 30, 32, 31, 33, 35, 34, 36, 38, 37, 39, 41, 40, 42, 44, 43, 45, 47, 46, 48, 50, 49, 51, 53, 52, 54, 56, 55, 57, 59, 58, 60, 62, 61, 63, 65, 64, 66, 68, 67, 69, 71, 70

Offset: 0

Reinhard Zumkeller, Oct 15 2001

a(a(n)) = n (a self-inverse permutation).
Take natural numbers, exchange trisections starting with 1 and 2.
Lodumo_3 of A080425. - Philippe Deléham, Apr 26 2009
From Franck Maminirina Ramaharo, Jul 27 2018: (Start)
The sequence is A008585 interleaved with A016789 and A016777.
a(n) is also obtained as follows: write n in base 3; if the rightmost digit is '1', then replace it with '2' and vice versa; convert back to decimal. For example a(14) = a('11'2') = '11'1' = 13 and a(10) = a('10'1') = '10'2' = 11. (End)
A permutation of the nonnegative integers partitioned into triples [3*k-3, 3*k-1, 3*k-2] for k > 0. - Guenther Schrack, Feb 05 2020

			From _Franck Maminirina Ramaharo_, Jul 27 2018: (Start)
Interleave 3 sequences:
A008585: 0.....3.....6.....9.......12.......15........
A016789: ..2.....5.....8.....11.......14.......17.....
A016777: ....1.....4.....7......10.......13.......16..
(End)

Muniru A Asiru, Table of n, a(n) for n = 0..3000
Eric Weisstein's World of Mathematics, Alternating Permutations.
Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).
Index entries for sequences that are permutations of the natural numbers.

Cf. A001477, A004442, A074066, A080425, A080782, A102283, A143097.

a:=[0,2,1,3];; for n in [5..100] do a[n]:=a[n-1]+a[n-3]-a[n-4]; od; a; # Muniru A Asiru, Jul 27 2018

[2*n - 3 - 3*((n-2) div 3): n in [0..80]]; // Vincenzo Librandi, Aug 05 2018

A064429:=n->2*n-3-3*floor((n-2)/3): seq(A064429(n), n=0..100); # Wesley Ivan Hurt, Nov 30 2013

Table[2 n - 3 - 3 Floor[(n - 2)/3], {n, 0, 100}] (* Wesley Ivan Hurt, Nov 30 2013 *)
{#+1,#-1,#}[[Mod[#,3,1]]]&/@Range[0, 100] (* Federico Provvedi, May 11 2021 *)
LinearRecurrence[{1,0,1,-1},{0,2,1,3},80] (* or *) {#[[1]],#[[3]],#[[2]]}&/@Partition[Range[0,80],3]//Flatten (* Harvey P. Dale, Mar 28 2025 *)

a(n) = 2*n-3-3*((n-2)\3); \\ Altug Alkan, Oct 06 2017

a(n) = A080782(n+1) - 1.
a(n) = n - 2*sin(4*Pi*n/3)/sqrt(3). - Jaume Oliver Lafont, Dec 05 2008
a(n) = A001477(n) + A102283(n). - Jaume Oliver Lafont, Dec 05 2008
a(n) = lod_3(A080425(n)). - Philippe Deléham, Apr 26 2009
G.f.: x*(2 - x + 2*x^2)/((1 + x + x^2)*(1 - x)^2 ). - R. J. Mathar, Feb 20 2011
a(n) = 2*n - 3 - 3*floor((n-2)/3). - Wesley Ivan Hurt, Nov 30 2013
a(n) = a(n-1) + a(n-3) - a(n-4) for n > 3. - Wesley Ivan Hurt, Oct 06 2017
E.g.f.: x*exp(x) + (2*sin((sqrt(3)*x)/2))/(exp(x/2)*sqrt(3)). - Franck Maminirina Ramaharo, Jul 27 2018
From Guenther Schrack, Feb 05 2020: (Start)
a(n) = a(n-3) + 3 with a(0)=0, a(1)=2, a(2)=1 for n > 2;
a(n) = n + (w^(2*n) - w^n)*(1 + 2*w)/3 where w = (-1 + sqrt(-3))/2. (End)
Sum_{n>=1} (-1)^n/a(n) = log(2)/3. - Amiram Eldar, Jan 31 2023

A115302 Permutation of natural numbers generated by 3-rowed array shown below.

Original entry on oeis.org

1, 4, 2, 7, 5, 3, 10, 8, 6, 13, 11, 9, 16, 14, 12, 19, 17, 15, 22, 20, 18, 25, 23, 21, 28, 26, 24, 31, 29, 27, 34, 32, 30, 37, 35, 33, 40, 38, 36, 43, 41, 39, 46, 44, 42, 49, 47, 45, 52, 50, 48, 55, 53, 51, 58, 56, 54, 61, 59, 57, 64, 62, 60

Offset: 1

Giovanni Teofilatto, Mar 05 2006

4 7 10 13 16 19 22 25...a(n)=3n+1 => A016777
5 8 11 14 17 20 23 26...a(n)=3n+2 => A016789
6 9 12 15 18 21 24 27...a(n)=3n => A008585
Reversing the direction of the diagonals gives A143097. - Jeremy Gardiner, Oct 14 2012.

M. Cerasoli, F. Eugeni and M. Protasi, Elementi di Matematica Discreta, Bologna 1988
Emanuele Munarini and Norma Zagaglia Salvi, Matematica Discreta,UTET, CittaStudiEdizioni, Milano 1997

Cf. A088520, A090243, A090298, A143097.

For n > 1, a(n+5) = a(n) + 6, iff a(n+5)=1.

A143100 A007318 * A143098.

Original entry on oeis.org

1, 3, 4, 6, 13, 30, 64, 129, 256, 510, 1021, 2046, 4096, 8193, 16384, 32766, 65533, 131070, 262144, 524289, 1048576, 2097150, 4194301, 8388606, 16777216, 33554433, 67108864, 134217726, 268435453, 536870910, 1073741824, 2147483649, 4294967296, 8589934590

Offset: 1

Gary W. Adamson, Jul 24 2008

nonn

			a(4) = 6 = (1, 3, 3, 1) dot (1, 2, -1, 2) = (1 + 6 - 3 + 2).

Nathaniel Johnston, Table of n, a(n) for n = 1..500

Cf. A143097, A143098, A143099.

A143098 := proc(n) if(n=1)then return 1: elif(n mod 3 = 0)then return -1: else return 2: fi: end: A143100 := proc(n) return add(binomial(n-1,k-1)*A143098(k),k=1..n): end: seq(A143100(n),n=1..34); # Nathaniel Johnston, Apr 30 2011

Binomial transform of A143098: (1, 2, -1, 2, 2, -1, 2, 2, -1, 2, 2, ...).
From R. J. Mathar, Jul 31 2008: (Start)
G.f.: (3x^3 - 2x^2 - x + 1)*x/((x^2-x+1)*(2x-1)*(x-1)).
a(n) = -1 + 2^(n-1) + A057079(n-1). (End)

A281409 Lexicographically first sequence of distinct terms, beginning with a(1)=5, with the property that each triple of consecutive terms contains a term that divides the sum of the other two terms.

Original entry on oeis.org

5, 1, 2, 3, 4, 7, 9, 8, 10, 6, 14, 16, 12, 20, 28, 24, 13, 11, 15, 18, 21, 33, 27, 30, 19, 41, 22, 25, 47, 36, 58, 50, 42, 23, 61, 31, 32, 63, 65, 64, 43, 85, 44, 91, 45, 17, 40, 57, 74, 97, 51, 37, 60, 83, 106, 129, 152, 175, 109, 66, 35, 39, 82, 113, 133, 93

Offset: 1

Rémy Sigrist, Jan 21 2017

nonn

The initial term a(1)=5 seems to be the least one that leads to a sequence that is not ultimately linear.
The variant with:
- a(1)=1 matches A000027,
- a(1)=2 matches A181440,
- a(1)=3 starts with 3, 1, 2, and then matches A000027,
- a(1)=4 starts with 4, 1, 2, and then matches A143097,
- a(1)=6 starts with 6, 1, 2, 3, 4, 5, and then matches A000027,
- a(1)=9 starts with 9, 1, 2, 3, 4, 5, 6, 7, 8, 13, 11, 12, 10, and then matches A143097.
Conjecturally, all other variants are not ultimately linear.

			The first terms, alongside the indexes of the terms that divide the sum of the other two terms within the n-th triple of consecutive terms, are:
   n  a(n)   Indexes
  --  ----   -------
   1    5    2, 3
   2    1    1, 2, 3
   3    2    2
   4    3    3
   5    4    1
   6    7    3
   7    9    1
   8    8    1, 3
   9   10    1, 2
  10    6    1
  11   14    1
  12   16    1, 2
  13   12    1, 2
  14   20    3
  15   28    3
  16   24    1
  17   13    1
  18   11    1
  19   15    2
  20   18    1
  21   21    3
  22   33    3
  23   27    3
  24   30    1
  25   19    2

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A281409

Cf. A000027, A143097, A181440, A278962, A281408.

A236348 Expansion of (1 - x + 2x^2 + x^3) / ((1 - x) (1 - x^3)) in powers of x.

Original entry on oeis.org

1, 0, 2, 4, 3, 5, 7, 6, 8, 10, 9, 11, 13, 12, 14, 16, 15, 17, 19, 18, 20, 22, 21, 23, 25, 24, 26, 28, 27, 29, 31, 30, 32, 34, 33, 35, 37, 36, 38, 40, 39, 41, 43, 42, 44, 46, 45, 47, 49, 48, 50, 52, 51, 53, 55, 54, 56, 58, 57, 59, 61, 60, 62, 64, 63, 65, 67

Offset: 0

Michael Somos, Jan 23 2014

An order 2 permutation of nonnegative integers.

			G.f. = 1 + 2*x^2 + 4*x^3 + 3*x^4 + 5*x^5 + 7*x^6 + 6*x^7 + 8*x^8 + 10*x^9 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

Cf. A143097.

[n-1+((n-1) mod 3) : n in [0..100]]; // Wesley Ivan Hurt, Aug 21 2014

I:=[1,0,2,4]; [n le 4 select I[n] else Self(n-1)+Self(n-3)-Self(n-4): n in [1..80]]; // Vincenzo Librandi, Sep 28 2017

Table[n - 1 + Mod[n - 1, 3], {n, 0, 100}] (* Wesley Ivan Hurt, Aug 21 2014 *)
LinearRecurrence[{1, 0, 1, -1}, {1, 0, 2, 4}, 80] (* or *) CoefficientList[Series[(1 - x + 2 x^2 + x^3) / ((1 - x) (1 -x^3)), {x, 0, 80}], x] (* Vincenzo Librandi, Sep 28 2017 *)

PARI
```
{a(n) = (n-1) % 3 + n-1 }
```

G.f.: (1 - x + 2*x^2 + x^3) / ((1 - x) * (1 - x^3)).
First difference is period 3 sequence [-1, 2, 2, ...].
a(n) = a(n-1) + a(n-3) - a(n-4). a(4-n) = 4-a(n).
0 = a(n)*(-a(n+1) + a(n+3)) + a(n+1)*(a(n+1) - a(n+2)) + a(n+2)*(a(n+2) - a(n+3)) for all n in Z.
a(n) = A143097(n) if n>1.
a(n) = n - 1 + mod(n-1, 3). - Wesley Ivan Hurt, Aug 21 2014
a(n) = n + (2/sqrt(3))*sin(2*(n+1)*Pi/3). - Wesley Ivan Hurt, Sep 26 2017
Sum_{n>=2} (-1)^n/a(n) = 2*Pi/(3*sqrt(3)) + log(2)/3 - 1. - Amiram Eldar, Sep 10 2023

A143103 Row sums of triangle A143102.

Original entry on oeis.org

1, 5, 17, 29, 54, 96, 138, 202, 292, 382, 503, 659, 815, 1011

Offset: 1

Gary W. Adamson, Jul 24 2008

nonn

			a(4) = 29 = sum of row 4 terms of triangle A143102: (3 + 7 + 9 + 10).
a(4) = 29 since A143097 = (1, 2, 4, 3, 5,...) with n*A143097(n) = (1, 4, 12, 12, 25, 42,...) and 29 = (1 + 4 + 12 + 12).

Cf. A143097, A143102.

cp's OEIS Frontend

A143098 First differences of A143097.

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A143099 A007318 * A143097.

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A143101 Partial sums of A143097.

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A143102 Triangle read by rows, A000012 * (A143097 * 0^(n-k)) * A000012, 1<=k<=n.

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A064429 a(n) = floor(n / 3) * 3 + sign(n mod 3) * (3 - n mod 3).

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A115302 Permutation of natural numbers generated by 3-rowed array shown below.

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A143100 A007318 * A143098.

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A281409 Lexicographically first sequence of distinct terms, beginning with a(1)=5, with the property that each triple of consecutive terms contains a term that divides the sum of the other two terms.

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A236348 Expansion of (1 - x + 2x^2 + x^3) / ((1 - x) (1 - x^3)) in powers of x.