A014681 -id:A014681 - cp's OEIS frontend

A101677 a(n) = a(n-1) - 2a(n-2) + 2a(n-3) - 2a(n-4) + 2a(n-5) - a(n-6).

1, 1, -1, -2, -2, -2, -1, -1, -3, -4, -4, -4, -3, -3, -5, -6, -6, -6, -5, -5, -7, -8, -8, -8, -7, -7, -9, -10, -10, -10, -9, -9, -11, -12, -12, -12, -11, -11, -13, -14, -14, -14, -13, -13, -15, -16, -16, -16, -15, -15, -17, -18, -18, -18, -17, -17, -19, -20, -20, -20, -19, -19, -21, -22, -22, -22, -21, -21, -23, -24, -24, -24, -23, -23, -25, -26, -26, -26, -25, -25, -27

Offset: 0

Paul Barry, Dec 11 2004

Partial sums of A101676, second partial sums of A101675.

G. C. Greubel, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-2,2,-1)

m:=100; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-x-x^2)/((1-x)^2*(1+x^2+x^4)))); // G. C. Greubel, Sep 07 2018

LinearRecurrence[{2, -2, 2, -2, 2, -1},{1, 1, -1, -2, -2, -2},81] (* Ray Chandler, Sep 03 2015 *)
CoefficientList[Series[(1-x-x^2)/((1-x)^2(1+x^2+x^4)),{x,0,80}],x] (* Harvey P. Dale, Dec 02 2021 *)

x='x+O('x^100); Vec((1-x-x^2)/((1-x)^2*(1+x^2+x^4))) \\ G. C. Greubel, Sep 07 2018

G.f.: (1-x-x^2)/((1-x)^2*(1+x^2+x^4)).
a(n) = 2*sqrt(3)*sin(2*Pi*n/3)/9 + cos(Pi*n/3) + sin(Pi*n/3)/sqrt(3) - n/3.
a(3*(n+1)) = -A014681(n+1); a(3*n) = a(3*n+1) = 0^n -A014681(n); a(3*n+2) = -(n+1).

A135682 a(n)=n if n=1 or if n=prime. Otherwise, n=4 if n is even and n=7 if n is odd.

Original entry on oeis.org

1, 2, 3, 4, 5, 4, 7, 4, 7, 4, 11, 4, 13, 4, 7, 4, 17, 4, 19, 4, 7, 4, 23, 4, 7, 4, 7, 4, 29, 4, 31, 4, 7, 4, 7, 4, 37, 4, 7, 4, 41, 4, 43, 4, 7, 4, 47, 4, 7, 4, 7, 4, 53, 4, 7, 4, 7, 4, 59, 4, 61, 4, 7, 4, 7, 4, 67, 4, 7, 4, 71, 4, 73

Offset: 1

Mohammad K. Azarian, Dec 01 2007

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A000027; A000040; A135679; A135680; A014681; A014682; A014683; A014684; A014685; A014686; A014687; A014688; A014689; A014690; A135681.

a[n_] := If[PrimeQ[n] || n == 1, n, If[EvenQ[n], 4, 7] ]; Table[a[n], {n,1,25}] (* G. C. Greubel, Oct 26 2016 *)

A135684 a(n)=11 if n is a prime number. Otherwise, a(n)=n.

Original entry on oeis.org

1, 11, 11, 4, 11, 6, 11, 8, 9, 10, 11, 12, 11, 14, 15, 16, 11, 18, 11, 20, 21, 22, 11, 24, 25, 26, 27, 28, 11, 30, 11, 32, 33, 34, 35, 36, 11, 38, 39, 40, 11, 42, 11, 44, 45, 46, 11, 48, 49, 50, 51, 52, 11, 54, 55, 56, 57, 58, 11

Offset: 1

Mohammad K. Azarian, Dec 01 2007

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000027; A000040; A135679; A135680; A014681; A014682; A014683; A014684; A014685; A014686; A014687; A014688; A014689; A014690; A135681; A135682; A135683.

[IsPrime(n) select 11 else n: n in [1..70]]; // Vincenzo Librandi, Feb 22 2013

Table[If[PrimeQ[n], 11, n], {n, 70}] (* Vincenzo Librandi, Feb 22 2013 *)

A167419 Exchange adjacent nonprimes and primes.

Original entry on oeis.org

2, 1, 4, 3, 6, 5, 8, 7, 9, 11, 10, 13, 12, 14, 15, 17, 16, 19, 18, 20, 21, 23, 22, 24, 25, 26, 27, 29, 28, 31, 30, 32, 33, 34, 35, 37, 36, 38, 39, 41, 40, 43, 42, 44, 45, 47, 46, 48, 49, 50, 51, 53, 52, 54, 55, 56, 57, 59, 58, 61, 60, 62, 63, 64, 65, 67, 66, 68, 69, 71, 70, 73, 72

Offset: 1

Giovanni Teofilatto, Nov 03 2009

If we have consecutive numbers, one prime and one nonprime, swap them. So after the initial 2,1,4,3,6,5,8,7 we have "if n is prime, a(n) = n-1; if n+1 is prime, a(n) = n+1, otherwise a(n) = n".

Stefano Spezia, Table of n, a(n) for n = 1..10000

Cf. A014681.

a[n_]:=If[PrimeQ[n],n-1,If[PrimeQ[n+1],n+1,n]]; Join[{2,1,4,3,6,5,8,7},Array[a,65,9]] (* Stefano Spezia, May 05 2023 *)

Edited by Franklin T. Adams-Watters, Nov 04 2009

42 and 64 inserted by Stefano Spezia, May 05 2023

A167542 Natural numbers, swapped in pairs, with decimal digits reversed.

Original entry on oeis.org

2, 1, 4, 3, 6, 5, 8, 7, 1, 9, 21, 11, 41, 31, 61, 51, 81, 71, 2, 91, 22, 12, 42, 32, 62, 52, 82, 72, 3, 92, 23, 13, 43, 33, 63, 53, 83, 73, 4, 93, 24, 14, 44, 34, 64, 54, 84, 74, 5, 94, 25, 15, 45, 35, 65, 55, 85, 75, 6, 95, 26, 16, 46, 36, 66, 56, 86, 76, 7, 96, 27, 17, 47, 37, 67

Offset: 1

Giovanni Teofilatto, Nov 06 2009

read("transforms") ; A014681 := proc(n) option remember; if n <= 3 then op(n+1,[0,2,1,4]) ; else procname(n-1)+procname(n-2)-procname(n-3) ; end if; end proc: A167542 := proc(n) digrev(A014681(n)) ; end proc: seq(A167542(n),n=1..120) ; # R. J. Mathar, Jan 30 2010

All terms from a(18) on corrected by R. J. Mathar, Jan 30 2010

A317630 Lexicographically first sequence of different terms, starting with a(1) = 0 and showing a 1-step roller coaster of terms together with a 1-step roller coaster of digits (see the Comments section).

Original entry on oeis.org

0, 2, 1, 4, 3, 6, 5, 8, 7, 80, 9, 13, 12, 15, 14, 17, 16, 19, 18, 24, 23, 26, 25, 28, 27, 34, 29, 36, 35, 38, 37, 45, 39, 47, 46, 49, 48, 57, 56, 59, 58, 68, 67, 120, 10, 21, 20, 31, 30, 40, 32, 42, 41, 50, 43, 52, 51, 54, 53, 61, 60, 63, 62, 65, 64, 71, 70, 73, 72, 75, 74, 81, 76, 83, 82, 85, 84, 87, 86, 91, 90, 93, 92, 95

Offset: 1

Eric Angelini and Jean-Marc Falcoz, Aug 02 2018

Every term a(n) > 0 of the sequence stands between two terms bigger than a(n) or between two terms smaller than a(n); this is also true for every digit d > 0 of the sequence that stands between two digits bigger than d or between two digits smaller than d.

			The sequence starts with 0,2,1,4,3,6,5,8,7,80,9,13,12,15,14,17,16,19...
If we consider the terms, we see indeed that 0 < 2 > 1 < 4 > 3 < 6 > 5 < 8 > 7 < 80 > 9 < 13 > 12 < 15 > 14 < 17 > 16 < 19... and if we consider the digits, we see also that 0 < 2 > 1 < 4 > 3 < 6 > 5 < 8 > 7 < 8 > 0 < 9 > 1 < 3 > 1 < 2 > 1 < 5 > 1 < 4 > 1 < 7 > 1 < 6 > 1 < 9...
So no matter the elements considered (terms, digits), those elements seem to ride on a 1-step roller-coaster: up, down, up, down, up, etc.

Lars Blomberg, Table of n, a(n) for n = 1..10000

Cf. A014681 (a 1-step roller coaster of terms only) and A317548 (a 1-step roller coaster of digits only).

A354522 Square array A(n, k), n, k >= 0, read by antidiagonals; A(n, k) = g(f(n) + f(k)) where f denotes A001057 and g denotes its inverse.

Original entry on oeis.org

0, 1, 1, 2, 3, 2, 3, 0, 0, 3, 4, 5, 4, 5, 4, 5, 2, 1, 1, 2, 5, 6, 7, 6, 7, 6, 7, 6, 7, 4, 3, 0, 0, 3, 4, 7, 8, 9, 8, 9, 8, 9, 8, 9, 8, 9, 6, 5, 2, 1, 1, 2, 5, 6, 9, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 8, 7, 4, 3, 0, 0, 3, 4, 7, 8, 11, 12, 13, 12, 13, 12, 13, 12, 13, 12, 13, 12, 13, 12

Offset: 0

Rémy Sigrist, Sep 14 2022

This sequence is directly related to A355278.
The function f is a bijection from the nonnegative integers to the integers (Z).
The nonnegative integers, together with (x,y) -> A(x,y), form an abelian group isomorph to the additive group Z (f and g act as isomorphisms).
As a consequence, each row and each column is a permutation of the nonnegative integers.

			Array A(n, k) begins:
  n\k |  0   1   2   3   4   5   6   7   8   9  10  11  12
  ----+---------------------------------------------------
    0 |  0   1   2   3   4   5   6   7   8   9  10  11  12
    1 |  1   3   0   5   2   7   4   9   6  11   8  13  10
    2 |  2   0   4   1   6   3   8   5  10   7  12   9  14
    3 |  3   5   1   7   0   9   2  11   4  13   6  15   8
    4 |  4   2   6   0   8   1  10   3  12   5  14   7  16
    5 |  5   7   3   9   1  11   0  13   2  15   4  17   6
    6 |  6   4   8   2  10   0  12   1  14   3  16   5  18
    7 |  7   9   5  11   3  13   1  15   0  17   2  19   4
    8 |  8   6  10   4  12   2  14   0  16   1  18   3  20
    9 |  9  11   7  13   5  15   3  17   1  19   0  21   2
   10 | 10   8  12   6  14   4  16   2  18   0  20   1  22
   11 | 11  13   9  15   7  17   5  19   3  21   1  23   0
   12 | 12  10  14   8  16   6  18   4  20   2  22   0  24

Cf. A001057, A014601, A014681, A047264, A047521, A355278, A357144.

f(n) = - (-1)^n * ((n+1)\2)
g(n) = if (n<=0, -2*n, 2*n-1)
A(n, k) = g(f(n) + f(k))

A355278(n+1, k+1) = prime(1 + A(n, k)) (where prime(m) denotes the m-th prime number).
A(n, k) = A(k, n).
A(n, 0) = n.
A(n, A014681(n)) = 0.
A(m, A(n, k)) = A(A(m, n), k).
A(n, n) = A014601(n).
A(n, A(n, n)) = A047264(n+1).
A(A(n, n), A(n, n)) = A047521(n+1).

A132293 Maximal number of right angles in an n-gon.

Original entry on oeis.org

1, 4, 3, 4, 5, 6

Offset: 3

Asher Stuhlman (asherstuhlman(AT)gmail.com), Nov 06 2007

A proper definition is needed for this sequence.
Conjecture from R. J. Mathar, Mar 07 2008, Apr 21 2008: (Start)
The correct sequence is a(n)=A014681(n-1), because a polyomino with a zigzag "stair" shape along a diagonal provides a solution where the number of right angles equals the number of edges:
┌─────────┐
|. . . . .|
|         ┘
|. . . . .
|
|. .|
| ┌─┘
|.|
└─┘
(End)
Possibly an incorrect version of A049008. - Michal Paulovic, Sep 27 2023

			The first (n=3) integer in this sequence is 1 because a triangle cannot have more than one right angle.

Cf. A049008.

cp's OEIS Frontend

A101677 a(n) = a(n-1) - 2*a(n-2) + 2*a(n-3) - 2*a(n-4) + 2*a(n-5) - a(n-6).

Views

Author

Keywords

Comments

Links

Programs

Magma

Mathematica

PARI

Formula

A135682 a(n)=n if n=1 or if n=prime. Otherwise, n=4 if n is even and n=7 if n is odd.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

A135684 a(n)=11 if n is a prime number. Otherwise, a(n)=n.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

A167419 Exchange adjacent nonprimes and primes.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Extensions

A167542 Natural numbers, swapped in pairs, with decimal digits reversed.

Views

Author

Keywords

Programs

Maple

Extensions

A317630 Lexicographically first sequence of different terms, starting with a(1) = 0 and showing a 1-step roller coaster of terms together with a 1-step roller coaster of digits (see the Comments section).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A354522 Square array A(n, k), n, k >= 0, read by antidiagonals; A(n, k) = g(f(n) + f(k)) where f denotes A001057 and g denotes its inverse.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Formula

A132293 Maximal number of right angles in an n-gon.

Views

Author

Keywords

Comments

Examples

Crossrefs

A101677 a(n) = a(n-1) - 2a(n-2) + 2a(n-3) - 2a(n-4) + 2a(n-5) - a(n-6).