A014681 -id:A014681 - cp's OEIS frontend

A124349 Numbers of directed Hamiltonian cycles on the n-prism graph.

6, 12, 10, 16, 14, 20, 18, 24, 22, 28, 26, 32, 30, 36, 34, 40, 38, 44, 42, 48, 46, 52, 50, 56, 54, 60, 58, 64, 62, 68, 66, 72, 70, 76, 74, 80, 78, 84, 82, 88, 86, 92, 90, 96, 94, 100, 98, 104, 102, 108, 106, 112, 110, 116, 114, 120, 118, 124, 122, 128, 126, 132, 130

Offset: 3

Eric W. Weisstein, Oct 26 2006

Eric Weisstein's World of Mathematics, Hamiltonian Cycle.
Eric Weisstein's World of Mathematics, Prism Graph.
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A014681, A124350, A270273.

[2*n+(1-(n mod 2))*4: n in [3..80]]; // Vincenzo Librandi, Jan 26 2016

seq( 2*n + (1-(n mod 2))*4, n=3..100); # Robert Israel, Mar 14 2016

Table[2 n + (1 - Mod[n, 2]) 4, {n, 3, 100}] (* Vincenzo Librandi, Jan 26 2016 *)

Vec(2*x^3*(3+3*x-4*x^2)/((1-x)^2*(1+x)) + O(x^100)) \\ Altug Alkan, Mar 14 2016

a(n) = 2*n + (1-(n mod 2))*4.
From Colin Barker, Aug 22 2012: (Start)
a(n) = a(n-1)+a(n-2)-a(n-3).
G.f.: 2*x^3*(3+3*x-4*x^2)/((1-x)^2*(1+x)). (End)
a(n) = 2*A014681(n+1). - R. J. Mathar, Jan 25 2016
E.g.f.: 2*(2 + x)*cosh(x) + 2*x*sinh(x) - 2*(2 + x + 2*x^2). - Stefano Spezia, Jan 28 2024

Name clarified by Andrew Howroyd, Mar 14 2016

A162853 Take the binary representation of n. Reduce by one digit every run (completely of either 0's or 1's) of an even number of digits. Increase by one digit every run of an odd number of digits in the binary representation of n (where this added digit has the same value that makes up the rest of the run's digits). a(n) = the decimal equivalent of the result.

Original entry on oeis.org

0, 3, 12, 1, 6, 51, 4, 15, 48, 27, 204, 25, 2, 19, 60, 7, 24, 195, 108, 13, 102, 819, 100, 207, 16, 11, 76, 9, 30, 243, 28, 63, 192, 99, 780, 97, 54, 435, 52, 111, 816, 411, 3276, 409, 50, 403, 828, 103, 8, 67, 44, 5, 38, 307, 36, 79, 240, 123, 972, 121, 14, 115, 252, 31, 96

Offset: 0

Leroy Quet, Jul 14 2009

This is a self-inverse permutation of the nonnegative integers.
Clarification: The consecutive "runs" (mentioned in the definition) alternate between those completely of 1's and those completely of 0's.
In the binary representation of n, replace each run of length r by a run of length A014681(r). - Rémy Sigrist, Oct 09 2018

			152 in binary is 10011000. There is a run of one 1, followed by a run of two 0's, followed by a run of two 1's, followed by a run of three 0's. We reduce the two runs of two digits each to one digit; and we add a digit (a 1) to the first run of one 1, and a digit (a 0) to the last run of three 0's, to get 11010000. So a(152) is the decimal equivalent of this, which is 208.

Cf. A014681, A266150, A266151.

rerun := proc(L) if nops(L) mod 2 = 0 then subsop(1=NULL,L) ; else [op(L),op(1,L)] ; fi; end: Lton := proc(L) local i; add( op(i,L)*2^(i-1),i=1..nops(L)) ; end: A162853 := proc(n) local strt,en,L,dgs,i; strt := 1; en := -1; L := [] ; dgs := convert(n,base,2) ; for i from 2 to nops(dgs) do if op(i,dgs) <> op(i-1,dgs) then en := i-1 ; L := [op(L), op(rerun([op(strt..en,dgs)])) ] ; strt := i; fi; od: en := nops(dgs) ; L := [op(L), op(rerun([op(strt..en,dgs)])) ] ; Lton(L) ; end: seq(A162853(n),n=1..100) ; # R. J. Mathar, Aug 01 2009

Table[FromDigits[Flatten[If[OddQ[Length[#]],Join[{First[#]},#],Drop[#,1]]& /@Split[ IntegerDigits[ n,2]]], 2],{n,70}] (* Harvey P. Dale, Jun 20 2011 *)

a(n) = if (n==0, 0, my (b=n%2, r=valuation(n+b,2), rr=if (r%2, r+1, r-1)); (a(n\2^r)+b)*2^rr-b) \\ Rémy Sigrist, Oct 09 2018

a(n) = A266150(A266151(n)) = A266151(A266150(n)) for any n > 0. - Rémy Sigrist, Oct 09 2018

Extended beyond a(13) by R. J. Mathar, Aug 01 2009

a(0) added by Rémy Sigrist, Oct 09 2018

A163543 The relative direction (0=straight ahead, 1=turn right, 2=turn left) taken by the type I Hilbert's Hamiltonian walk A163359 at the step n.

Original entry on oeis.org

2, 2, 1, 0, 1, 1, 2, 2, 1, 1, 0, 1, 2, 2, 0, 2, 1, 1, 2, 0, 2, 2, 1, 1, 2, 2, 0, 2, 1, 1, 0, 0, 1, 1, 2, 0, 2, 2, 1, 1, 2, 2, 0, 2, 1, 1, 2, 0, 2, 2, 1, 0, 1, 1, 2, 2, 1, 1, 0, 1, 2, 2, 1, 0, 1, 1, 2, 0, 2, 2, 1, 1, 2, 2, 0, 2, 1, 1, 0, 1, 2, 2, 1, 0, 1, 1, 2, 2, 1, 1, 0, 1, 2, 2, 0, 0, 2, 2, 1, 0, 1, 1

Offset: 1

Antti Karttunen, Aug 01 2009

nonn

a(16*n) = a(256*n) for all n.

A. Karttunen, Table of n, a(n) for n = 1..4096

a(n) = A014681(A163542(n)). See also A163541.

HC = {
L[n_ /; IntegerQ[n/2]] :> {F[n], L[n], L[n + 1], R[n + 2]},
R[n_ /; IntegerQ[(n + 1)/2]] :> {F[n], R[n], R[n + 3], L[n + 2]},
R[n_ /; IntegerQ[n/2]] :> {L[n], R[n + 1], R[n], F[n + 3]},
L[n_ /; IntegerQ[(n + 1)/2]] :> {R[n], L[n + 3], L[n], F[n + 1]},
F[n_ /; IntegerQ[n/2]] :> {L[n], R[n + 1], R[n], L[n + 3]},
F[n_ /; IntegerQ[(n + 1)/2]] :> {R[n], L[n + 3], L[n], R[n + 1]}};
a[1] = F[0]; Map[(a[n_ /; IntegerQ[(n - #)/16] ] := Part[Flatten[a[(n + 16 - #)/16] /. HC /. HC],#]) &, Range[16]];
Part[a[#] & /@ Range[4^4] /. {L[] -> 2, R[] -> 1, F[] -> 0}, 2 ;; -1] (* _Bradley Klee, Aug 06 2015 *)

(define (A163543 n) (A163241 (modulo (- (A163541 (1+ n)) (A163541 n)) 4)))

a(n) = A163241((A163541(n+1)-A163541(n)) modulo 4).

A061800 a(n) = n + (-1)^(n mod 3).

Original entry on oeis.org

1, 0, 3, 4, 3, 6, 7, 6, 9, 10, 9, 12, 13, 12, 15, 16, 15, 18, 19, 18, 21, 22, 21, 24, 25, 24, 27, 28, 27, 30, 31, 30, 33, 34, 33, 36, 37, 36, 39, 40, 39, 42, 43, 42, 45, 46, 45, 48, 49, 48, 51, 52, 51, 54, 55, 54, 57, 58, 57, 60, 61, 60, 63, 64, 63, 66, 67, 66, 69, 70, 69, 72

Offset: 0

Olivier Gérard, Jun 22 2001

The arithmetic function v_3(n,1) as defined in A289187. - Robert Price, Aug 22 2017; corrected by Ridouane Oudra, Dec 28 2024

			a(4) = 4 + (-1)^1 = 3.

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

Cf. A004442, A014681, A289187.

[n+(-1)^(n mod 3): n in [0..80]]; // Vincenzo Librandi, Aug 23 2017

A061800:=n->n+(-1)^(n mod 3): seq(A061800(n), n=0..150); # Wesley Ivan Hurt, Oct 07 2017

Table[n + (-1)^Mod[n, 3], {n, 0, 80}] (* Vincenzo Librandi, Aug 23 2017 *)

a(n) = { n + (-1)^(n%3) } \\ Harry J. Smith, Jul 28 2009

O.g.f.: (1-x+3*x^2)/((-1+x)^2*(1+x+x^2)). - R. J. Mathar, Apr 02 2008
a(n) = (3*n + 1 - 4*cos(2*(n+2)*Pi/3))/3. - Wesley Ivan Hurt, Sep 26 2017
a(n) = a(n-1) + a(n-3) - a(n-4) for n > 3. - Wesley Ivan Hurt, Oct 07 2017
a(n) = n + 2 - gcd(n+2,3). - Ridouane Oudra, Dec 28 2024
Sum_{n>=2} (-1)^n/a(n) = Pi/(3*sqrt(3)) + log(2) - 1. - Amiram Eldar, Jan 15 2025

A065168 Permutation t->t-1 of Z, folded to N.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Oct 19 2001

This permutation consists of just one cycle, which is infinite.

Index entries for linear recurrences with constant coefficients, signature (1,1,-1).
Index entries for sequences that are permutations of the natural numbers.

Inverse permutation to A065164.

Obtained by composing permutations A065190 and A014681.

a:= n-> n-2*(-1)^n +`if`(n=2, 1, 0):
seq(a(n), n=1..80); # Alois P. Heinz, Mar 07 2012

Join[{3, 1}, LinearRecurrence[{1, 1, -1}, {5, 2, 7}, 100]] (* Jean-François Alcover, Feb 28 2016 *)

Let f: Z -> N be given by f(z) = 2z if z>0 else 2|z|+1, with inverse g(z) = z/2 if z even else (1-z)/2. Then a(n) = f(g(n)-1).
G.f.: x*(3-2*x+x^4+x^2-x^3) / ((x+1)*(x-1)^2). - Alois P. Heinz, Mar 07 2012
Sum_{n>=1} (-1)^n/a(n) = 2 - log(2). - Amiram Eldar, Aug 08 2023

A086970 Fix 1, then exchange the subsequent odd numbers in pairs.

Original entry on oeis.org

1, 5, 3, 9, 7, 13, 11, 17, 15, 21, 19, 25, 23, 29, 27, 33, 31, 37, 35, 41, 39, 45, 43, 49, 47, 53, 51, 57, 55, 61, 59, 65, 63, 69, 67, 73, 71, 77, 75, 81, 79, 85, 83, 89, 87, 93, 91, 97, 95, 101, 99, 105, 103, 109, 107, 113, 111, 117, 115, 121, 119

Offset: 0

Paul Barry, Jul 26 2003

Partial sums are A086955.

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A004442, A014681, A065190.

[1] cat [2*n+1-2*(-1)^n: n in [1..70]]; // Vincenzo Librandi, Jun 21 2017

Join[{1}, LinearRecurrence[{1, 1, -1}, {5, 3, 9}, 60]] (* Vincenzo Librandi, Jun 21 2017 *)

Vec((1+4*x-3*x^2+2*x^3)/((1+x)*(1-x)^2) + O(x^100)) \\ Michel Marcus, Jun 21 2017

G.f.: (1+4*x-3*x^2+2*x^3)/((1+x)*(1-x)^2).
a(n) = n + abs(2 - (n + 1)*(-1)^n). - Lechoslaw Ratajczak, Dec 09 2016
a(n) = 2*A065190(n+1)-1 and a(n) = 2*A014681(n)+1. - Michel Marcus, Dec 10 2016
From Guenther Schrack, Jun 09 2017: (Start)
a(n) = 2*n + 1 - 2*(-1)^n for n > 0.
a(n) = 2*n + 1 - 2*cos(n*Pi) for n > 0.
a(n) = 4*n - a(n-1) for n > 1.
Linear recurrence: a(n) = a(n-1) + a(n-2) - a(n-3) for n > 3.
First differences: 2 - 4*(-1)^n for n > 1; -(-1)^n*A010696(n) for n > 1.
a(n) = A065164(n+1) + n for n > 0.
a(A014681(n)) = A005408(n) for n >= 0.
a(A005408(A014681(n)) for n >= 0.
a(n) = A005408(A103889(n)) for n >= 0.
A103889(a(n)) = 2*A065190(n+1) for n >= 0.
a(2*n-1) = A004766(n) for n > 0.
a(2*n+2) = A004767(n) for n >= 0. (End)

A124356 Number of (directed) Hamiltonian cycles on the Moebius ladder graph M_n (for n>=4).

Original entry on oeis.org

2, 8, 6, 12, 10, 16, 14, 20, 18, 24, 22, 28, 26, 32, 30, 36, 34, 40, 38, 44, 42, 48, 46, 52, 50, 56, 54, 60, 58, 64, 62, 68, 66, 72, 70, 76, 74, 80, 78, 84, 82, 88, 86, 92, 90, 96, 94, 100, 98, 104, 102, 108, 106, 112, 110, 116, 114, 120, 118, 124, 122, 128, 126, 132

Offset: 1

Eric W. Weisstein, Nov 05 2006

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Hamiltonian Cycle
Eric Weisstein's World of Mathematics, Moebius Ladder
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

[2*(n+1)+2*(-1)^n: n in [1..70]]; // Vincenzo Librandi, Aug 11 2011

a(n odd) = 2*n, a(n even) = 2*n+4.
a(n) = 2(n+1)+2(-1)^n. - Paul Barry, Feb 17 2008
a(n) -a(n-4) = 8. - Paul Curtz, Apr 19 2011
a(n) = +1*a(n-1) +1*a(n-2) -1*a(n-3). - Joerg Arndt, Apr 22 2011
G.f.: 2*x*(1+3*x-2*x^2)/((1-x)^2*(1+x)). - Colin Barker, Jan 23 2012
a(n) = 2*A014681(n+1). - R. J. Mathar, Nov 27 2015

A135681 a(n)=n if n=1 or if n=prime. Otherwise, n=4 if n is even and n=1 if n is odd.

Original entry on oeis.org

1, 2, 3, 4, 5, 4, 7, 4, 1, 4, 11, 4, 13, 4, 1, 4, 17, 4, 19, 4, 1, 4, 23, 4, 1, 4, 1, 4, 29, 4, 31, 4, 1, 4, 1, 4, 37, 4, 1, 4, 41, 4, 43, 4, 1, 4, 47, 4, 1, 4, 1, 4, 53, 4, 1, 4, 1, 4, 59, 4, 61, 4, 1, 4, 1, 4, 67, 4, 1, 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. A135679; A135580; A014681; A014682; A014683; A014684; A014685; A014686; A014687; A014688; A014689; A014690.

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

A163537 The relative direction (0=straight ahead, 1=turn right, 2=turn left) of the Peano curve A163336 at point n.

Original entry on oeis.org

0, 2, 2, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 2, 2, 0, 0, 0, 0, 2, 2, 0, 1, 1, 0, 2, 2, 0, 1, 1, 0, 2, 2, 0, 0, 0, 0, 2, 2, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 2, 2, 0, 1, 1, 0, 2, 2, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 2, 2, 0, 0, 0, 0, 2, 2, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 2, 2, 0, 0, 0, 0, 2, 2, 0, 1, 1, 0, 0, 0, 0, 1, 1

Offset: 1

Antti Karttunen, Aug 01 2009

nonn

a(9*n) = a(81*n) for all n.

Antti Karttunen, Table of n, a(n) for n = 1..6561

Cf. A163535 (direction), A163536 (transposed relative).

(define (A163537 n) (A163241 (modulo (- (A163535 (1+ n)) (A163535 n)) 4)))

a(n) = A163241((A163535(n+1)-A163535(n)) modulo 4).

a(n) = A014681(A163536(n)).

Name corrected by Kevin Ryde, Aug 29 2020

A263449 Permutation of the natural numbers: [4k+1, 4k+4, 4k+3, 4k+2, ...].

Original entry on oeis.org

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

Offset: 0

Wesley Ivan Hurt, Oct 18 2015

From Franklin T. Adams-Watters, Jul 13 2017: (Start)
For this to be a permutation, it should have offset one, not zero.
With offset 1, a(n) is the smallest positive integer == n (mod 2) with a(n) != a(n-1) + 1. (End)

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-1).
Index entries for sequences that are permutations of the natural numbers.

Cf. A002162, A005408, A014681.

[n+1+(1-(-1)^n)*(-1)^(n*(n-1) div 4) : n in [0..100]];

/* By definition: */ &cat[[4*k+1,4*k+4,4*k+3,4*k+2]: k in [0..20]]; // Bruno Berselli, Oct 19 2015

A263449:=n->n+1+(1-(-1)^n)*(-1)^(n*(n-1)/2): seq(A263449(n), n=0..100);

Table[n + 1 + (1 - (-1)^n) (-1)^(n (n - 1)/2), {n, 0, 100}] (* or *) LinearRecurrence[{2,-2,2,-1}, {1, 4, 3, 2}, 70]

Vec((1+2*x-3*x^2+2*x^3)/((x-1)^2*(1+x^2)) + O(x^100)) \\ Altug Alkan, Oct 19 2015

a(n) = n+1+I*((-I)^n-I^n) \\ Colin Barker, Oct 27 2015

G.f.: (1+2*x-3*x^2+2*x^3)/((x-1)^2*(1+x^2)).
a(n) = 2*a(n-1)-2*a(n-2)+2*a(n-3)-a(n-4) for n>3.
a(n) = n+1+(1-(-1)^n)*(-1)^(n*(n-1)/2).
a(2n) = A005408(n), a(2n+1) = 2*A014681(n+1).
a(n) = n+1+i*((-i)^n-i^n), where i=sqrt(-1). - Colin Barker, Oct 27 2015
a(n) = 4*ceiling(n/4) - (n mod 4) + 1. - Wesley Ivan Hurt, Nov 07 2015
Sum_{n>=0} (-1)^n/a(n) = log(2) (A002162). - Amiram Eldar, Nov 28 2023

cp's OEIS Frontend

A124349 Numbers of directed Hamiltonian cycles on the n-prism graph.

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A163543 The relative direction (0=straight ahead, 1=turn right, 2=turn left) taken by the type I Hilbert's Hamiltonian walk A163359 at the step n.

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A061800 a(n) = n + (-1)^(n mod 3).

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A065168 Permutation t->t-1 of Z, folded to N.

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A086970 Fix 1, then exchange the subsequent odd numbers in pairs.

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A124356 Number of (directed) Hamiltonian cycles on the Moebius ladder graph M_n (for n>=4).

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