A065186 a(1)=1, a(2)=3, a(3)=5, a(4)=2, a(5)=4; for n > 5, a(n) = a(n-5) + 5.

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

Offset: 1

Author

Antti Karttunen, Oct 19 2001

Keywords

• nonn
• easy

Comments

"Greedy Dragons" permutation of the natural numbers, inverse of A065187.

This permutation is produced by a simple greedy algorithm: walk along each successive antidiagonal of an infinite array and place a Shoogi dragon piece (i.e., the "promoted" rook, Ryuu, that moves like a chess rook, but can also move one square diagonally) in the first available position where it is not threatened by any dragon already placed.

I.e., this permutation satisfies the condition that p(i+1) != p(i)+-1 for all i.

Alternatively, this is obtained directly if n-1 is converted to base 5, the least significant digit is doubled (modulo 5, i.e., 0->0, 1->2, 2->4, 3->1, 4->3) and one is added back to the resulting number.

a(1) = 1, a(n) = smallest number such that no two successive terms differ by 1 and no two terms are equal. - Amarnath Murthy, May 06 2003

This is also the lexicographic first positive sequence such that the distance between any subsequent terms, |a(n+1)-a(n)|, is a prime number and no number occurs twice, with a(1) = 1: A variant of A277618, which obeys the same rules but starts with a(0) = 0; and of A277617, which is defined similarly with squares > 1 instead of primes. - M. F. Hasler, Oct 23 2016

Links

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

Crossrefs

"Greedy Queens" and "Quintal Queens" permutations: A065188, A065257.

Cf. A065186.

Programs

Maple

[seq(GreedyDragonsDirect(j),j=1..125)]; GreedyDragonsDirect := n -> n + ((n-1) mod 5) - 5*(floor((n-1 mod 5)/3));
Or empirically, by using the algorithm given at A065188: GreedyDragons := upto_n -> PM2PL(GreedyNonThreateningPermutation(upto_n,1,1),upto_n);

Mathematica

RecurrenceTable[{a[1] == 1, a[2] == 3, a[3] == 5, a[4] == 2, a[5] == 4, a[n] == a[n - 5] + 5}, a, {n, 80}] (* or *) LinearRecurrence[{1, 0, 0, 0, 1, -1}, {1, 3, 5, 2, 4, 6}, 80] (* Harvey P. Dale, Mar 11 2012 *)
Flatten[Table[5n + {1, 3, 5, 2, 4}, {n, 0, 14}]] (* Alonso del Arte, Jul 25 2017 *)

PARI

{ for (n=1, 1000, if (n>5, a=a5 + 5; a5=a4; a4=a3; a3=a2; a2=a1; a1=a, if (n==1, a=a5=1, if (n==2, a=a4=3, if (n==3, a=a3=5, if (n==4, a=a2=2, a=a1=4))))); write("b065186.txt", n, " ", a) ) } \\ Harry J. Smith, Oct 13 2009

PARI

n=1;v=[n];while(n<200,if(isprime(abs(n-v[#v]))&&!vecsearch(vecsort(v),n),v=concat(v,n);n=1);n++);v \\ Derek Orr, Jul 24 2017

PARI

a(n) = n--; [1,3,5,2,4][n%5+1]+5*(n\5) \\ David A. Corneth, Jul 24 2017

PARI

first(n) = my(v = [1,3,5,2,4]); if(n < 5, return(vector(n, i, v[i])), v = concat(v, vector(n-5))); for(i=6, n, v[i]=5 + v[i-5]); v \\ David A. Corneth, Jul 24 2017

PARI

nxt(n) = if(n%5, n+2, n-3) \\ David A. Corneth, Jul 24 2017

Formula

a(n) = n + ((n-1) mod 5) - 5*(floor(((n-1) mod 5)/3)).

G.f.: x*(x^5 + 2*x^4 - 3*x^3 + 2*x^2 + 2*x + 1)/((x - 1)*(x^5 - 1))

a(n) = a(n-1) + a(n-5) - a(n-6), with n>6, a(1)=1, a(2)=3, a(3)=5, a(4)=2, a(5)=4, a(6)=6. - Harvey P. Dale, Mar 11 2012