

A065186


a(1)=1, a(2)=3, a(3)=5, a(4)=2, a(5)=4; for n > 5, a(n) = a(n5) + 5.


6



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
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OFFSET

1,2


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 n1 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.  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).


FORMULA

a(n) = n + ((n1) mod 5)  5*(floor(((n1) 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(n1) + a(n5)  a(n6), 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


MAPLE

[seq(GreedyDragonsDirect(j), j=1..125)]; GreedyDragonsDirect := n > n + ((n1) mod 5)  5*(floor((n1 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 *)


PROG

(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(nv[#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(n5))); for(i=6, n, v[i]=5 + v[i5]); v \\ David A. Corneth, Jul 24 2017
(PARI) nxt(n) = if(n%5, n+2, n3) \\ David A. Corneth, Jul 24 2017


CROSSREFS

"Greedy Queens" and "Quintal Queens" permutations: A065188, A065257.
Cf. A065186.
Sequence in context: A222601 A104807 A131793 * A210521 A219249 A203553
Adjacent sequences: A065183 A065184 A065185 * A065187 A065188 A065189


KEYWORD

nonn,easy


AUTHOR

Antti Karttunen, Oct 19 2001


STATUS

approved



