A200975 Numbers on the diagonals in Ulam's spiral.

1, 3, 5, 7, 9, 13, 17, 21, 25, 31, 37, 43, 49, 57, 65, 73, 81, 91, 101, 111, 121, 133, 145, 157, 169, 183, 197, 211, 225, 241, 257, 273, 289, 307, 325, 343, 361, 381, 401, 421, 441, 463, 485, 507, 529, 553, 577, 601, 625, 651, 677, 703, 729, 757, 785, 813, 841, 871, 901

Offset

1,2

Comments

All entries are odd.

From Bob Selcoe, Oct 22 2014: (Start)

The following hold:

1. a(n) = (2k + 1)^2 when n = 4k + 1, k >= 0

2. a(n) = 4*k^2 + 1 when n = 4k - 1, k > 0

3 a(n) = k^2 + k + 1 when n = 2k, k > 0.

Conjecture 1: there must be at least one prime in [a(n), a(n+1)] inclusive.

Conjecture 2: generally, when j is in [(2m-1)^2+1, (2m+1)^2] inclusive, there must be at least one prime in [j-2m-1, j] inclusive. If true, then Conjecture 1 is true; also suggests A248623, A248835 and Oppermann's conjecture (see A002620) likely are true. (End)

Links

Todd Silvestri, Table of n, a(n) for n = 1..1000

Project Euler, Problem 28: Number spiral diagonals

Index entries for linear recurrences with constant coefficients, signature (2,-1,0,1,-2,1).

Formula

a(4n) = 4n^2 + 2n + 1; a(4n+1) = 4n^2 + 4n + 1; a(4n+2) = 4n^2 + 6n + 3; a(4n+3) = 4n^2 + 8n + 5. [corrected by James Mitchell, Dec 31 2017]

G.f.: -x*(1+x+x^5-x^4) / ( (1+x)*(x^2+1)*(x-1)^3 ). - R. J. Mathar, Nov 28 2011

a(n) = (2*n*(n+2)+(-1)^n-4*sin((Pi*n)/2)+7)/8 = (A249356(n)+7)/8. - Todd Silvestri, Oct 25 2014

a(n) = floor_(n*(n+2)/4) + floor_(n(mod 4)/3) + 1. - Bob Selcoe, Oct 27 2014

Example

The numbers between ** are in this sequence.
.
  *21*--22---23---24--*25*
    |
    |
   20   *7*---8---*9*--10
    |    |              |
    |    |              |
   19    6   *1*---2   11
    |    |         |    |
    |    |         |    |
   18   *5*---4---*3*  12
    |                   |
    |                   |
  *17*--16---15---14--*13*

Mathematica

Sort@ Flatten@ Table[4n^2 + (2j - 4)n + 1, {j, 0, 3}, {n, 16}] (* Robert G. Wilson v, Jul 10 2014 *)
a[n_Integer/; n>0]:=Quotient[2 n (n+2)+(-1)^n-4 Mod[n^2 (3 n+2), 4, -1]+7, 8] (* Todd Silvestri, Oct 25 2014 *)

Prog

(Python)
# prints all numbers on the diagonals of a sq*sq spiral
sq = 5
d = 1
while 2*d - 1 < sq:
    print(4*d*d - 4*d +1)
    print(4*d*d - 4*d +1 + 1* 2* d)
    print(4*d*d - 4*d +1 + 2* 2* d)
    print(4*d*d - 4*d +1 + 3* 2* d)
    d += 1
print(sq*sq)
(PARI) al(n)=local(r=vector(n), j); r[1]=1; for(k=2, n, r[k]=r[k-1]+(k+2)\4*2); r /* Franklin T. Adams-Watters, Nov 26 2011 */

Crossrefs

Cf. A016754, A054554, A053755, and A054569 interleaved, A002620,

Cf. A121658 (complementary)

Sequence in context: A222314 A228232 A182058 * A370452 A058871 A126278

Adjacent sequences: A200972 A200973 A200974 * A200976 A200977 A200978

Keyword

nonn,easy

Author

Ismael Bouya, Nov 25 2011

Extensions

Edited with more terms by Franklin T. Adams-Watters, Nov 26 2011

Status

approved