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A200975 Numbers on the diagonals in Ulam's spiral. 4
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 (list; graph; refs; listen; history; text; internal format)
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 * A058871 A126278 A121259

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

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Last modified September 24 06:13 EDT 2021. Contains 347623 sequences. (Running on oeis4.)