A357745 Numbers on the 8 main spokes of a square spiral with 1 in the center.

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 15, 17, 19, 21, 23, 25, 28, 31, 34, 37, 40, 43, 46, 49, 53, 57, 61, 65, 69, 73, 77, 81, 86, 91, 96, 101, 106, 111, 116, 121, 127, 133, 139, 145, 151, 157, 163, 169, 176, 183, 190, 197, 204, 211, 218, 225, 233, 241, 249, 257, 265, 273

Offset: 1

Author

Karl-Heinz Hofmann, Dec 22 2022

Keywords

• nonn
• easy

Comments

The 8 main spokes are (with 1 in the center, 2 to the east, 3 to the northeast): east: A054552; northeast: A054554; north: A054556; northwest: A053755; west: A054567; southwest: A054569; south: A033951; southeast: A016754.

Alternatively the 8 main spokes are pairwise part of the 4 main axes: horizontal: A317186; vertical: A267682; diagonal: A002061; antidiagonal: A080335.

And lastly the 4 main axes are giving two main crosses: Horizontal-vertical cross: A039823; Diagonal-antidiagonal cross: A200975.

Examples

			See visualization in links.
		

Links

• Karl-Heinz Hofmann, Table of n, a(n) for n = 1..10000
• Karl-Heinz Hofmann, Visualization of the first few terms
• Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,0,0,0,1,-2,1).

Crossrefs

Cf. A054552, A054554, A054556, A053755, A054567, A054569, A033951, A016754.

Cf. A317186, A267682, A002061, A080335, A039823, A200975.

Programs

Mathematica

Rest@ CoefficientList[Series[x (1 - x^8 + x^9)/((1 - x)^3*(1 + x) (1 + x^2) (1 + x^4)), {x, 0, 63}], x] (* Michael De Vlieger, Dec 29 2022 *)
a[n_] := BitShiftRight[(n + 3)^2, 4] + Boole[BitAnd[n, 7] != 1]; Array[a, 65] (* Amiram Eldar, Dec 30 2022, after the PARI code *)
LinearRecurrence[{2,-1,0,0,0,0,0,1,-2,1},{1,2,3,4,5,6,7,8,9,11},70] (* Harvey P. Dale, Jul 13 2025 *)

PARI

a(n) = sqr(n+3)>>4 + (bitand(n,7)!=1); \\ Kevin Ryde, Dec 30 2022

Python

def A357745(n): return ((n+3)**2 >> 4) + 1 if n % 8 != 1 else (n+3)**2 >> 4

Formula

G.f.: x*(1-x^8+x^9)/((1-x)^3*(1+x)*(1+x^2)*(1+x^4)). - Joerg Arndt, Dec 29 2022

a(n) = floor((n+3)^2 / 16) + (1 if n != 1 mod 8). - Kevin Ryde, Dec 30 2022