A054552 -id:A054552 - cp's OEIS frontend

A257171 Sum of numbers on n-th segment of Ulam's spiral.

1, 5, 9, 13, 27, 36, 62, 78, 120, 145, 207, 243, 329, 378, 492, 556, 702, 783, 965, 1065, 1287, 1408, 1674, 1818, 2132, 2301, 2667, 2863, 3285, 3510, 3992, 4248, 4794, 5083, 5697, 6021, 6707, 7068, 7830, 8230, 9072, 9513, 10439, 10923, 11937, 12466, 13572, 14148, 15350

Offset: 0

Kival Ngaokrajang, Apr 17 2015

From Ulam's spiral, consider successive segments s(0) = [1]; s(1) = [2, 3]; s(2) = [4, 5]; s(3) = [6, 7]; s(4) = [8, 9, 10] and so on. a(n) is sum of numbers of the segment s(n). The first differences are A000290 interleaved with 2*A002061. See illustration in the links.

Michael De Vlieger, Table of n, a(n) for n = 0..10000
Kival Ngaokrajang, Illustration of initial terms
Index entries for linear recurrences with constant coefficients, signature (1,3,-3,-3,3,1,-1).

Cf. A054552, A002061, A000290.

f[n_] := Block[{t = {5, 9}}, For[i = 3, i <= n, i++, If[OddQ@ i, AppendTo[t, t[[i - 1]] + ((i - 1)/2 + 1)^2], AppendTo[t, t[[i - 1]] + 2 ((i - 1)^2/4 + (i - 1) + 7/4)]]]; {1}~Join~t]; f@ 48(* Michael De Vlieger, Apr 17 2015 *)

a(n) = if(n<=0, 1, if(n<=1, 5, if(n<=2, 9, if(Mod(n,2)==0, a(n-1)+2*((n-1)^2/4+(n-1)+7/4), a(n-1)+((n-1)/2+1)^2))))
for (n=0, 100, print1(a(n),", "))

Vec((2*x^8-3*x^7-5*x^6+9*x^5+5*x^4-8*x^3+x^2+4*x+1)/((x-1)^4*(x+1)^3) + O(x^100)) \\ Colin Barker, Apr 18 2015

a(0) = 1; a(1) = 5; a(2) = 9; for n >= 3, a(n) = a(n-1)+((n-1)/2+1)^2, if n = even; otherwise a(n) = a(n-1)+2*((n-1)^2/4+(n-1)+7/4).
From Colin Barker, Apr 17 2015: (Start)
a(n) = (n^3+5*n^2+14*n+16)/8 for n even and n>1.
a(n) = (n^3+4*n^2+11*n+8)/8 for n odd and n>1.
G.f.: (2*x^8-3*x^7-5*x^6+9*x^5+5*x^4-8*x^3+x^2+4*x+1) / ((x-1)^4*(x+1)^3).
(End)

A308385 a(n) is the last square visited by fers moves on a spirally numbered (2n-1) X (2n-1) board, moving to the lowest available unvisited square at each step.

Original entry on oeis.org

1, 3, 15, 29, 61, 87, 139, 177, 249, 299, 391, 453, 565, 639, 771, 857, 1009, 1107, 1279, 1389, 1581, 1703, 1915, 2049, 2281, 2427, 2679, 2837, 3109, 3279, 3571, 3753, 4065, 4259, 4591, 4797, 5149, 5367, 5739, 5969, 6361, 6603, 7015, 7269, 7701, 7967, 8419

Offset: 1

Sangeet Paul, May 23 2019

A 5 X 5 board, for example, is numbered with the square spiral:
.
  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
.
A fers is a (1,1)-leaper and can move one square diagonally.

Colin Barker, Table of n, a(n) for n = 1..1000
Stephen Emmerson and Geoff Foster, A glossary of fairy chess definitions, British Chess Problem Society, 2018.
Wikipedia, Ferz
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A054552, A054556, A316667, A316884.

[(3/2)*(5+(-1)^n) - (10+(-1)^n)*n + 4*n^2: n in [1..50]]; // Vincenzo Librandi, Aug 01 2019

Table[(3/2) (5 + (-1)^n) - (10 + (-1)^n) n + 4 n^2, {n, 60}] (* Vincenzo Librandi, Aug 01 2019 *)

Vec(x*(1 + 2*x + 10*x^2 + 10*x^3 + 9*x^4) / ((1 - x)^3*(1 + x)^2) + O(x^40)) \\ Colin Barker, May 23 2019

a(n) = (4n^2-9n+6)*[n is odd] + (4n^2-11n+9)*[n is even] where [] is the Iverson bracket.
a(n) = A054556(n)*[n is odd] + (A054552(n)+1)*[n is even] where [] is the Iverson bracket.
a(n) = A316884(n^2)*[n is odd] + A316884(n^2-n)*[n is even] where [] is the Iverson bracket.
From Colin Barker, May 23 2019: (Start)
G.f.: x*(1 + 2*x + 10*x^2 + 10*x^3 + 9*x^4) / ((1 - x)^3*(1 + x)^2).
a(n) = (3/2)*(5+(-1)^n) - (10+(-1)^n)*n + 4*n^2.
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n>4.
(End)
E.g.f.: (1/2)*exp(-x)*(3 + 2*x + exp(2*x)*(15 - 12*x + 8*x^2)) - 9. - Stefano Spezia, Aug 17 2019

A336335 a(n) is the index of the first occurrence of the Euclidean distance prime(n) from a point on a square spiral to its starting point at 1.

Original entry on oeis.org

11, 28, 50, 176, 452, 536, 848, 1388, 2048, 1682, 3752, 4784, 6272, 7268, 8696, 7938, 13748, 14210, 17756, 19952, 11888, 24728, 27308, 25322, 20456, 38888, 42128, 45476, 32792, 49826, 64136, 68252, 43698, 76868, 77930, 90752, 69216, 105788, 111056, 108354, 127628

Offset: 1

Hugo Pfoertner, Jul 24 2020

nonn

			  37--36--35--34--33--32--31
   |                       |
  38  17--16--15--14--13  30  ...
   |   |               |   |   |
  39  18   5---4---3  12  29  54
   |   |   |       |   |   |   |
  40  19   6   1---2 d=2 d=3  53
   |   |   |           |   |   |
  41  20   7---8---9--10  27  52
   |   |                   |   |
  42  21--22--23--24--25--26  51
   |                           |
  43--44--45--46--47--48--49-d=5
.
a(1) = 11 is the index of the first occurrence of distance d = 2 = prime(1) from the start of the spiral.
a(2) = 28 is the index of the first occurrence of distance d = 3 = prime(2) from the start of the spiral.
Distances of the form 4*k+1 corresponding to Pythagorean primes A002144 occur earlier than on the East spoke of the square spiral, dependent on the decomposition of p^2 into two squares. prime(3)^2 = 4^2 + 3^2 leads to index a(3) = 50 in the spiral.

Hugo Pfoertner, Table of n, a(n) for n = 1..1000

Cf. A002144, A002145, A054552, A174344, A268038, A274923, A336336.

a(n) = A054552(prime(n)) if prime(n) != 1 mod 4.

A357281 The numbers of a square spiral with 1 in the center, lying at integer points of the right branch of the parabola y=n^2.

Original entry on oeis.org

1, 9, 79, 355, 1077, 2581, 5299, 9759, 16585, 26497, 40311, 58939, 83389, 114765, 154267, 203191, 262929, 334969, 420895, 522387, 641221, 779269, 938499, 1120975, 1328857, 1564401, 1829959, 2127979, 2461005, 2831677, 3242731, 3696999, 4197409

Offset: 0

Nicolay Avilov, Sep 22 2022

On a square spiral with 1 in the center is a parabola y=n^2. The coordinate system is defined by the On and Oy axes, which intersect at the center of the spiral. The points of the parabola with integer coordinates located on the right branch of the parabola y=n^2 determine the terms of the sequence. In the attached figure, the terms are highlighted in red cells.

			a(0) = 1;
a(4) = 4*4^4 + 3*4^2 + 4 + 1 = 1024 + 48 + 5 = 1077.

Nicolay Avilov, Figure illustrating the terms a(0) - a(3)
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A033951 (x axes), A054552 (y axes).

Cf. A000583, A056108.

LinearRecurrence[{5,-10,10,-5,1},{1,9,79,355,1077},40] (* Harvey P. Dale, Oct 15 2023 *)

a(n) = 4*n^4 + 3*n^2 + n + 1.
a(n) = 4*A000583(n) + A056108(n).
G.f.: (1 + 4*x + 44*x^2 + 40*x^3 + 7*x^4)/(1 - x)^5. - Stefano Spezia, Sep 24 2022

A357744 a(n) is the least k such that prime(n) * k occurs in one of the eight main spokes of a square spiral with 1 in the center.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 25, 1, 17, 1, 59, 1, 13, 37, 1, 4, 3, 13, 5, 1, 21, 8, 2, 4, 1, 131, 3, 1, 2, 1, 1, 1, 2, 37, 4, 13, 58, 7, 1, 34, 1, 7, 23, 4, 1, 29, 1, 251, 1, 5, 25, 3, 13, 1, 7, 30, 1, 311, 31, 38, 3, 49, 3, 6, 5, 37, 19, 16, 7, 5, 149, 3, 1, 7, 419, 1, 1, 91, 10, 2

Offset: 1

Karl-Heinz Hofmann, Dec 01 2022

nonn

Numbers on the spokes of the spiral are A357745.
a(n) = 1 when prime(n) is directly on a main spoke.
a(n) <= prime(n) since odd squares are on the southeast spoke (A016754).

Karl-Heinz Hofmann, Table of n, a(n) for n = 1..10000
Karl-Heinz Hofmann, Graphic example.
Karl-Heinz Hofmann, Graphic together with the first 10000 primes.

Cf. A000040, A054552, A054554, A054556, A053755, A054567.

Cf. A054569, A033951, A016754, A357745.

from sympy import sieve
A357744, A357745, aupto = [], [], 82
for n in range (1, sieve[aupto]**2):
    A357745.append(((n+3)**2 >> 4) + 1 if n % 8 != 1 else (n+3)**2 >> 4)
for p in sieve[1:aupto + 1]:
    k = 1
    while (p*k) not in A357745: k += 1
    A357744.append(k)
print(A357744)

A386486 a(0) = 1; thereafter a(n) = 4n^2 - 3n + 2.

Original entry on oeis.org

1, 3, 12, 29, 54, 87, 128, 177, 234, 299, 372, 453, 542, 639, 744, 857, 978, 1107, 1244, 1389, 1542, 1703, 1872, 2049, 2234, 2427, 2628, 2837, 3054, 3279, 3512, 3753, 4002, 4259, 4524, 4797, 5078, 5367, 5664, 5969, 6282, 6603, 6932, 7269, 7614, 7967, 8328, 8697, 9074, 9459, 9852, 10253, 10662, 11079, 11504, 11937, 12378

Offset: 0

N. J. A. Sloane, Aug 27 2025

Differs from A001107, A054552, and A343560 only by a small constant, but has its own entry because of an important geometric application (which will be added soon).

Paolo Xausa, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A001107, A054552, A343560.

A386486[n_] := If[n == 0, 1, (4*n - 3)*n + 2]; Array[A386486, 100, 0] (* or *)
LinearRecurrence[{3, -3, 1}, {1, 3, 12, 29}, 100] (* Paolo Xausa, Aug 27 2025 *)

From Elmo R. Oliveira, Sep 02 2025: (Start)
G.f.: (1 + 6*x^2 + x^3)/(1 - x)^3.
E.g.f.: exp(x)*(2 + x + 4*x^2) - 1.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3. (End)

cp's OEIS Frontend

A257171 Sum of numbers on n-th segment of Ulam's spiral.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A308385 a(n) is the last square visited by fers moves on a spirally numbered (2n-1) X (2n-1) board, moving to the lowest available unvisited square at each step.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A336335 a(n) is the index of the first occurrence of the Euclidean distance prime(n) from a point on a square spiral to its starting point at 1.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A357281 The numbers of a square spiral with 1 in the center, lying at integer points of the right branch of the parabola y=n^2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A357744 a(n) is the least k such that prime(n) * k occurs in one of the eight main spokes of a square spiral with 1 in the center.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Python

A386486 a(0) = 1; thereafter a(n) = 4*n^2 - 3*n + 2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A386486 a(0) = 1; thereafter a(n) = 4n^2 - 3n + 2.