A054554 -id:A054554 - cp's OEIS frontend

A143861 Ulam's spiral (NNE spoke).

1, 14, 59, 136, 245, 386, 559, 764, 1001, 1270, 1571, 1904, 2269, 2666, 3095, 3556, 4049, 4574, 5131, 5720, 6341, 6994, 7679, 8396, 9145, 9926, 10739, 11584, 12461, 13370, 14311, 15284, 16289, 17326, 18395, 19496, 20629, 21794, 22991, 24220

Offset: 1

Vladimir Joseph Stephan Orlovsky, Sep 03 2008

Stanislaw M. Ulam was doodling during the presentation of a "long and very boring paper" at a scientific meeting in 1963. The spiral is its result. Note that conforming to trigonometric conventions, the spiral begins on the abscissa and rotates counterclockwise. Other spirals, orientations, direction of rotation and initial values exist, even in the OEIS.

Also sequence found by reading the segment (1, 14) together with the line from 14, in the direction 14, 59, ..., in the square spiral whose vertices are the generalized decagonal numbers A074377. - Omar E. Pol, Nov 05 2012

Chris K. Caldwell & G. L. Honaker, Jr., Prime Curios! The Dictionary of Prime Number Trivia, CreateSpace, Sept 2009, pp. 2-3.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Martin Gardner, Mathematical Recreations: The Remarkable Lore of the Prime Number, Scientific American 210 3: 120 - 128.
Hermetic Systems, Prime Number Spiral
OEIS wiki, Ulam spiral
Ivars Peterson's MathTrek, Prime Spirals, Science News, May 3 2002.
Robert Sacks, Number Spiral
Scientific American, Cover page of the March 1964
Eric Weisstein's World of Mathematics, Prime Spiral
Wikipedia, Ulam spiral
Wikipedia, Boxing the compass
Robert G. Wilson v, Ulam's spiral
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A016754, A033638, A033951, A053755, A054552, A054554, A054556, A054567, A054569, A073337, A143838, A143839, A143854, A143855, A143856, A143859, A143860.

List([1..40], n-> ((32*n-35)^2 +55)/64); # G. C. Greubel, Nov 09 2019

[((32*n-35)^2 +55)/64: n in [1..40]]; // G. C. Greubel, Nov 09 2019

seq( ((32*n-35)^2 +55)/64, n=1..40); # G. C. Greubel, Nov 09 2019

(* From Robert G. Wilson v, Oct 29 2011 *)
f[n_]:= 16n^2 -35n +20; Array[f, 40]
LinearRecurrence[{3,-3,1}, {1,14,59}, 40]
FoldList[#1 + #2 &, 1, 32Range@ 10 - 19] (* End *)
((32*Range[40] -35)^2 +55)/64 (* G. C. Greubel, Nov 09 2019 *)

a(n)=16*n^2-35*n+20 \\ Charles R Greathouse IV, Oct 29 2011

[((32*n-35)^2 +55)/64 for n in (1..40)] # G. C. Greubel, Nov 09 2019

a(n) = 16*n^2 - 35*n + 20. - R. J. Mathar, Sep 08 2008
G.f.: x*(1 + 11*x + 20*x^2)/(1-x)^3. - Colin Barker, Aug 03 2012
E.g.f.: -20 + (20 - 19*x + 16*x^2)*exp(x). - G. C. Greubel, Nov 09 2019

A172979 Primes with locations of right angle turns in Ulam square spiral (primes in A033638).

Original entry on oeis.org

2, 3, 5, 7, 13, 17, 31, 37, 43, 73, 101, 157, 197, 211, 241, 257, 307, 401, 421, 463, 577, 601, 677, 757, 1123, 1297, 1483, 1601, 1723, 2551, 2917, 2971, 3137, 3307, 3541, 3907, 4357, 4423, 4831, 5113, 5477, 5701, 6007, 6163, 6481, 7057, 8011, 8101, 8191, 8837, 9901

Offset: 1

Michel Lagneau, Nov 21 2010

nonn

Except for the first term, 2, these are the primes on the main diagonals of the Ulam spiral. - Robert G. Wilson v, Jul 10 2014

Primes p for which floor(sqrt(p)) | (p-1). - Davide Rotondo, Jun 06 2025

			Ulam square spiral = 7 8 9 / 6 1 2 / 5 4 3 /...; changes of direction (right-angle)
  for the primes at 2 3 5 7 ...

Cf. A033638, A054554, A053755, A054569, A016754, A078784.

with(numtheory): a0:=1:for n from 1 to 200 do : a1:=a0+floor(n/2):a0:=a1:if
  type(a1,prime)=true then printf(`%d, `,a1):else fi:od:

Select[ Sort@ Flatten@ Table[ 4n^2 + (2j - 4)n + 1, {j, 0, 3}, {n, 55}], PrimeQ]

for(n=0,10^3, my(t=n^2\4+1); if(isprime(t),print1(t,", "))); \\ Joerg Arndt, Jul 12 2014

A073338 Positive values of n for which 4n^2-10n+7 is prime.

Original entry on oeis.org

2, 3, 4, 9, 10, 12, 15, 18, 22, 30, 31, 36, 37, 39, 40, 46, 51, 52, 54, 57, 60, 61, 67, 72, 73, 75, 78, 79, 82, 85, 88, 96, 103, 106, 109, 117, 124, 141, 145, 148, 156, 166, 177, 180, 186, 192, 193, 199, 204, 219, 225, 228, 229, 246, 249, 264, 267, 268, 270, 277

Offset: 1

Zak Seidov, Aug 25 2002

			n=10 is in the sequence since 4n^2-10n+7=307 is prime.

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A054554, A073337.

Mathematica
```
Select[Range[300], PrimeQ[4#^2-10#+7]&]
```

Edited by Dean Hickerson, Aug 28 2002

A081353 Diagonal of square maze arrangement of natural numbers A081349.

Original entry on oeis.org

3, 5, 13, 19, 31, 41, 57, 71, 91, 109, 133, 155, 183, 209, 241, 271, 307, 341, 381, 419, 463, 505, 553, 599, 651, 701, 757, 811, 871, 929, 993, 1055, 1123, 1189, 1261, 1331, 1407, 1481, 1561, 1639, 1723, 1805, 1893, 1979, 2071, 2161, 2257, 2351, 2451, 2549

Offset: 0

Paul Barry, Mar 19 2003

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A081349, A081350, A081351, A081352.
Cf. A109613, A137932.
Bisections are in A054554, A125202.

[(n + 1)*(n + 2) + (-1)^n: n in [0..50]]; // Vincenzo Librandi, Sep 06 2011

A081353:=n->(n+1)*(n+2)+(-1)^n: seq(A081353(n), n=0..100); # Wesley Ivan Hurt, Aug 09 2015

Table[(n + 1) (n + 2) + (-1)^n, {n, 0, 70}] (* Wesley Ivan Hurt, Aug 09 2015 *)
LinearRecurrence[{2,0,-2,1},{3,5,13,19},50] (* Harvey P. Dale, Aug 02 2021 *)

a(n) = (n+1)*(n+2)+(-1)^n = 2*binomial(n+2,2)+(-1)^n.
G.f.: (3-x)*(1+x^2)/((1-x)^3*(1+x)). [Colin Barker, Sep 03 2012]
From Wesley Ivan Hurt, Aug 09 2015: (Start)
a(n) = 2*a(n-1)-2*a(n-3)+a(n-4), n>4.
a(n) = n^2+3n+3 if n is even, otherwise n^2+3n+1.
a(n) = A137932(n+3) - A109613(n+1). (End)

A254527 Total number of points on a sphere when both poles are on an x by x grid where x=8*n+1.

Original entry on oeis.org

6, 26, 62, 114, 182, 266, 366, 482, 614, 762, 926, 1106, 1302, 1514, 1742, 1986, 2246, 2522, 2814, 3122, 3446, 3786, 4142, 4514, 4902, 5306, 5726, 6162, 6614, 7082, 7566, 8066, 8582, 9114, 9662, 10226, 10806, 11402, 12014, 12642, 13286, 13946, 14622, 15314

Offset: 1

Thomas Olson, Jan 31 2015

Maximum number of regions formed by n circles and n ellipses in the plane. - Ivan N. Ianakiev, Sep 21 2019

Number of points on a sphere whose longitude and latitude are both multiples of (90 degrees)/n, including the poles. - Jianing Song, Aug 28 2022

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A054554, A051890.

Table[8*n^2  - 4*n + 2,{n,1,44}] (* Ivan N. Ianakiev, Sep 21 2019 *)

vector(50, n, 8*n^2 - 4*n + 2) \\ Michel Marcus, Feb 08 2015

Vec(-2*x*(x+1)*(x+3)/(x-1)^3 + O(x^100)) \\ Colin Barker, Aug 09 2015

a(n) = 8*n^2 - 4*n + 2.
From Colin Barker, Aug 09 2015: (Start)
a(n) = 2*A054554(n+1).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>3.
G.f.: -2*x*(x+1)*(x+3) / (x-1)^3.
(End)
E.g.f.: -2 + exp(x)*(2 + 4*x + 8*x^2). - Stefano Spezia, Sep 21 2019
a(n) = A051890(2*n). - Jianing Song, Aug 28 2022

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

Original entry on oeis.org

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

Karl-Heinz Hofmann, Dec 22 2022

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.

			See visualization in 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).

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

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

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 *)

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

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

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

A185669 a(n) = 4n^2 + 3n + 2.

Original entry on oeis.org

2, 9, 24, 47, 78, 117, 164, 219, 282, 353, 432, 519, 614, 717, 828, 947, 1074, 1209, 1352, 1503, 1662, 1829, 2004, 2187, 2378, 2577, 2784, 2999, 3222, 3453, 3692, 3939, 4194, 4457, 4728, 5007, 5294, 5589, 5892, 6203, 6522, 6849, 7184, 7527, 7878, 8237, 8604, 8979, 9362, 9753, 10152, 10559, 10974, 11397, 11828

Offset: 0

Paul Curtz, Feb 09 2011

Natural numbers A000027 written clockwise as a square spiral:
.
  43--44--45--46--47--48--49
   |
  42  21--22--23--24--25--26
   |   |                   |
  41  20   7---8---9--10  27
   |   |   |           |   |
  40  19   6   1---2  11  28
   |   |   |       |   |   |
  39  18   5---4---3  12  29
   |   |               |   |
  38  17--16--15--14--13  30
   |                       |
  37--36--35--34--33--32--31
.
Walking in straight lines away from the center:
1,  2, 11, ... = A054552(n)     = 1 -3*n+4*n^2,
1,  8, 23, ... = A033951(n)     = 1 +3*n+4*n^2,
1,  3, 13, ... = A054554(n+1)   = 1 -2*n-4*n^2,
1,  7, 21, ... = A054559(n+1)   = 1 +2*n+4*n^2,
1,  4, 15, ... = A054556(n+1)   = 1   -n+4*n^2,
1,  6, 19, ... = A054567(n+1)   = 1   +n+4*n^2,
1,  5, 17, ... = A053755(n)     = 1     +4*n^2,
1,  9, 25, ... = A016754(n)     = 1 +4*n+4*n^2 = (1+2*n)^2,
2,  8, 22, ... = 2*A084849(n)   = 2 +2*n+4*n^2,
2, 12, 30, ... = A002939(n+1)   = 2 +6*n+4*n^2,
2,  9, 24, ... = a(n)           = 2 +3*n+4*n^2,
2, 10, 26, ... = A069894(n)     = 2 +4*n+4*n^2,
3, 11, 27, ... = A164897(n)     = 3 +4*n+4*n^2,
3, 12, 29, ... = A054552(n+1)+1 = 3 +5*n+4*n^2,
3, 14, 33, ... = A033991(n+1)   = 3 +7*n+4*n^2,
3, 15, 35, ... = A000466(n+1)   = 3 +8*n+4*n^2,
4, 14, 32, ... = 2*A130883(n+1) = 4 +6*n+4*n^2,
4, 16, 36, ... = A016742(n+1)   = 4 +8*n+4*n^2 = (2+2*n)^2,
5, 18, 39, ... = A007742(n+1)   = 5 +9*n+4*n^2,
5, 19, 41, ... = A125202(n+2)   = 5+10*n+4*n^2.

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

[2+3*n+4*n^2: n in [0..80]];  // Vincenzo Librandi, Feb 09 2011

Table[4n^2 + 3n + 2, {n,0,50}] (* G. C. Greubel, Jul 09 2017 *)
LinearRecurrence[{3,-3,1},{2,9,24},60] (* Harvey P. Dale, Aug 11 2021 *)

a(n)=4*n^2+3*n+2 \\ Charles R Greathouse IV, Apr 14 2014

a(n) = a(n-1) + 8*n - 1.
a(n) = 2*a(n-1) - a(n-2) + 8.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f.: (2 +3*x +3*x^2)/(1-x)^3 . - R. J. Mathar, Feb 11 2011
a(n) = A033954(n) + 2. - Bruno Berselli, Apr 10 2011
E.g.f.: (4*x^2 + 7*x + 2)*exp(x). - G. C. Greubel, Jul 09 2017

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)

A386670 Number of ternary strings of length 2*n that have more 0's than the combined number of 1's and 2's.

Original entry on oeis.org

0, 1, 9, 73, 577, 4521, 35313, 275577, 2150721, 16793929, 131230609, 1026283545, 8032614625, 62921342953, 493262044977, 3869724080313, 30379987189377, 238661880787593, 1876072096450257, 14756076838714713, 116126703647975457, 914363729294862633, 7203083947383222897

Offset: 0

Enrique Navarrete, Jul 28 2025

nonn

			a(1)=1 since the string of length 2 is 00.
a(2)=9 since the strings of length 4 are the 4 permutations of 0001, the 4 permutations of 0002, and 0000.
a(4)=577 since the strings of length 8 are (number of permutations in parentheses): 00000001 (8), 00000002 (8), 00000011 (28), 00000012 (56), 00000022 (28), 00000111 (56), 00000112 (168), 00000122 (168), 00000222 (56), 00000000 (1).

Cf. A054554, A059304, A128417, A128418, A243019.

a(n) = Sum_{k=1..n} 2^(n-k)*binomial(2*n,n-k).
a(n) = Sum_{k=1..n} A128417(n,k).
G.f.: (1-4*x-sqrt(1-8*x))/(sqrt(1-8*x)*(sqrt(1-8*x)+12*x-1)).
a(n) = A128418(n) - A059304(n).

A161372 In Ulam's spiral starting at 101, take the elements not used so far from the two spokes SW, NE, SW, NE, SW, NE ...

Original entry on oeis.org

107, 101, 121, 103, 143, 113, 173, 131, 211, 157, 257, 191, 311, 233, 373, 283, 443, 341, 521, 407, 607, 481, 701, 563, 803, 653, 913, 751, 1031, 857, 1157, 971, 1291, 1093, 1433, 1223, 1583, 1361, 1741, 1507, 1907, 1661, 2081, 1823, 2263, 1993, 2453

Offset: 1

Milton L. Brown (miltbrown(AT)earthlink.net), Jun 08 2009

NE to SW Diagonal of Ulam's Spiral, with 101 at center.

The sequence did not match the original definition. A working definition might be a(2n)=4n^2-10n+107, a(2n-1)=4n^2+2n+101, a(n)=n^2-n+105+2*(-1)^n*(1-2*n), but this seems to be unrelated to Ulam spirals. [R. J. Mathar, Jun 11 2009]

			SW-NE diagonal in:
  137 136 135 134 133 132 131
  138 117 116 115 114 113 130
  139 118 105 104 103 112 129
  140 119 106 101 102 111 128
  141 120 107 108 109 110 127
  142 121 122 123 124 125 126
  143 144 145 146 147 148 149

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

Cf. A054569 (SW spoke), A054554 (NE spoke).

CoefficientList[Series[(107*x^4-6*x^3-194*x^2-6*x+107)/((1+x)^2*(1-x)^3), {x,0,32}], x] (* Georg Fischer, Dec 03 2024 *)

Edited by Georg Fischer, Dec 03 2024

cp's OEIS Frontend

A143861 Ulam's spiral (NNE spoke).

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A172979 Primes with locations of right angle turns in Ulam square spiral (primes in A033638).

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A073338 Positive values of n for which 4n^2-10n+7 is prime.

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A081353 Diagonal of square maze arrangement of natural numbers A081349.

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A254527 Total number of points on a sphere when both poles are on an x by x grid where x=8*n+1.

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

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A185669 a(n) = 4*n^2 + 3*n + 2.

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A185669 a(n) = 4n^2 + 3n + 2.