A054552 -id:A054552 - cp's OEIS frontend

A308215 a(n) is the multiplicative inverse of A001844(n+1) modulo A001844(n); where A001844 is the sequence of centered square numbers.

0, 2, 12, 11, 39, 28, 82, 53, 141, 86, 216, 127, 307, 176, 414, 233, 537, 298, 676, 371, 831, 452, 1002, 541, 1189, 638, 1392, 743, 1611, 856, 1846, 977, 2097, 1106, 2364, 1243, 2647, 1388, 2946, 1541, 3261, 1702, 3592, 1871, 3939, 2048

Offset: 0

Daniel Hoyt, May 15 2019

nonn

The sequence explores the relationship between the terms of A001844, the sums of consecutive squares. The sequence is an interleaving of A054552 (a number spiral arm) and (A001844-n). The gap between the lower values of A308215 and the upper values of A308217 increase by 3n; each successive gap increasing by 6.

Daniel Hoyt, Graph of A308215 and A308217 in relation to A001844

Cf. A001844, A033951, A054552, A308217.

f(n) = 2*n*(n+1)+1; \\ A001844
a(n) = lift(1/Mod(f(n+1), f(n))); \\ Michel Marcus, May 16 2019

import gmpy2
sos = [] # sum of squares
a=0
b=1
for i in range(50):
    c = a**2 + b**2
    sos.append(c)
    a +=1
    b +=1
ls = []
for i in range(len(sos)-1):
    c = gmpy2.invert(sos[i+1],sos[i])
    ls.append(int(c))
print(ls)

a(n) satisfies a(n)*(2*n*(n+1)+1) == 1 (mod 2*n*(n-1)+1).
Conjectures from Colin Barker, May 16 2019: (Start)
G.f.: x*(2 + 12*x + 5*x^2 + 3*x^3 + x^4 + x^5) / ((1 - x)^3*(1 + x)^3).
a(n) = (3 + (-1)^n + 2*(2+(-1)^n)*n + 2*(3+(-1)^n)*n^2) / 4 for n>0.
a(n) = 3*a(n-2) - 3*a(n-4) + a(n-6) for n>6.
(End)

A308217 a(n) is the multiplicative inverse of A001844(n) modulo A001844(n+1); where A001844 is the sequence of centered square numbers.

Original entry on oeis.org

1, 8, 2, 23, 3, 46, 4, 77, 5, 116, 6, 163, 7, 218, 8, 281, 9, 352, 10, 431, 11, 518, 12, 613, 13, 716, 14, 827, 15, 946, 16, 1073, 17, 1208, 18, 1351, 19, 1502, 20, 1661, 21, 1828, 22, 2003, 23, 2186, 24, 2377, 25, 2576, 26, 2783, 27, 2998, 28, 3221, 29, 3452

Offset: 0

Daniel Hoyt, May 15 2019

The sequence explores the relationship between the terms of A001844, the sums of consecutive squares. The sequence is an interleaving of A033951 (a number spiral arm) and the natural numbers. The gap between the lower values of A308215 and the upper values of A308217 increase by 3n; each successive gap increasing by 6.

Robert Israel, Table of n, a(n) for n = 0..10000
Daniel Hoyt, Graph of A308215 and A308217 in relation to A001844
Index entries for linear recurrences with constant coefficients, signature (0,3,0,-3,0,1).

Cf. A001844, A054552, A033951, A308215.

A001844:= n -> 2*n*(n+1)+1:
seq(1/A001844(n) mod A001844(n+1),n=0..100); # Robert Israel, Aug 11 2019

LinearRecurrence[{0, 3, 0, -3, 0, 1}, {1, 8, 2, 23, 3, 46}, 30] (* Georg Fischer, Dec 06 2024 *)

f(n) = 2*n*(n+1)+1; \\ A001844
a(n) = lift(1/Mod(f(n), f(n+1))); \\ Michel Marcus, May 16 2019

a(n) satisfies a(n)*(2*n*(n-1)+1) == 1 (mod 2*n*(n+1)+1).
Conjectures from Colin Barker, May 16 2019: (Start)
G.f.: (1 + 8*x - x^2 - x^3 + x^5) / ((1 - x)^3*(1 + x)^3).
a(n) = 3*a(n-2) - 3*a(n-4) + a(n-6) for n>5.
a(n) = (9 - 5*(-1)^n + (8-6*(-1)^n)*n - 2*(-1+(-1)^n)*n^2) / 4. (End)
From Robert Israel, Aug 11 2019: (Start)
a(n) = 1 + n/2 if n is even, since 0 < 1+n/2 < A001844(n+1) and (1+n/2)*A001844(n)-1 = (n/2)*A001844(n+1).
a(n) = n^2 + 7/2*(n+1) if n is odd, since 0 < n^2+7/2*(n+1) < A001844(n+1) and (n^2+7/2*(n+1))*A001844(n)-1 = (n^2+3*k/2+1/2)*A001844(n+1).
Colin Barker's conjectures easily follow. (End)
E.g.f.: ((2 + 9*x)*cosh(x) + (7 + x + 2*x^2)*sinh(x))/2. - Stefano Spezia, Dec 06 2024

A143861 Ulam's spiral (NNE spoke).

Original entry on oeis.org

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

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

A078784 Primes on axis of Ulam square spiral (with rows ... / 7 8 9 / 6 1 2 / 5 4 3 / ... ) with origin at (1).

Original entry on oeis.org

2, 11, 19, 23, 53, 61, 127, 139, 151, 163, 233, 281, 431, 541, 613, 743, 827, 977, 1009, 1279, 1621, 1871, 2003, 2281, 2377, 2731, 3109, 3221, 3511, 3571, 3631, 3691, 4001, 4129, 4523, 4591, 5077, 6361, 6521, 7789, 7877, 8419, 9851, 10151, 10973, 11503, 11719, 11827, 12377, 12601, 12713, 13399

Offset: 1

Donald S. McDonald, Jan 10 2003

nonn

Quadrants are numbered clockwise: 4=north, 1=east, 2=south, 3=west. The spiral numbers falling on axes (whether prime or not) are 4=north (2n+1)^2-n, 1=east (2n+1)^2+n+1, 2=south (2n)^2-(n-1), 3=west (2n)^2+n+1.
Primes to the left, right, above or below the 1 in the example in A054552.
This is the union of the primes in A168022, A168023, A168025 and A168027. - R. J. Mathar, Jul 11 2014

			For n=0, quadrant = 1, a(1) =  2, distance = 1;
for n=1, quadrant = 1, a(2) = 11, distance = 2;
for n=2, quadrant = 3, a(3) = 19, distance = 2.

Cf. A039823, A054552, A054556, A054567, A033951, A172979.

Select[ Sort@ Flatten@ Table[ 4n^2 + (2j - 3)n + 1, {j, 0, 3}, {n, 58}], PrimeQ] (* Robert G. Wilson v, Jul 10 2014 *)

Primes in A039823(n) = ceiling((n^2 + n + 2)/4). - Georg Fischer, Dec 04 2024

a(12) onward from Robert G. Wilson v, Jul 10 2014

A108781 Expansion of sqrt((1-x+8*x^2)/(1-x)^3).

Original entry on oeis.org

1, 1, 5, 9, 5, -7, 5, 73, 69, -295, -571, 1321, 4613, -4167, -32635, -1783, 211141, 200601, -1229243, -2468375, 6117509, 22557305, -21519611, -176980023, -13664955, 1234115673, 1250908869, -7608051031, -16094268667, 39424807225, 153179607429, -139981878647, -1243776268859

Offset: 0

N. J. A. Sloane and Nadia Heninger, Jun 28 2005

sign

N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, arXiv:math/0509316 [math.NT], 2005-2006.
N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.

Square root of g.f. for A054552.

m:=40; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!(Sqrt((1-x+8*x^2)/(1-x)^3))); // Vincenzo Librandi, Jan 25 2020

CoefficientList[Series[Sqrt[(1-x+8x^2)/(1-x)^3],{x,0,50}],x] (* Harvey P. Dale, Dec 24 2018 *)

D-finite with recurrence: n*a(n) +(-2*n+1)*a(n-1) +(9*n-25)*a(n-2) +4*(-2*n+5)*a(n-3)=0. - R. J. Mathar, Jan 24 2020

A185413 A000027 written clockwise as spiral and read counterclockwise.

Original entry on oeis.org

2, 1, 4, 3, 12, 11, 10, 9, 8, 7, 6, 5, 16, 15, 14, 13, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 36, 35, 34, 33, 32, 31, 56, 55, 54, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 64, 63, 62, 61, 60, 59, 58, 57

Offset: 1

Paul Curtz, Feb 08 2011

A permutation of the natural numbers. Variant of A090861.

			  21--22--23--24--25--26
   |                   |
  20   7---8---9--10  27
   |   |           |   |
  19   6   1---2  11  28
   |   |       |   |   |
  18   5---4---3  12  29
   |               |   |
  17--16--15--14--13  30
                       |
  36--35--34--33--32--31

Alois P. Heinz, Table of n, a(n) for n = 1..1024
Index to sequences related to permutations of the integers

Cf. A185586, A054552.

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

A241807 Numerators of c(n) = (n^2+n+2)/((n+1)(n+2)(n+3)) as defined in A241269.

Original entry on oeis.org

1, 1, 2, 7, 11, 2, 11, 29, 37, 23, 28, 67, 79, 23, 53, 121, 137, 77, 86, 191, 211, 29, 127, 277, 301, 163, 176, 379, 407, 109, 233, 497, 529, 281, 298, 631, 667, 88, 371, 781, 821, 431, 452, 947, 991, 259, 541, 1129, 1177, 613, 638

Offset: 0

Jean-François Alcover and Paul Curtz, Apr 29 2014

The subsequence 1, 23, 77, 163, 281, 431, 613, 827, ..., with indices congruent to 1 mod 8, is 16n^2+6n+1, that is, A000124(8n+1)/2 or A014206(8n+1)/4. Its second differences are constant: (16n^2+6n+1)'' = 32.

The sequence A014206/A241807 is integral and consists of the 16-periodic sequence (2, 4, 4, 2, 2, 16, 4, 2, 2, 4, 4, 2, 2, 8, 4, 2, ...).

			1/3, 1/6, 2/15, 7/60, 11/105, 2/21, 11/126, 29/360, 37/495, 23/330, ...

Cf. A000124, A014206, A241269, A188135, A054552, A185438.

Table[(n^2+n+2)/((n+1)*(n+2)*(n+3)) // Numerator, {n, 0, 50}]

a(n) = A014206(n)/period 16: repeat 2, 4, 4, 2, 2, 16, 4, 2, 2, 4, 4, 2, 2, 8, 4, 2 (conjectured).
a(4k)     = 8*k^2 +2*k +1,
a(4k+2)   = 4*k^2 +5*k +2,
a(4k+3)   = 8*k^2 +14*k +7,
a(8k+1)   = 16*k^2 +6*k +1,
a(16k+5)  = 16*k^2 +11*k +2,
a(16k+13) = 32*k^2 + 54*k +23.

A248825 a(n) = n^2 + 1 - (-1)^n.

Original entry on oeis.org

0, 3, 4, 11, 16, 27, 36, 51, 64, 83, 100, 123, 144, 171, 196, 227, 256, 291, 324, 363, 400, 443, 484, 531, 576, 627, 676, 731, 784, 843, 900, 963, 1024, 1091, 1156, 1227, 1296, 1371, 1444, 1523, 1600, 1683, 1764, 1851, 1936, 2027, 2116

Offset: 0

Paul Curtz, Oct 15 2014

Also, A016742 and A164897 interleaved.
See the spiral in Example field of A054552: after 0, the sequence is given by the terms of the semidiagonals 4, 16, 36, 64, 100, ... and 3, 11, 27, 51, 83, ... sorted into ascending order.
Primes of the sequence are in A056899.

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

Cf. A000290, A008586, A008602, A010673, A016742, A042963, A056899, A164897, A166519, A168277, A248800.

[n^2+1-(-1)^n: n in [0..60]]; // Vincenzo Librandi, Oct 16 2014

Table[n^2 + 1 - (-1)^n, {n, 0, 60}] (* Vincenzo Librandi, Oct 16 2014 *)
LinearRecurrence[{2,0,-2,1},{0,3,4,11},60] (* Harvey P. Dale, Jun 30 2019 *)

vector(100,n,(n-1)^2+1+(-1)^n) \\ Derek Orr, Oct 15 2014

[n^2+1-(-1)^n for n in (0..60)] # Bruno Berselli, Oct 16 2014

a(n) = a(-n) = 2*a(n-1) - 2*(n-3) + a(n-4).
a(n) = n^2 + A010673(n) = (n+1)^2 - A168277(n+1).
a(n+1) = A248800(n) + A042963(n+1) = a(n) + A166519(n).
a(n+2) = a(n) + 4*n.
a(n+5) = a(n-5) + A008602(n).
G.f.: x*(3 - 2*x + 3*x^2)/((1 + x)*(1 - x)^3). - Bruno Berselli, Oct 15 2014
Sum_{n>=1} 1/a(n) = Pi^2/24 + tanh(Pi/sqrt(2))*Pi/(4*sqrt(2)). - Amiram Eldar, Aug 21 2022

Edited by Bruno Berselli, Oct 16 2014

cp's OEIS Frontend

A308215 a(n) is the multiplicative inverse of A001844(n+1) modulo A001844(n); where A001844 is the sequence of centered square numbers.

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A308217 a(n) is the multiplicative inverse of A001844(n) modulo A001844(n+1); where A001844 is the sequence of centered square numbers.

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A143861 Ulam's spiral (NNE spoke).

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

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A078784 Primes on axis of Ulam square spiral (with rows ... / 7 8 9 / 6 1 2 / 5 4 3 / ... ) with origin at (1).

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A108781 Expansion of sqrt((1-x+8*x^2)/(1-x)^3).

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A185413 A000027 written clockwise as spiral and read counterclockwise.

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

A241807 Numerators of c(n) = (n^2+n+2)/((n+1)(n+2)(n+3)) as defined in A241269.