A002943 -id:A002943 - cp's OEIS frontend

A033954 Second 10-gonal (or decagonal) numbers: n(4n+3).

0, 7, 22, 45, 76, 115, 162, 217, 280, 351, 430, 517, 612, 715, 826, 945, 1072, 1207, 1350, 1501, 1660, 1827, 2002, 2185, 2376, 2575, 2782, 2997, 3220, 3451, 3690, 3937, 4192, 4455, 4726, 5005, 5292, 5587, 5890, 6201, 6520, 6847, 7182, 7525, 7876, 8235

Offset: 0

N. J. A. Sloane

Same as A033951 except start at 0. See example section.

Bisection of A074377. Also sequence found by reading the line from 0, in the direction 0, 22, ... and the line from 7, in the direction 7, 45, ..., in the square spiral whose vertices are the generalized 10-gonal numbers A074377. - Omar E. Pol, Jul 24 2012

			  36--37--38--39--40--41--42
   |                       |
  35  16--17--18--19--20  43
   |   |               |   |
  34  15   4---5---6  21  44
   |   |   |       |   |   |
  33  14   3   0===7==22==45==76=>
   |   |   |   |   |   |
  32  13   2---1   8  23
   |   |           |   |
  31  12--11--10---9  24
   |                   |
  30--29--28--27--26--25

S. M. Ellerstein, The square spiral, J. Recreational Mathematics 29 (#3, 1998) 188; 30 (#4, 1999-2000), 246-250.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 2nd ed., 1994, p. 99.

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Emilio Apricena, A version of the Ulam spiral.
Leo Tavares, Illustration: V numbers
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A002620, A033951.
Sequences from spirals: A001107, A002939, A007742, A033951, A033952, A033953, A033954, A033989, A033990, A033991, A002943, A033996, A033988.
Sequences on the four axes of the square spiral: Starting at 0: A001107, A033991, A007742, A033954; starting at 1: A054552, A054556, A054567, A033951.
Sequences on the four diagonals of the square spiral: Starting at 0: A002939 = 2*A000384, A016742 = 4*A000290, A002943 = 2*A014105, A033996 = 8*A000217; starting at 1: A054554, A053755, A054569, A016754.
Sequences obtained by reading alternate terms on the X and Y axes and the two main diagonals of the square spiral: Starting at 0: A035608, A156859, A002378 = 2*A000217, A137932 = 4*A002620; starting at 1: A317186, A267682, A002061, A080335.
Second n-gonal numbers: A005449, A014105, A147875, A045944, A179986, this sequence, A062728, A135705.
Cf. A060544.

List([0..50], n-> n*(4*n+3)) # G. C. Greubel, May 24 2019

[n*(4*n+3): n in [0..50]]; // G. C. Greubel, May 24 2019

Table[n(4n+3),{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{0,7,22},50] (* Harvey P. Dale, May 06 2018 *)

PARI
```
a(n)=4*n^2+3*n
```

[n*(4*n+3) for n in (0..50)] # G. C. Greubel, May 24 2019

a(n) = A001107(-n) = A074377(2*n).
G.f.: x*(7+x)/(1-x)^3. - Michael Somos, Mar 03 2003
a(n) = a(n-1) + 8*n - 1 with a(0)=0. - Vincenzo Librandi, Jul 20 2010
For n>0, Sum_{j=0..n} (a(n) + j)^4 + (4*A000217(n))^3 = Sum_{j=n+1..2n} (a(n) + j)^4; see also A045944. - Charlie Marion, Dec 08 2007, edited by Michel Marcus, Mar 14 2014
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) with a(0) = 0, a(1) = 7, a(2) = 22. - Philippe Deléham, Mar 26 2013
a(n) = A118729(8n+6). - Philippe Deléham, Mar 26 2013
a(n) = A002943(n) + n = A007742(n) + 2n = A016742(n) + 3n = A033991(n) + 4n = A002939(n) + 5n = A001107(n) + 6n = A033996(n) - n. - Philippe Deléham, Mar 26 2013
Sum_{n>=1} 1/a(n) = 4/9 + Pi/6 - log(2) = 0.2748960394827980081... . - Vaclav Kotesovec, Apr 27 2016
E.g.f.: exp(x)*x*(7 + 4*x). - Stefano Spezia, Jun 08 2021
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/(3*sqrt(2)) + log(2)/3 - 4/9 - sqrt(2)*arcsinh(1)/3. - Amiram Eldar, Nov 28 2021
For n>0, (a(n)^2 + n)/(a(n) + n) = (4*n + 1)^2/4, a ratio of two squares. - Rick L. Shepherd, Feb 23 2022
a(n) = A060544(n+1) - A000217(n+1). - Leo Tavares, Mar 31 2022

A054569 a(n) = 4n^2 - 6n + 3.

Original entry on oeis.org

1, 7, 21, 43, 73, 111, 157, 211, 273, 343, 421, 507, 601, 703, 813, 931, 1057, 1191, 1333, 1483, 1641, 1807, 1981, 2163, 2353, 2551, 2757, 2971, 3193, 3423, 3661, 3907, 4161, 4423, 4693, 4971, 5257, 5551, 5853, 6163, 6481, 6807, 7141, 7483, 7833, 8191

Offset: 1

Enoch Haga, G. L. Honaker, Jr., Apr 10 2000

Move in 1-7 direction in a spiral organized like A068225 etc.
Third row of A082039. - Paul Barry, Apr 02 2003
Inverse binomial transform of A036826. - Paul Barry, Jun 11 2003
Equals the "middle sequence" T(2*n,n) of the Connell sequence A001614 as a triangle. - Johannes W. Meijer, May 20 2011
Ulam's spiral (SW spoke). - Robert G. Wilson v, Oct 31 2011

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Robert G. Wilson v, Cover of the March 1964 issue of Scientific American
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000384, A033951, A054552, A054554, A054556, A054567, A054568, A068225.
Sequences on the four axes of the square spiral: Starting at 0: A001107, A033991, A007742, A033954; starting at 1: A054552, A054556, A054567, A033951.
Sequences on the four diagonals of the square spiral: Starting at 0: A002939 = 2*A000384, A016742 = 4*A000290, A002943 = 2*A014105, A033996 = 8*A000217; starting at 1: A054554, A053755, A054569, A016754.
Sequences obtained by reading alternate terms on the X and Y axes and the two main diagonals of the square spiral: Starting at 0: A035608, A156859, A002378 = 2*A000217, A137932 = 4*A002620; starting at 1: A317186, A267682, A002061, A080335.

List([1..50], n-> 4*n^2-6*n+3); # G. C. Greubel, Jul 04 2019

[4*n^2-6*n+3: n in [1..50]]; // G. C. Greubel, Jul 04 2019

f[n_]:= 4*n^2-6*n+3; Array[f, 50] (* Vladimir Joseph Stephan Orlovsky, Sep 02 2008 *)
LinearRecurrence[{3,-3,1},{1,7,21},50] (* Harvey P. Dale, Nov 17 2012 *)

a(n)=4*n^2-6*n+3 \\ Charles R Greathouse IV, Sep 24 2015

[4*n^2-6*n+3 for n in (1..50)] # G. C. Greubel, Jul 04 2019

a(n+1) = 4*n^2 + 2*n + 1. - Paul Barry, Apr 02 2003
a(n) = 4*n^2 - 6*n+3 - 3*0^n (with leading zero). - Paul Barry, Jun 11 2003
Binomial transform of [1, 6, 8, 0, 0, 0, ...]. - Gary W. Adamson, Dec 28 2007
a(n) = 8*n + a(n-1) - 10 (with a(1)=1). - Vincenzo Librandi, Aug 07 2010
From Colin Barker, Mar 23 2012: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f.: x*(1+x)*(1+3*x)/(1-x)^3. (End)
a(n) = A000384(n) + A000384(n-1). - Bruce J. Nicholson, May 07 2017
E.g.f.: -3 + (3 - 2*x + 4*x^2)*exp(x). - G. C. Greubel, Jul 04 2019
Sum_{n>=1} 1/a(n) = A339237. - R. J. Mathar, Jan 22 2021

Edited by Frank Ellermann, Feb 24 2002

A035608 Expansion of g.f. x(1 + 3x)/((1 + x)*(1 - x)^3).

Original entry on oeis.org

0, 1, 5, 10, 18, 27, 39, 52, 68, 85, 105, 126, 150, 175, 203, 232, 264, 297, 333, 370, 410, 451, 495, 540, 588, 637, 689, 742, 798, 855, 915, 976, 1040, 1105, 1173, 1242, 1314, 1387, 1463, 1540, 1620, 1701, 1785, 1870, 1958, 2047, 2139, 2232, 2328, 2425, 2525, 2626

Offset: 0

N. J. A. Sloane

Maximum value of Voronoi's principal quadratic form of the first type when variables restricted to {-1,0,1}. - Michael Somos, Mar 10 2004
This is the main row of a version of the "square spiral" when read alternatively from left to right (see link). See also A001107, A007742, A033954, A033991. It is easy to see that the only prime in the sequence is 5. - Emilio Apricena (emilioapricena(AT)yahoo.it), Feb 08 2009
From Mitch Phillipson, Manda Riehl, Tristan Williams, Mar 06 2009: (Start)
a(n) gives the number of elements of S_2 \wr C_k that avoid the pattern 12, using the following ordering:
In S_j, a permutation p avoids a pattern q if it has no subsequence that is order-isomorphic to q. For example, p avoids the pattern 132 if it has no subsequence abc with a < c < b. We extend this notion to S_j \wr C_n as follows. Element psi =[ alpha_1^beta_1, ... alpha_j^beta_j ] avoids tau = [ a_1 ... a_m ] (tau in S_m) if psi' = [ alpha_1*beta_1 ... alpha_j*beta_j ] avoids tau in the usual sense. For n=2, there are 5 elements of S_2 \wr C_2 that avoid the pattern 12. They are: [ 2^1,1^1 ], [ 2^2,1^1 ], [ 2^2,1^2 ], [ 2^1,1^2 ], [ 1^2,2^1 ].
For example, if psi = [2^1,1^2], then psi'=[2,2] which avoids tau=[1,2] because no subsequence ab of psi' has a < b. (End)

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 115.

William A. Tedeschi, Table of n, a(n) for n = 0..10000
Emilio Apricena, A version of the Ulam spiral (This is called the square spiral. - _N. J. A. Sloane_, Jul 27 2018)
Emanuele Munarini, Topological indices for the antiregular graphs, Le Mathematiche, Vol. 76, No. 1 (2021), see p. 283.
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Partial sums of A042948.
Cf. A002265, A004526, A006578, A011848, A133983, A139592, A173562, A253146.
Sequences on the four axes of the square spiral: Starting at 0: A001107, A033991, A007742, A033954; starting at 1: A054552, A054556, A054567, A033951.
Sequences on the four diagonals of the square spiral: Starting at 0: A002939 = 2*A000384, A016742 = 4*A000290, A002943 = 2*A014105, A033996 = 8*A000217; starting at 1: A054554, A053755, A054569, A016754.
Sequences obtained by reading alternate terms on the X and Y axes and the two main diagonals of the square spiral: Starting at 0: A035608, A156859, A002378 = 2*A000217, A137932 = 4*A002620; starting at 1: A317186, A267682, A002061, A080335.

[n^2 + n - 1 - Floor((n-1)/2): n in [0..25]]; // G. C. Greubel, Oct 29 2017

A035608:=n->floor((n + 1/4)^2): seq(A035608(n), n=0..100); # Wesley Ivan Hurt, Oct 29 2017

Table[n^2 + Floor[n/2], {n, 0, 100}] (* Vladimir Joseph Stephan Orlovsky, Apr 12 2011 *)
CoefficientList[Series[x (1 + 3 x)/((1 + x) (1 - x)^3), {x, 0, 60}], x] (* or *) LinearRecurrence[{2, 0, -2, 1}, {0, 1, 5, 10}, 60] (* Harvey P. Dale, Feb 21 2013 *)

PARI
```
a(n)=n^2+n-1-(n-1)\2
```

a(n) = n^2 + n - 1 - floor((n-1)/2).
a(n) = A011848(2*n+1).
a(n) = A002378(n) - A004526(n+1). - Reinhard Zumkeller, Jan 27 2010
a(n) = 2*A006578(n) - A002378(n)/2 = A139592(n)/2. - Reinhard Zumkeller, Feb 07 2010
a(n) = A002265(n+2) + A173562(n). - Reinhard Zumkeller, Feb 21 2010
a(n) = floor((n + 1/4)^2). - Reinhard Zumkeller, Jan 27 2010
a(n) = (-1)^n*Sum_{i=0..n} (-1)^i*(2*i^2 + 3*i + 1). Omits the leading 0. - William A. Tedeschi, Aug 25 2010
a(n) = n^2 + floor(n/2), from Mathematica section. - Vladimir Joseph Stephan Orlovsky, Apr 12 2011
a(0)=0, a(1)=1, a(2)=5, a(3)=10; for n > 3, a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4). - Harvey P. Dale, Feb 21 2013
For n > 1: a(n) = a(n-2) + 4*n - 3; see also row sums of triangle A253146. - Reinhard Zumkeller, Dec 27 2014
a(n) = 3*A002620(n) + A002620(n+1). - R. J. Mathar, Jul 18 2015
From Amiram Eldar, Mar 20 2022: (Start)
Sum_{n>=1} 1/a(n) = 4 - 2*log(2) - Pi/3.
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*Pi/3 - 4*(1-log(2)). (End)
E.g.f.: (x*(2*x + 3)*cosh(x) + (2*x^2 + 3*x - 1)*sinh(x))/2. - Stefano Spezia, Apr 24 2024

A054554 a(n) = 4n^2 - 10n + 7.

Original entry on oeis.org

1, 3, 13, 31, 57, 91, 133, 183, 241, 307, 381, 463, 553, 651, 757, 871, 993, 1123, 1261, 1407, 1561, 1723, 1893, 2071, 2257, 2451, 2653, 2863, 3081, 3307, 3541, 3783, 4033, 4291, 4557, 4831, 5113, 5403, 5701, 6007, 6321, 6643, 6973, 7311, 7657, 8011, 8373, 8743

Offset: 1

Enoch Haga, G. L. Honaker, Jr., Apr 10 2000

Move in 1-3 direction in a spiral organized like A068225 etc.
Equals binomial transform of [1, 2, 8, 0, 0, 0, ...]. - Gary W. Adamson, May 03 2008
Ulam's spiral (NE spoke). - Robert G. Wilson v, Oct 31 2011
Number of ternary strings of length 2*(n-1) that have one or no 0's, one or no 1's, and an even number of 2's. For n=2, the 3 strings of length 2 are 01, 10 and 22. For n=3, the 13 strings of length 4 are the 12 permutations of 0122 and 2222. - Enrique Navarrete, Jul 25 2025

Ivan Panchenko, Table of n, a(n) for n = 1..1000
James Grime and Brady Haran, Prime Spirals, Numberphile video (2013).
Scientific American, Cover of the March 1964 issue
Leo Tavares, Illustration: Hexagonal Dual Rays
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A054553, A068225, A054552, A054556, A054567, A054569, A033951.
Cf. A014105.
Sequences on the four axes of the square spiral: Starting at 0: A001107, A033991, A007742, A033954; starting at 1: A054552, A054556, A054567, A033951.
Sequences on the four diagonals of the square spiral: Starting at 0: A002939 = 2*A000384, A016742 = 4*A000290, A002943 = 2*A014105, A033996 = 8*A000217; starting at 1: A054554, A053755, A054569, A016754.
Sequences obtained by reading alternate terms on the X and Y axes and the two main diagonals of the square spiral: Starting at 0: A035608, A156859, A002378 = 2*A000217, A137932 = 4*A002620; starting at 1: A317186, A267682, A002061, A080335.
Cf. A003215, A227776.

A054554:=n->4*n^2 -10*n + 7; seq(A054554(k),k=1..100); # Wesley Ivan Hurt, Nov 05 2013

f[n_] := 4n^2 -10n + 7; Array[f, 40] (* Vladimir Joseph Stephan Orlovsky, Sep 01 2008 *)
LinearRecurrence[{3,-3,1},{1,3,13},60] (* Harvey P. Dale, Jan 20 2025 *)

a(n)=4*n^2-10*n+7 \\ Charles R Greathouse IV, Nov 05 2013

a(n) = 8*n + a(n-1) - 14 with n > 1, a(1)=1. - Vincenzo Librandi, Aug 07 2010
G.f.: -x*(7*x^2+1)/(x-1)^3. - Colin Barker, Sep 21 2012
For n > 2, a(n) = A014105(n) + A014105(n-1). - Bruce J. Nicholson, May 07 2017
From Leo Tavares, Feb 21 2022: (Start)
a(n) = A003215(n-2) + 2*A000217(n-1). See Hexagonal Dual Rays illustration in links.
a(n) = A227776(n-1) - 4*A000217(n-1). (End)
a(k+1) = 4k^2 - 2k + 1 in the Numberphile video. - Frank Ellermann, Mar 11 2020
E.g.f.: exp(x)*(7 - 6*x + 4*x^2) - 7. - Stefano Spezia, Apr 24 2024

Edited by Frank Ellermann, Feb 24 2002

A054000 a(n) = 2*n^2 - 2.

Original entry on oeis.org

0, 6, 16, 30, 48, 70, 96, 126, 160, 198, 240, 286, 336, 390, 448, 510, 576, 646, 720, 798, 880, 966, 1056, 1150, 1248, 1350, 1456, 1566, 1680, 1798, 1920, 2046, 2176, 2310, 2448, 2590, 2736, 2886, 3040, 3198, 3360, 3526, 3696, 3870, 4048, 4230, 4416

Offset: 1

Asher Auel, Jan 12 2000

a(n) is the number of edges in (n+1) X (n+1) square grid with all horizontal, vertical and great diagonal segments filled in.
Nonnegative X values of integer solutions to the equation 2*X^3 + 4*X^2 = Y^2. To find Y values: b(n) = 2*n*(2*n^2 - 2). - Mohamed Bouhamida, Nov 06 2007
Second term of an arithmetic progression of 5 numbers with common difference 2n+1. The sum of squares of such 5 terms equals the sum of squares of 5 consecutive numbers starting a(n) + 2n + 1. - Carmine Suriano, Oct 16 2013
For m > 2, a(m-1) = 2*m*(m-2) is the number of Hamiltonian circuits on an m-gonal bipyramid with labeled vertices. - Stanislav Sykora, Jul 22 2014
a(n+1), n >= 0, appears also as the third member of the quartet [p0(n), p1(n), a(n+1), p3(n)] of the square of [n, n+1, n+2, n+3] in the Clifford algebra Cl_2 for n >= 0. p0(n) = -A147973(n+3), p1(n) = A046092(n) and p3(n) = A139570(n). See a comment on A147973, also with a reference. - Wolfdieter Lang, Oct 15 2014
From Bui Quang Tuan, Mar 31 2015: (Start)
For n >= 2, a(n) is the total sum of all numbers on the perimeter of a square consisting of n columns, each of which contains n numbers 1, 2, 3, ..., n.
Here is an example with n = 5:
  1 1 1 1 1
  2 2 2 2 2
  3 3 3 3 3
  4 4 4 4 4
  5 5 5 5 5
where 1+1+1+1+1 + 2+2 + 3+3 + 4+4 + 5+5+5+5+5 = 48 = a(5).
(End)
Nonnegative k such that k/2+1 is a square. - Bruno Berselli, Apr 10 2018

			For n=5, a(5)=48 and 37^2 + 48^2 + 59^2 + 70^2 + 81^2 = 59^2 + 60^2 + 61^2 + 62^2 + 63^2. - _Carmine Suriano_, Oct 16 2013

G. C. Greubel, Table of n, a(n) for n = 1..5000
Milan Janjić, Pascal Matrices and Restricted Words, J. Int. Seq., Vol. 21 (2018), Article 18.5.2.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

a(n) = A100345(n+1, n-4) for n>2.

Cf. A000217, A001082, A002378, A002943, A005563, A028347, A036666, A046092, A056220, A062717, A067725, A087475.

Maple
```
[ seq(2*n^2 - 2, n=1..60) ];
```

2 Range[50]^2 - 2 (* or *) LinearRecurrence[{3, -3, 1}, {0, 6, 16}, 50] (* Harvey P. Dale, Feb 03 2012 *)
CoefficientList[Series[2 x (3 - x) / (1 - x)^3, {x, 0, 50}], x] (* Vincenzo Librandi, Apr 01 2015 *)

a(n)=2*n^2-2 \\ Charles R Greathouse IV, Sep 24 2015

a(n) = 4*n + a(n-1) - 2, with n>1, a(1)=0. - Vincenzo Librandi, Aug 06 2010
a(1)=0, a(2)=6, a(3)=16; for n>3, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Feb 03 2012
a(n) = (n+i)^2 + (n-i)^2, where i=sqrt(-1). - Bruno Berselli, Jan 23 2014
a(n) = 1*A000290(n-1) + 2*A000217(n-1) + 3*A001477(n-1). - J. M. Bergot, Apr 23 2014
G.f.: 2*x^2*(3 - x)/(1 - x)^3. - Vincenzo Librandi, Apr 01 2015
E.g.f.: 2*(x^2 + x -1)*exp(x) + 2. - G. C. Greubel, Jul 13 2017
a(n) + a(n+2) = A005843(n+1)^2. - Ezhilarasu Velayutham, May 30 2019
From Amiram Eldar, Dec 09 2021: (Start)
Sum_{n>=2} 1/a(n) = 3/8.
Sum_{n>=2} (-1)^n/a(n) = 1/8. (End)

A054556 a(n) = 4n^2 - 9n + 6.

Original entry on oeis.org

1, 4, 15, 34, 61, 96, 139, 190, 249, 316, 391, 474, 565, 664, 771, 886, 1009, 1140, 1279, 1426, 1581, 1744, 1915, 2094, 2281, 2476, 2679, 2890, 3109, 3336, 3571, 3814, 4065, 4324, 4591, 4866, 5149, 5440, 5739, 6046, 6361, 6684, 7015, 7354, 7701, 8056, 8419, 8790

Offset: 1

Enoch Haga, G. L. Honaker, Jr., Apr 10 2000

Move in 1-4 direction in a spiral organized like A068225 etc.
Equals binomial transform of [1, 3, 8, 0, 0, 0, ...]. - Gary W. Adamson, Apr 30 2008
Ulam's spiral (N spoke). - Robert G. Wilson v, Oct 31 2011
Also, numbers of the form m*(4*m+1)+1 for nonpositive m. - Bruno Berselli, Jan 06 2016

Ivan Panchenko, Table of n, a(n) for n = 1..1000
Franck Ramaharo, Statistics on some classes of knot shadows, arXiv:1802.07701 [math.CO], 2018.
Robert G. Wilson v, Cover of the March 1964 issue of Scientific American
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A054555, A068225, A054552, A054554, A054567, A054569, A033951.
Cf. A266883: m*(4*m+1)+1 for m = 0,-1,1,-2,2,-3,3,...
Sequences on the four axes of the square spiral: Starting at 0: A001107, A033991, A007742, A033954; starting at 1: A054552, A054556, A054567, A033951.
Sequences on the four diagonals of the square spiral: Starting at 0: A002939 = 2*A000384, A016742 = 4*A000290, A002943 = 2*A014105, A033996 = 8*A000217; starting at 1: A054554, A053755, A054569, A016754.
Sequences obtained by reading alternate terms on the X and Y axes and the two main diagonals of the square spiral: Starting at 0: A035608, A156859, A002378 = 2*A000217, A137932 = 4*A002620; starting at 1: A317186, A267682, A002061, A080335.
Cf. A001105, A058331, A130883.

[4*n^2-9*n+6 : n in [1..50]]; // Vincenzo Librandi, Mar 10 2018

a:=n->4*n^2-9*n+6: seq(a(n),n=1..50); # Muniru A Asiru, Mar 09 2018

a[n_] := 4*n^2 - 9*n + 6; Array[a, 40] (* Vladimir Joseph Stephan Orlovsky, Sep 01 2008 *)
LinearRecurrence[{3,-3,1},{1,4,15},50] (* Harvey P. Dale, Sep 06 2015 *)
CoefficientList[Series[-(6x^2 + x + 1)/(x - 1)^3, {x, 0, 49}], x] (* Robert G. Wilson v, Mar 12 2018 *)

a(n)=4*n^2-9*n+6 \\ Charles R Greathouse IV, Sep 24 2015

a(n)^2 = Sum_{i = 0..2*(4*n-5)} (4*n^2-13*n+9+i)^2*(-1)^i = ((n-1)*(4*n-5)+1)^2. - Bruno Berselli, Apr 29 2010
From Harvey P. Dale, Aug 21 2011: (Start)
a(0)=1, a(1)=4, a(2)=15; for n > 2, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f.: -x*(6*x^2+x+1)/(x-1)^3. (End)
From Franck Maminirina Ramaharo, Mar 09 2018: (Start)
a(n) = binomial(2*n - 2, 2) + 2*(n - 1)^2 + 1.
a(n) = A000384(n-1) + A058331(n-1).
a(n) = A130883(n-1) + A001105(n-1). (End)
E.g.f.: exp(x)*(6 - 5*x + 4*x^2) - 6. - Stefano Spezia, Apr 24 2024

Edited by Frank Ellermann, Feb 24 2002

Incorrect formula deleted by N. J. A. Sloane, Aug 02 2009

A054567 a(n) = 4n^2 - 7n + 4.

Original entry on oeis.org

1, 6, 19, 40, 69, 106, 151, 204, 265, 334, 411, 496, 589, 690, 799, 916, 1041, 1174, 1315, 1464, 1621, 1786, 1959, 2140, 2329, 2526, 2731, 2944, 3165, 3394, 3631, 3876, 4129, 4390, 4659, 4936, 5221, 5514, 5815, 6124, 6441, 6766, 7099, 7440, 7789, 8146, 8511, 8884

Offset: 1

Enoch Haga, G. L. Honaker, Jr., Apr 10 2000

The number 1 is placed in the middle of a sheet of squared paper and the numbers 2, 3, 4, 5, 6, etc. are written in a clockwise spiral around 1, as in A068225 etc. This sequence is read off along one of the rays from 1.
Ulam's spiral (W spoke of A054552). - Robert G. Wilson v, Oct 31 2011
Also, numbers of the form m*(4*m+1)+1 for nonnegative m. - Bruno Berselli, Jan 06 2016
The sequence forms the 1x2 diagonal of the square maze arrangement in A081344. - Jarrod G. Sage, Jul 17 2024

Ivan Panchenko, Table of n, a(n) for n = 1..1000
Robert G. Wilson v, Cover of the March 1964 issue of Scientific American
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A054566, A068225, A054552, A054554, A054556, A054569, A033951, A204674.
Cf. A266883: m*(4*m+1)+1 for m = 0,-1,1,-2,2,-3,3,...
Sequences on the four axes of the square spiral: Starting at 0: A001107, A033991, A007742, A033954; starting at 1: A054552, A054556, A054567, A033951.
Sequences on the four diagonals of the square spiral: Starting at 0: A002939 = 2*A000384, A016742 = 4*A000290, A002943 = 2*A014105, A033996 = 8*A000217; starting at 1: A054554, A053755, A054569, A016754.
Sequences obtained by reading alternate terms on the X and Y axes and the two main diagonals of the square spiral: Starting at 0: A035608, A156859, A002378 = 2*A000217, A137932 = 4*A002620; starting at 1: A317186, A267682, A002061, A080335.

Table[4 n^2 - 7 n + 4, {n, 100}] (* Vladimir Joseph Stephan Orlovsky, Sep 01 2008 *)

Vec(-x*(4*x^2+3*x+1)/(x-1)^3 + O(x^100)) \\ Colin Barker, Oct 25 2014

a(n) = 8*n+a(n-1)-11 for n>1, a(1)=1. - Vincenzo Librandi, Aug 07 2010
a(n) = A204674(n-1) / n. - Reinhard Zumkeller, Jan 18 2012
From Colin Barker, Oct 25 2014: (Start)
a(n) = 3*a(n-1)-3*a(n-2)+a(n-3).
G.f.: -x*(4*x^2+3*x+1) / (x-1)^3. (End)
E.g.f.: exp(x)*(4 - 3*x + 4*x^2) - 4. - Stefano Spezia, Apr 24 2024
a(n) = A016742(n-1) + n. - Jarrod G. Sage, Jul 17 2024

Edited by Frank Ellermann, Feb 24 2002

Typo fixed by Charles R Greathouse IV, Oct 28 2009

A017137 a(n) = 8*n + 6.

Original entry on oeis.org

6, 14, 22, 30, 38, 46, 54, 62, 70, 78, 86, 94, 102, 110, 118, 126, 134, 142, 150, 158, 166, 174, 182, 190, 198, 206, 214, 222, 230, 238, 246, 254, 262, 270, 278, 286, 294, 302, 310, 318, 326, 334, 342, 350, 358, 366, 374, 382, 390, 398, 406, 414, 422, 430

Offset: 0

N. J. A. Sloane, Dec 11 1996

First differences of A002943. - Aaron David Fairbanks, May 13 2014

			G.f. = 6 + 14*x + 22*x^2 + 30*x^3 + 38*x^4 + 46*x^5 + 54*x^6 + 62*x^7 + ...

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

Cf. A000290, A002943, A004767, A004770, A008590, A017101, A017113, A017245, A156676.

Cf. A016825, A017629, A047595 (complement), A089911.

a017137 = (+ 6) . (* 8)  -- Reinhard Zumkeller, Jul 05 2013

[8*n+6: n in [0..60]]; // Vincenzo Librandi, Jun 07 2011

A017137:=n->8*n+6; seq(A017137(n), n=0..50); # Wesley Ivan Hurt, May 13 2014

Range[6, 1000, 8] (* Vladimir Joseph Stephan Orlovsky, May 27 2011 *)
8Range[0,60]+6 (* or *) LinearRecurrence[{2,-1},{6,14},60] (* Harvey P. Dale, Nov 14 2021 *)

a(n) = 8*n+6; \\ Michel Marcus, Sep 17 2015

Vec((6+2*x)/(1-x)^2 + O(x^100)) \\ Altug Alkan, Oct 23 2015

a(n) = 2*A004767(n) = A000290(A017245(n)) - A156676(n+1). - Reinhard Zumkeller, Jul 13 2010
a(n) = 2*a(n-1) - a(n-2). - Vincenzo Librandi, Jun 07 2011
A089911(3*a(n)) = 4. - Reinhard Zumkeller, Jul 05 2013
From Michael Somos, May 15 2014: (Start)
G.f.: (6 + 2*x)/(1 - x)^2.
E.g.f.: (6 + 8*x)*exp(x). (End)
Sum_{n>=0} (-1)^n/a(n) = (Pi + log(3-2*sqrt(2)))/(8*sqrt(2)). - Amiram Eldar, Dec 11 2021
a(n) = A016825(2*n+1). - Elmo R. Oliveira, Apr 12 2025

A080335 Diagonal in square spiral or maze arrangement of natural numbers.

Original entry on oeis.org

1, 5, 9, 17, 25, 37, 49, 65, 81, 101, 121, 145, 169, 197, 225, 257, 289, 325, 361, 401, 441, 485, 529, 577, 625, 677, 729, 785, 841, 901, 961, 1025, 1089, 1157, 1225, 1297, 1369, 1445, 1521, 1601, 1681, 1765, 1849, 1937, 2025, 2117, 2209, 2305, 2401, 2501

Offset: 0

Paul Barry, Mar 19 2003

Interleaves the odd squares A016754 with (1+4n^2), A053755.
Squares of positive integers (plus 1 if n is odd). - Wesley Ivan Hurt, Oct 10 2013
a(n) is the maximum total number of queens that can coexist without attacking each other on an [n+3] X [n+3] chessboard, when the lone queen is in the most vulnerable position on the board. Specifically, the lone queen will placed in any center position, facing an opponent's "army" of size a(n)-1 == A137932(n+2). - Bob Selcoe, Feb 12 2015
a(n) is also the edge chromatic number of the complement of the (n+2) X (n+2) rook graph. - Eric W. Weisstein, Jan 31 2024

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Edge Chromatic Number
Eric Weisstein's World of Mathematics, Rook Complement Graph
Eric Weisstein's World of Mathematics, Rook Graph
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A081347, A081348.
Sequences on the four axes of the square spiral: Starting at 0: A001107, A033991, A007742, A033954; starting at 1: A054552, A054556, A054567, A033951.
Sequences on the four diagonals of the square spiral: Starting at 0: A002939 = 2*A000384, A016742 = 4*A000290, A002943 = 2*A014105, A033996 = 8*A000217; starting at 1: A054554, A053755, A054569, A016754.
Sequences obtained by reading alternate terms on the X and Y axes and the two main diagonals of the square spiral: Starting at 0: A035608, A156859, A002378 = 2*A000217, A137932 = 4*A002620; starting at 1: A317186, A267682, A002061, A080335.

[(3+4*n+2*n^2-(-1)^n)/2: n in [0..50]]; // Vincenzo Librandi, Sep 06 2011

A080335:=n->(n mod 2) + (n+1)^2; seq(A080335(k),k=0..49); # Wesley Ivan Hurt, Oct 10 2013

With[{nn = 60}, Riffle[Range[1, nn, 2]^2, 4 Range[nn]^2 + 1]] (* Harvey P. Dale, Jan 29 2012 *)
LinearRecurrence[{2, 0, -2, 1}, {1, 5, 9, 17}, 60] (* Harvey P. Dale, Jan 29 2012 *)
Table[(3 + 4 n + 2 n^2 - (-1)^n)/2, {n, 0, 50}] (* Wesley Ivan Hurt, Oct 10 2013 *)
Table[Mod[n, 2] + (n + 1)^2, {n, 0, 20}] (* Eric W. Weisstein, Jan 31 2024 *)

a(n) = (3 + 4*n + 2*n^2 - (-1)^n)/2.
a(2*n) = A016754(n), a(2*n+1) = A053755(n+1).
E.g.f.: exp(x)*(2 + 3*x + x^2) - cosh(x). The sequence 1,1,5,9,... is given by n^2+(1+(-1)^n)/2 with e.g.f. exp(1+x+x^2)*exp(x)-sinh(x). - Paul Barry, Sep 02 2003 and Sep 19 2003
a(0)=1, a(1)=5, a(2)=9, a(3)=17, a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4). - Harvey P. Dale, Jan 29 2012
a(n)+(-1)^n = A137928(n+1). - Philippe Deléham, Feb 17 2012
G.f.: (1 + 3*x - x^2 + x^3)/((1-x)^3*(1+x)). - Colin Barker, Mar 18 2012
a(n) = A000035(n) + A000290(n+1). - Wesley Ivan Hurt, Oct 10 2013
From Bob Selcoe, Feb 12 2015: (Start)
a(n) = A137932(n+2) + 1.
a(n) = (n+1)^2 when n is even; a(n) = (n+1)^2 + 1 when n is odd.
a(n) = A002378(n+2) - A047238(n+3) + 1.
(End)
Sum_{n>=0} 1/a(n) = Pi*coth(Pi/2)/4 + Pi^2/8 - 1/2. - Amiram Eldar, Jul 07 2022

A137932 Terms in an n X n spiral that do not lie on its principal diagonals.

Original entry on oeis.org

0, 0, 0, 4, 8, 16, 24, 36, 48, 64, 80, 100, 120, 144, 168, 196, 224, 256, 288, 324, 360, 400, 440, 484, 528, 576, 624, 676, 728, 784, 840, 900, 960, 1024, 1088, 1156, 1224, 1296, 1368, 1444, 1520, 1600, 1680, 1764, 1848, 1936, 2024, 2116, 2208, 2304, 2400, 2500, 2600, 2704, 2808

Offset: 0

William A. Tedeschi, Feb 29 2008

The count of terms not on the principal diagonals is always even.
The last digit is the repeating pattern 0,0,0,4,8,6,4,6,8,4, which is palindromic if the leading 0's are removed, 4864684.
The sum of the last digits is 40, which is the count of the pattern times 4.
A 4 X 4 spiral is the only spiral, aside from a 0 X 0, whose count of terms that do not lie on its principal diagonals equal the count of terms that do [A137932(4) = A042948(4)] making the 4 X 4 the "perfect spiral".
Yet another property is mod(a(n), A042948(n)) = 0 iff n is even. This is a large family that includes the 4 X 4 spiral.
a(n) is the maximum number of queens of one color that can coexist without attacking one queen of the opponent's color on an [n+1] X [n+1] chessboard, when the lone queen is in the most vulnerable position on the board, i.e., on a center square. - Bob Selcoe, Feb 12 2015
Also the circumference of the (n-1) X (n-1) grid graph for n > 2. - Eric W. Weisstein, Mar 25 2018
Also the crossing number of the complete bipartite graph K_{5,n}. - Eric W. Weisstein, Sep 11 2018

			a(0) = 0^2 - (2(0) - mod(0,2)) = 0.
a(3) = 3^2 - (2(3) - mod(3,2)) = 4.

Enrique Pérez Herrero, Table of n, a(n) for n = 0..5000
Kival Ngaokrajang, Illustration of initial terms.
Eric Weisstein's World of Mathematics, Complete Bipartite Graph.
Eric Weisstein's World of Mathematics, Graph Circumference.
Eric Weisstein's World of Mathematics, Graph Crossing Number.
Eric Weisstein's World of Mathematics, Grid Graph.
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A042948.
Sequences on the four axes of the square spiral: Starting at 0: A001107, A033991, A007742, A033954; starting at 1: A054552, A054556, A054567, A033951.
Sequences on the four diagonals of the square spiral: Starting at 0: A002939 = 2*A000384, A016742 = 4*A000290, A002943 = 2*A014105, A033996 = 8*A000217; starting at 1: A054554, A053755, A054569, A016754.
Sequences obtained by reading alternate terms on the X and Y axes and the two main diagonals of the square spiral: Starting at 0: A035608, A156859, A002378 = 2*A000217, A137932 = 4*A002620; starting at 1: A317186, A267682, A002061, A080335.

A137932:=n->2*floor((n-1)^2/2); seq(A137932(n), n=0..50); # Wesley Ivan Hurt, Apr 19 2014

Table[2 Floor[(n - 1)^2/2], {n, 0, 20}] (* Enrique Pérez Herrero, Jul 04 2012 *)
2 Floor[(Range[20] - 1)^2/2] (* Eric W. Weisstein, Sep 11 2018 *)
Table[n^2 - 2 n + (1 - (-1)^n)/2, {n, 20}] (* Eric W. Weisstein, Sep 11 2018 *)
LinearRecurrence[{2, 0, -2, 1}, {0, 0, 4, 8}, 20] (* Eric W. Weisstein, Sep 11 2018 *)
CoefficientList[Series[-((4 x^2)/((-1 + x)^3 (1 + x))), {x, 0, 20}], x] (* Eric W. Weisstein, Sep 11 2018 *)

A137932(n)={ return(n^2 - (2*n-n%2))} ;
print(vector(30,n,A137932(n-1))); /* R. J. Mathar, May 12 2014 */

Python
```
a = lambda n: n**2 - (2*n - (n%2))
```

a(n) = n^2 - (2*n - mod(n,2)) = n^2 - A042948(n).
a(n) = 2*A007590(n-1). - Enrique Pérez Herrero, Jul 04 2012
G.f.: -4*x^3 / ( (1+x)*(x-1)^3 ). a(n) = 4*A002620(n-1). - R. J. Mathar, Jul 06 2012
From Bob Selcoe, Feb 12 2015: (Start)
a(n) = (n-1)^2 when n is odd; a(n) = (n-1)^2 - 1 when n is even.
a(n) = A002378(n) - A047238(n+1). (End)
From Amiram Eldar, Mar 20 2022: (Start)
Sum_{n>=3} 1/a(n) = Pi^2/24 + 1/4.
Sum_{n>=3} (-1)^(n+1)/a(n) = Pi^2/24 - 1/4. (End)
E.g.f.: x*(x - 1)*cosh(x) + (x^2 - x + 1)*sinh(x). - Stefano Spezia, Oct 17 2022

cp's OEIS Frontend

A033954 Second 10-gonal (or decagonal) numbers: n*(4*n+3).

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A054569 a(n) = 4*n^2 - 6*n + 3.

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A035608 Expansion of g.f. x*(1 + 3*x)/((1 + x)*(1 - x)^3).

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A054554 a(n) = 4*n^2 - 10*n + 7.

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A054000 a(n) = 2*n^2 - 2.

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A054556 a(n) = 4*n^2 - 9*n + 6.

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A033954 Second 10-gonal (or decagonal) numbers: n(4n+3).

A054569 a(n) = 4n^2 - 6n + 3.

A035608 Expansion of g.f. x(1 + 3x)/((1 + x)*(1 - x)^3).

A054554 a(n) = 4n^2 - 10n + 7.

A054556 a(n) = 4n^2 - 9n + 6.

A054567 a(n) = 4n^2 - 7n + 4.