A014105 -id:A014105 - cp's OEIS frontend

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

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

A084849 a(n) = 1 + n + 2*n^2.

Original entry on oeis.org

1, 4, 11, 22, 37, 56, 79, 106, 137, 172, 211, 254, 301, 352, 407, 466, 529, 596, 667, 742, 821, 904, 991, 1082, 1177, 1276, 1379, 1486, 1597, 1712, 1831, 1954, 2081, 2212, 2347, 2486, 2629, 2776, 2927, 3082, 3241, 3404, 3571, 3742, 3917, 4096, 4279, 4466

Offset: 0

Paul Barry, Jun 09 2003

Equals (1, 2, 3, ...) convolved with (1, 2, 4, 4, 4, ...). a(3) = 22 = (1, 2, 3, 4) dot (4, 4, 2, 1) = (4 + 8 + 6 + 4). - Gary W. Adamson, May 01 2009
a(n) is also the number of ways to place 2 nonattacking bishops on a 2 X (n+1) board. - Vaclav Kotesovec, Jan 29 2010
Partial sums are A174723. - Wesley Ivan Hurt, Apr 16 2016
Also the number of irredundant sets in the n-cocktail party graph. - Eric W. Weisstein, Aug 09 2017

G. C. Greubel, Table of n, a(n) for n = 0..5000
W. Burrows and C. Tuffley, Maximising common fixtures in a round robin tournament with two divisions, arXiv:1502.06664 [math.CO], 2015.
Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]
Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.
Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph.D. Thesis, Waterford Institute of Technology, 2011.
Eric Weisstein's World of Mathematics, Cocktail Party Graph.
Eric Weisstein's World of Mathematics, Irredundant Set.
Wikipedia, Alexander polynomial and Seifert surface. [See _Peter Bala_'s comment.]
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000027, A000217, A001844, A004767 (first differences), A014105, A058331, A060884, A100036, A100037, A100038, A100039, A100040, A100041, A131901, A134082, A174723, A177342.

[1+n+2*n^2 : n in [0..100]]; // Wesley Ivan Hurt, Apr 15 2016

A084849:=n->1+n+2*n^2: seq(A084849(n), n=0..100); # Wesley Ivan Hurt, Apr 15 2016

s = 1; lst = {s}; Do[s += n + 2; AppendTo[lst, s], {n, 1, 200, 4}]; lst (* Zerinvary Lajos, Jul 11 2009 *)
f[n_]:=(n*(2*n+1)+1);Table[f[n],{n,5!}] (* Vladimir Joseph Stephan Orlovsky, Feb 07 2010 *)
Table[1 + n + 2 n^2, {n, 0, 20}] (* Eric W. Weisstein, Aug 09 2017 *)
LinearRecurrence[{3, -3, 1}, {4, 11, 22}, {0, 20}] (* Eric W. Weisstein, Aug 09 2017 *)
CoefficientList[Series[(-1 - x - 2 x^2)/(-1 + x)^3, {x, 0, 20}], x] (* Eric W. Weisstein, Aug 09 2017 *)

a(n)=1+n+2*n^2 \\ Charles R Greathouse IV, Sep 24 2015

a(n) = A058331(n) + A000027(n).
G.f.: (1 + x + 2*x^2)/(1 - x)^3.
a(n) = A014105(n) + 1; A100035(a(n)) = 1. - Reinhard Zumkeller, Oct 31 2004
a(n) = ceiling((2*n + 1)^2/2) - n = A001844(n) - n. - Paul Barry, Jul 16 2006
From Gary W. Adamson, Oct 07 2007: (Start)
Row sums of triangle A131901.
(a(n): n >= 0) is the binomial transform of (1, 3, 4, 0, 0, 0, ...). (End)
Equals A134082 * [1,2,3,...]. -
a(n) = (1 + A000217(2*n-1) + A000217(2*n+1))/2. - Enrique Pérez Herrero, Apr 02 2010
a(n) = (A177342(n+1) - A177342(n))/2, with n > 0. - Bruno Berselli, May 19 2010
a(n) - 3*a(n-1) + 3*a(n-2) - a(n-3) = 0, with n > 2. - Bruno Berselli, May 24 2010
a(n) = 4*n + a(n-1) - 1 (with a(0) = 1). - Vincenzo Librandi, Aug 08 2010
With an offset of 1, the polynomial a(t-1) = 2*t^2 - 3*t + 2 is the Alexander polynomial (with negative powers cleared) of the 3-twist knot. The associated Seifert matrix S is [[-1,-1], [0,-2]]. a(n-1) = det(transpose(S) - n*S). Cf. A060884. - Peter Bala, Mar 14 2012
E.g.f.: (1 + 3*x + 2*x^2)*exp(x). - Ilya Gutkovskiy, Apr 16 2016

A014107 a(n) = n(2n-3).

Original entry on oeis.org

0, -1, 2, 9, 20, 35, 54, 77, 104, 135, 170, 209, 252, 299, 350, 405, 464, 527, 594, 665, 740, 819, 902, 989, 1080, 1175, 1274, 1377, 1484, 1595, 1710, 1829, 1952, 2079, 2210, 2345, 2484, 2627, 2774, 2925, 3080, 3239, 3402, 3569, 3740, 3915, 4094, 4277

Offset: 0

N. J. A. Sloane

Positive terms give a bisection of A000096. - Omar E. Pol, Dec 16 2016

G. C. Greubel, Table of n, a(n) for n = 0..5000
Emily Barnard and Nathan Reading, Coxeter-biCatalan combinatorics, arXiv:1605.03524 [math.CO], 2016. See p. 51.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000096, A000108, A014105, A033537, A097070, A100035, A100036, A100037, A100038, A100039, A100345, A002378, A000384.

A014107:=n->n*(2*n-3); seq(A014107(n), n=0..100); # Wesley Ivan Hurt, Nov 19 2013

Table[2n^2 - 3n, {n, 0, 49}] (* Alonso del Arte, Dec 15 2012 *)
LinearRecurrence[{3,-3,1},{0,-1,2},50] (* Harvey P. Dale, Sep 18 2018 *)

PARI
```
a(n)=n*(2*n-3)
```

a(n) = A100345(n, n - 3) for n > 2.
a(n) = A033537(n) - 8*n^2; A100035(a(n)) = 2 for n > 1. - Reinhard Zumkeller, Oct 31 2004
a(n) = A014106(-n) for all n in Z. - Michael Somos, Nov 06 2005
From Michael Somos, Nov 06 2005: (Start)
G.f.: x*(-1 + 5*x)/(1 - x)^3.
E.g.f: x*(-1 + 2*x)*exp(x). (End)
a(n) = A097070(n)/A000108(n - 2), n >= 2. - Philippe Deléham, Apr 12 2007
a(n) = 2*a(n-1) - a(n-2) + 4, n > 1; a(0) = 0, a(1) = -1, a(2) = 2. - Zerinvary Lajos, Feb 18 2008
a(n) = a(n-1) + 4*n - 5 with a(0) = 0. - Vincenzo Librandi, Nov 20 2010
a(n) = (2*n-1)*(n-1) - 1. Also, with an initial offset of -1, a(n) = (2*n-1)*(n+1) = 2*n^2 + n - 1. - Alonso del Arte, Dec 15 2012
(a(n) + 1)^2 + (a(n) + 2)^2 + ... + (a(n) + n)^2 = (a(n) + n + 1)^2 + (a(n) + n + 2)^2 + ... + (a(n) + 2n - 1)^2 starting with a(1) = -1. - Jeffreylee R. Snow, Sep 17 2013
a(n) = A014105(n-1) - 1 for all n in Z. - Michael Somos, Nov 23 2021
From Amiram Eldar, Feb 20 2022: (Start)
Sum_{n>=1} 1/a(n) = -2*(1 - log(2))/3.
Sum_{n>=1} (-1)^n/a(n) = Pi/6 + log(2)/3 + 2/3. (End)
For n > 0, A002378(a(n)) = A000384(n-1)*A000384(n). - Charlie Marion, May 21 2023

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

A130883 a(n) = 2*n^2 - n + 1.

Original entry on oeis.org

1, 2, 7, 16, 29, 46, 67, 92, 121, 154, 191, 232, 277, 326, 379, 436, 497, 562, 631, 704, 781, 862, 947, 1036, 1129, 1226, 1327, 1432, 1541, 1654, 1771, 1892, 2017, 2146, 2279, 2416, 2557, 2702, 2851, 3004, 3161, 3322, 3487, 3656, 3829, 4006, 4187, 4372, 4561

Offset: 0

Mohammad K. Azarian, Jul 26 2007

Maximum number of regions determined by n bent lines (or angular sectors). See Concrete Mathematics reference.
A "bent line" may also be regarded as a "long-legged letter V", meaning a letter V with both line segments extended to infinity.  See A117625 for the analogous sequence for a long-legged Z. - N. J. A. Sloane, Jun 18 2025
a(n)*Pi is the total length of half circle spiral after n rotations. It is formed as irregular spiral with two center points. At the 2nd stage, there are two alternatives: (1) select 2nd half circle radius, r2  = 2, the sequence will be A014105 or (2) select r2 = 0, the sequence will be A130883. See illustration in links. - Kival Ngaokrajang, Jan 19 2014
A128218(a(n)) = 2*n+1 and A128218(m) != 2*n+1 for m < a(n). - Reinhard Zumkeller, Jun 20 2015

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, 2nd ed., Addison-Wesley, Reading, MA, 1994, pp. 7-8, and Problem 1.18, pages 19 and 500.

G. C. Greubel, Table of n, a(n) for n = 0..5000
Dmitry Efimov, Hafnian of two-parameter matrices, arXiv:2101.09722 [math.CO], 2021.
Guo-Niu Han, Enumeration of Standard Puzzles
Guo-Niu Han, Enumeration of Standard Puzzles [Cached copy]
Ângela Mestre and José Agapito, Square Matrices Generated by Sequences of Riordan Arrays, J. Int. Seq., Vol. 22 (2019), Article 19.8.4.
Kival Ngaokrajang, Illustration of irregular spirals (center points: 1, 2)
Franck Ramaharo, Statistics on some classes of knot shadows, arXiv:1802.07701 [math.CO], 2018.
Franck Ramaharo, A generating polynomial for the pretzel knot, arXiv:1805.10680 [math.CO], 2018.
N. J. A. Sloane, Illustration for a(3) = 16
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000124, A000384, A014105, A084849, A128218, A128918, A152947, A270109.
See also A117625.
A row of the array in A386478.

a130883 = a128918 . (* 2)  -- Reinhard Zumkeller, Oct 27 2013

[2*n^2 - n + 1 : n in [0..50]]; // Wesley Ivan Hurt, Mar 25 2020

a[n_]:=2*n^2-n+1; (* or *) Array[ -#*(1-#*2)+1&,5!,0] (* Vladimir Joseph Stephan Orlovsky, Dec 21 2008 *)
LinearRecurrence[{3,-3,1},{1,2,7},50] (* Harvey P. Dale, Jul 20 2011 *)

a(n)=2*n^2-n+1 \\ Charles R Greathouse IV, Sep 24 2015

def A130883(n): return n*(2*n - 1) + 1 # Chai Wah Wu, May 24 2022

a(n) = a(n-1) + 4*n - 3 for n > 0, a(0)=1. - Vincenzo Librandi, Nov 23 2010
a(n) = A000124(2*n) - 2*n. - Geoffrey Critzer, Mar 30 2011
O.g.f.: (4*x^2-x+1)/(1-x)^3. - Geoffrey Critzer, Mar 30 2011
a(n) = 2*a(n-1) - a(n-2) + 4. - Eric Werley, Jun 27 2011
a(0)=1, a(1)=2, a(2)=7; for n > 2, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Jul 20 2011
a(n) = A128918(2*n). - Reinhard Zumkeller, Oct 27 2013
a(n) = 1 + A000384(n). - Omar E. Pol, Apr 27 2017
E.g.f.: (2*x^2 + x + 1)*exp(x). - G. C. Greubel, Jul 14 2017
a(n) = A152947(2*n+1). - Franck Maminirina Ramaharo, Jan 10 2018

A147875 Second heptagonal numbers: a(n) = n(5n+3)/2.

Original entry on oeis.org

0, 4, 13, 27, 46, 70, 99, 133, 172, 216, 265, 319, 378, 442, 511, 585, 664, 748, 837, 931, 1030, 1134, 1243, 1357, 1476, 1600, 1729, 1863, 2002, 2146, 2295, 2449, 2608, 2772, 2941, 3115, 3294, 3478, 3667, 3861, 4060, 4264, 4473, 4687, 4906, 5130, 5359, 5593

Offset: 0

Vladimir Joseph Stephan Orlovsky, Nov 16 2008

Zero followed by partial sums of A016897.
Apparently = every 2nd term of A111710 and A085787.
Bisection of A085787. Sequence found by reading the line from 0, in the direction 0, 13, ... and the line from 4, in the direction 4, 27, ..., in the square spiral whose vertices are the generalized heptagonal numbers A085787. - Omar E. Pol, Jul 18 2012
Numbers of the form m^2 + k*m*(m+1)/2: in this case is k=3. See also A254963. - Bruno Berselli, Feb 11 2015

			G.f. = 4*x + 13*x^2 + 27*x^3 + 46*x^4 + 70*x^5 + 99*x^6 + 133*x^7 + ... - _Michael Somos_, Jan 25 2019

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Leo Tavares, Illustration: Sliced Hexagons
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A016897, A111710, A000217, A085787, A224419 (positions of squares).
Second n-gonal numbers: A005449, A014105, A045944, A179986, A033954, A062728, A135705.
Cf. A000566.
Cf. A003215, A000096, A000290.

List([0..50], n-> n*(5*n+3)/2); # G. C. Greubel, Jul 04 2019

[n*(5*n+3)/2: n in [0..50]]; // G. C. Greubel, Jul 04 2019

Table[(n(5n+3))/2, {n, 0, 50}] (* or *) LinearRecurrence[{3, -3, 1}, {0, 4, 13}, 50] (* Harvey P. Dale, May 15 2013 *)

a(n)=n*(5*n+3)/2 \\ Charles R Greathouse IV, Sep 24 2015

[n*(5*n+3)/2 for n in (0..50)] # G. C. Greubel, Jul 04 2019

G.f.: x*(4+x)/(1-x)^3.
a(n) = Sum_{k=0..n-1} A016897(k).
a(n) - a(n-1) = 5*n -1. - Vincenzo Librandi, Nov 26 2010
G.f.: U(0) where U(k) = 1 + 2*(2*k+3)/(k + 2 - x*(k+2)^2*(k+3)/(x*(k+2)*(k+3) + (2*k+2)*(2*k+3)/U(k+1))); (continued fraction, 3-step). - Sergei N. Gladkovskii, Nov 14 2012
E.g.f.: U(0) where U(k) = 1 + 2*(2*k+3)/(k + 2 - 2*x*(k+2)^2*(k+3)/(2*x*(k+2)*(k+3) + (2*k+2)^2*(2*k+3)/U(k+1))); (continued fraction, 3rd kind, 3-step). - Sergei N. Gladkovskii, Nov 14 2012
a(n) = A130520(5n+3). - Philippe Deléham, Mar 26 2013
a(n) = A131242(10n+7)/2. - Philippe Deléham, Mar 27 2013
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(0)=0, a(1)=4, a(2)=13. - Harvey P. Dale, May 15 2013
Sum_{n>=1} 1/a(n) = 10/9 + sqrt(1 - 2/sqrt(5))*Pi/3 - 5*log(5)/6 + sqrt(5)*log((1 + sqrt(5))/2)/3 = 0.4688420784500060750083432... . - Vaclav Kotesovec, Apr 27 2016
a(n) = A000217(n) + A000217(2*n). - Bruno Berselli, Jul 01 2016
From Ilya Gutkovskiy, Jul 01 2016: (Start)
E.g.f.: x*(8 + 5*x)*exp(x)/2.
Dirichlet g.f.: (5*zeta(s-2) + 3*zeta(s-1))/2. (End)
a(n) = A000566(-n) for all n in Z. - Michael Somos, Jan 25 2019
From Leo Tavares, Feb 14 2022: (Start)
a(n) = A003215(n) - A000217(n+1). See Sliced Hexagons illustration in links.
a(n) = A000096(n) + 2*A000290(n). (End)

Edited by Klaus Brockhaus and R. J. Mathar, Nov 20 2008

New name from Bruno Berselli, Jan 13 2011

A016061 a(n) = n(n+1)(4*n+5)/6.

Original entry on oeis.org

0, 3, 13, 34, 70, 125, 203, 308, 444, 615, 825, 1078, 1378, 1729, 2135, 2600, 3128, 3723, 4389, 5130, 5950, 6853, 7843, 8924, 10100, 11375, 12753, 14238, 15834, 17545, 19375, 21328, 23408, 25619, 27965, 30450, 33078, 35853, 38779, 41860

Offset: 0

Robert G. Wilson v

Number of ZnS molecules in cluster of n layers in zinc blende crystal.
(Zinc sulfide crystallizes in two different forms: wurtzite and zinc blende, the latter is also spelled zincblende.) - Jonathan Vos Post, Jan 22 2013
The Kn4 triangle sums of the Connell-Pol triangle A159797 lead to the sequence given above. For the definitions of the Kn4 and other triangle sums see A180662. - Johannes W. Meijer, May 20 2011
If one generated primitive Pythagorean triangles (2n+1, 2n+3) the collective sum of their perimeters for each n is four times the numbers listed in this sequence. - J. M. Bergot, Jul 18 2011
a(n) is the number of 3-tuples (w,x,y) having all terms in {0,...,n} and nA000292(n)+A000292(n+1)=n^3. - Clark Kimberling, Jun 04 2012
Degrees of the Hilbert polynomials for B_3 and C_3, per p. 13 of Gashi et al. - Jonathan Vos Post, Dec 14 2013
Number of solutions to a + b = c + d when 0 < a <= k, 0 <= b, c, d <= k, k = 0, 1, 2, 3.... Taken from Step 1 2007 problem #1(i) using 4 digit balanced numbers. - Bobby Milazzo, Mar 09 2013
From J. M. Bergot, Jun 18 2013: (Start)
Consider the lower half, including the main diagonal, of the array in A144216 as a triangle.  The rows begin:
 0;
 1, 2;
 3, 4, 6;
 6, 7, 9, 12, ...
The sum of the terms in row(n) is a(n). (End)
This sequence is related to A008865 by a(n) = n*A008865(n+1) - Sum_{i=1..n} A008865(i) for n>0. - Bruno Berselli, Aug 06 2015

P. Jena and S. N. Behera, Clusters and Nanostructured Materials, Nova Science Publishers, 1996.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
David N. Blauch, Crystal Structure of Zinc Blende.
Qëndrim R. Gashi, Travis Schedler and David E. Speyer, Looping of the numbers game and the alcoved hypercube, arXiv:0909.5324v1 [math.RT], 2009.
Johan Kok, Introduction to total chromatic vertex stress of graphs, Open J. Disc. Appl. Math. (ODAM, 2023) Vol. 6, No. 2, 32-38. See p. 34.
T. P. Martin, Shells of atoms, Phys. Reports, Vol. 273 (1996), pp. 199-241, see p. 233.
Gloria Olive, Problem #504, Factorizations and Sums, Two-Year College Math. Jnl., Vol. 25 (1994), pp. 244-245.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Bisection of A002623.
Cf. A000292, A000330, A002412, A008865, A014105, A144216, A159797, A180662.
Row sums of triangle A120070.

I:=[0,3,13,34]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..40]]; // Vincenzo Librandi, Jul 25 2013

A016061 := proc(n)
    n*(n+1)*(4*n+5)/6 ;
end proc: # R. J. Mathar, Sep 26 2013

CoefficientList[Series[x (3 + x) / (1 - x)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Jul 25 2013 *)
Table[n(n+1)(4*n+5)/6, {n,0,100}] (* Wesley Ivan Hurt, Sep 25 2013 *)

v=vector(40,i,t(i)); s=0; forstep(i=2,40,2,s+=v[i]; print1(s","))

G.f.: x*(3+x)/(1-x)^4. - Paul Barry, Feb 27 2003
Partial sums of A014105. - Jon Perry, Jul 23 2003
a(n) = Sum_{i=0..n-1} 2*i^2 + i. - Jani Nurminen (slinky(AT)iki.fi), May 14 2006
a(n) = 2*n^3/3  +3*n^2/2 + 5*n/6. - Jonathan Vos Post, Dec 14 2013
a(n) = (4*n+5)/(2*n+1)*A000330(n). - Alexander R. Povolotsky, Mar 09 2013
a(n) = 4*a(n-1) -6*a(n-2) +4*a(n-3) -a(n-4). - Bobby Milazzo, Mar 10 2013
Sum_{n>=1} 1/a(n) = 12*Pi/5 + 72*log(2)/5 - 426/25. - Amiram Eldar, Jan 04 2022
E.g.f.: exp(x)*x*(18 + 21*x + 4*x^2)/6. - Stefano Spezia, Jul 31 2022

cp's OEIS Frontend

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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A084849 a(n) = 1 + n + 2*n^2.

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A014107 a(n) = n*(2*n-3).

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

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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.

A014107 a(n) = n(2n-3).

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

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

A147875 Second heptagonal numbers: a(n) = n(5n+3)/2.

A016061 a(n) = n(n+1)(4*n+5)/6.