A255840 -id:A255840 - cp's OEIS frontend

A085046 a(n) = n^2 - (1 + (-1)^n)/2.

1, 3, 9, 15, 25, 35, 49, 63, 81, 99, 121, 143, 169, 195, 225, 255, 289, 323, 361, 399, 441, 483, 529, 575, 625, 675, 729, 783, 841, 899, 961, 1023, 1089, 1155, 1225, 1295, 1369, 1443, 1521, 1599, 1681, 1763, 1849, 1935, 2025, 2115, 2209, 2303, 2401, 2499, 2601

Offset: 1

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), Jun 20 2003

Sequence pattern looks like this 1*1, 1*3, 3*3, 3*5, 5*5, 5*7, 7*7, 7*9, 9*9, 9*11, 11*11, ... = A109613(n-1)*A109613(n).
a(n+1) is the determinant of the n X n matrix M_(i, i)=3, M_(i, j)=2 if (i+j) is even, M_(i, j)=0 if (i+j) is odd. - Benoit Cloitre, Aug 06 2003
a(n) is also the longest path, in number of cells, between diagonally opposite corners of an n X n matrix if diagonal movement between adjacent cells is not allowed and no cell is used more than once. - Ray G. Opao, Jul 02 2007
(-1)^n*a(n) appears to be the Hankel transform of A141222. - Paul Barry, Jun 14 2008
Take an n X n square grid and add unit squares along each side except for the corners --> do this repeatedly along each side with the same restriction until no squares can be added. 4*a(n) is the total number of unit edges in each figure (see example and cf. A255840, A255876). - Wesley Ivan Hurt, Mar 09 2015

			4*a(n) is the number of unit edges in the pattern below (see comments).
                                                                 _
                                                               _|_|_
                            _              _ _               _|_|_|_|_
                          _|_|_          _|_|_|_           _|_|_|_|_|_|_
              _ _       _|_|_|_|_      _|_|_|_|_|_       _|_|_|_|_|_|_|_|_
    _        |_|_|     |_|_|_|_|_|    |_|_|_|_|_|_|     |_|_|_|_|_|_|_|_|_|
   |_|       |_|_|       |_|_|_|      |_|_|_|_|_|_|       |_|_|_|_|_|_|_|
                           |_|          |_|_|_|_|           |_|_|_|_|_|
                                          |_|_|               |_|_|_|
                                                                |_|
   n=1        n=2          n=3             n=4                  n=5
- _Wesley Ivan Hurt_, Mar 09 2015

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

Cf. A109613. [Bruno Berselli, Sep 17 2013]

Cf. A008811, A141222, A255840, A255876.

[n^2-(1+(-1)^n)/2 : n in [1..100]]; // Wesley Ivan Hurt, Mar 09 2015

A085046:=n->n^2-(1+(-1)^n)/2: seq(A085046(n), n=1..100); # Wesley Ivan Hurt, Mar 09 2015

Table[n^2-1/2 (1+(-1)^n), {n, 60}] (* Bruno Berselli, Sep 17 2013 *)
LinearRecurrence[{2,0,-2,1},{1,3,9,15},70] (* Harvey P. Dale, Oct 25 2015 *)

a(1) = 1, a(2) = 3, then a(2n) = (a(2n-1)*a(2n+1))^1/2 and a(2n+1) = {a(2n) + a(2n+2)}/2. Even-indexed terms are the geometric mean, and odd-indexed terms are the arithmetic mean, of their neighbors.
a(2n+1) = (2n+1)^2 and a(2n) = 4n^2 - 1.
a(n) = A008811(2n) - 1. - N. J. A. Sloane, Jun 12 2004
From Bruno Berselli, Sep 17 2013: (Start)
G.f.: x*(1 + x + 3*x^2 - x^3)/((1+x)*(1-x)^3).
a(n) = n^2 - (1 + (-1)^n)/2. (End)
a(1)=1, a(2)=3, a(3)=9, a(4)=15, a(n) = 2*a(n-1) + 0*a(n-2) -  2*a(n-3) + a(n-4). - Harvey P. Dale, Oct 25 2015
E.g.f.: 1 - cosh(x) + x*(1 + x)*(cosh(x) + sinh(x)). - Stefano Spezia, May 26 2021
Sum_{n>=1} 1/a(n) = Pi^2/8 + 1/2. - Amiram Eldar, Aug 25 2022

More terms from Benoit Cloitre, Aug 06 2003
Formula added in the first comment by Bruno Berselli, Sep 17 2013
Replaced name with Sep 17 2013 formula from Bruno Berselli [Wesley Ivan Hurt, May 17 2020]

A056640 At stage 1, start with a unit square. At each successive stage add 4*(n-1) new squares around outside with edge-to-edge contacts. Sequence gives number of squares (regardless of size) at n-th stage.

Original entry on oeis.org

1, 5, 18, 42, 83, 143, 228, 340, 485, 665, 886, 1150, 1463, 1827, 2248, 2728, 3273, 3885, 4570, 5330, 6171, 7095, 8108, 9212, 10413, 11713, 13118, 14630, 16255, 17995, 19856, 21840, 23953, 26197, 28578, 31098, 33763, 36575, 39540, 42660, 45941, 49385, 52998

Offset: 1

Robert G. Wilson v, Aug 21 2000

Number of unit squares at n-th stage = n^2 + (n-1)^2 (A001844).

First differences are in A255840. - Wesley Ivan Hurt, Mar 13 2015

Anthony Gardiner, "Mathematical Puzzling," Dover Publications, Inc., Mineola, NY., 1987, page 88.

Index entries for linear recurrences with constant coefficients, signature (3,-2,-2,3,-1).

Cf. A255840.

A056640:=n->(8*n^3-2*n+3-3*(-1)^n)/12: seq(A056640(n), n=1..50);

Table[(8*n^3 - 2*n + 3 - 3*(-1)^n)/12, {n, 30}] (* Wesley Ivan Hurt, Mar 13 2015 *)

Vec(x*(5*x^2+2*x+1)/((x-1)^4*(x+1)) + O(x^100)) \\ Colin Barker, Sep 29 2014

G.f.: x(5x^2+2x+1)/((1-x^2)(1-x)^3).
a(n) = (8*n^3-2*n+3-3*(-1)^n)/12. - Luce ETIENNE, Aug 21 2014
a(n) = 3*a(n-1)-2*a(n-2)-2*a(n-3)+3*a(n-4)-a(n-5). - Colin Barker, Sep 29 2014
G.f.: x*(5*x^2+2*x+1) / ((x-1)^4*(x+1)). - Colin Barker, Sep 29 2014

More terms from Colin Barker, Sep 29 2014

A255876 a(n) = (4n^2 + 4n - 3 - 3*(-1)^n)/2.

Original entry on oeis.org

4, 9, 24, 37, 60, 81, 112, 141, 180, 217, 264, 309, 364, 417, 480, 541, 612, 681, 760, 837, 924, 1009, 1104, 1197, 1300, 1401, 1512, 1621, 1740, 1857, 1984, 2109, 2244, 2377, 2520, 2661, 2812, 2961, 3120, 3277, 3444, 3609, 3784, 3957, 4140, 4321, 4512, 4701

Offset: 1

Wesley Ivan Hurt, Mar 08 2015

Take an n X n square grid and add unit squares along each side except for the corners --> do this repeatedly along each side with the same restriction until no squares can be added. a(n) gives the number of vertices in each figure (see example and cf. A255840).

			                                                                 _
                                                               _|_|_
                            _              _ _               _|_|_|_|_
                          _|_|_          _|_|_|_           _|_|_|_|_|_|_
              _ _       _|_|_|_|_      _|_|_|_|_|_       _|_|_|_|_|_|_|_|_
    _        |_|_|     |_|_|_|_|_|    |_|_|_|_|_|_|     |_|_|_|_|_|_|_|_|_|
   |_|       |_|_|       |_|_|_|      |_|_|_|_|_|_|       |_|_|_|_|_|_|_|
                           |_|          |_|_|_|_|           |_|_|_|_|_|
                                          |_|_|               |_|_|_|
                                                                |_|
   n=1        n=2          n=3             n=4                  n=5

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A000290 (squares), A085046, A198442, A255840.

[(4*n^2 + 4*n - 3 - 3*(-1)^n)/2 : n in [1..50]];

A255876:=n->(4*n^2 + 4*n - 3 - 3*(-1)^n)/2: seq(A255876(n), n=1..50);

CoefficientList[Series[(3 x^3 - 6 x^2 - x - 4)/((x + 1) (x - 1)^3), {x, 0, 50}], x]
LinearRecurrence[{2,0,-2,1},{4,9,24,37},60] (* Harvey P. Dale, Dec 26 2024 *)

vector(100,n,(4*n^2 + 4*n - 3 - 3*(-1)^n)/2) \\ Derek Orr, Mar 09 2015

G.f.: x*(3*x^3 - 6*x^2 - x - 4)/((x + 1)*(x - 1)^3).
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4).
a(n) = A000290(n+1) + 4*A198442(n).

A259486 a(n) = 3n^2 - 3n + 1 + 6floor((n-1)(n-2)/6).

Original entry on oeis.org

1, 7, 19, 43, 73, 109, 157, 211, 271, 343, 421, 505, 601, 703, 811, 931, 1057, 1189, 1333, 1483, 1639, 1807, 1981, 2161, 2353, 2551, 2755, 2971, 3193, 3421, 3661, 3907, 4159, 4423, 4693, 4969, 5257, 5551, 5851, 6163, 6481, 6805, 7141, 7483, 7831, 8191, 8557

Offset: 1

Wesley Ivan Hurt, Jun 28 2015

Start with the geometric picture for the centered hex numbers (A003215). Here, each hexagonal figure in the sequence is the aggregate of smaller unit hexes (with n hexes along each side). Then, when possible, add additional unit hexes to each side except for the corners --> do this repeatedly with the same restriction until no hexes can be added. a(n) gives the area of each figure (see example).

a(n) == 1 mod 6. - Robert Israel, Jun 29 2015

			-----------------------------------------------------------------------
Figure 1
-----------------------------------------------------------------------
                                                  __    __    __
                                                 /  \__/  \__/  \
                                                 \_*/  \__/  \*_/
                              __               __/  \__/  \__/  \__
                           __/  \__           /  \__/  \__/  \__/  \
            __          __/  \__/  \__        \__/  \__/  \__/  \__/
         __/  \__      /  \__/  \__/  \     __/  \__/  \__/  \__/  \__
.__     /  \__/  \     \__/  \__/  \__/    / *\__/  \__/  \__/  \__/* \
/  \    \__/  \__/     /  \__/  \__/  \    \__/  \__/  \__/  \__/  \__/
\__/    /  \__/  \     \__/  \__/  \__/       \__/  \__/  \__/  \__/
        \__/  \__/     /  \__/  \__/  \       /  \__/  \__/  \__/  \
           \__/        \__/  \__/  \__/       \__/  \__/  \__/  \__/
                          \__/  \__/             \__/  \__/  \__/
                             \__/                / *\__/  \__/* \
                                                 \__/  \__/  \__/
n=1         n=2               n=3                       n=4
-----------------------------------------------------------------------
Table 1
-----------------------------------------------------------------------
a(1) = 1                              =  1
a(2) = 3  + 2(2)                      =  7
a(3) = 5  + 2(3+4)                    =  19
a(4) = 7  + 2(4+5+6)          + 6(1)  =  43
a(5) = 9  + 2(5+6+7+8)        + 6(2)  =  73
a(6) = 11 + 2(6+7+8+9+10)     + 6(3)  =  109
a(7) = 13 + 2(7+8+9+10+11+12) + 6(5)  =  157
...

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

Cf. A003215 (hex numbers), A000969, A130518, A255840 (similar, with squares).

[3*n^2 - 3*n + 1 + 6*Floor((n-1)*(n-2)/6) : n in [1..100]];

I:=[1,7,19,43,73]; [n le 5 select I[n] else 2*Self(n-1)-Self(n-2)+Self(n-3)-2*Self(n-4)+Self(n-5): n in [1..60]]; // Vincenzo Librandi, Jul 14 2015

A259486:=n->3*n^2 - 3*n + 1 + 6*floor((n-1)*(n-2)/6): seq(A259486(n), n=1..100);

Table[3 n^2 - 3 n + 1 + 6 Floor[(n - 1) (n - 2)/6], {n, 50}] (* or *)
CoefficientList[Series[(1 + 5 x + 6 x^2 + 11 x^3 + x^4)/((1 - x)^3 (1 + x + x^2)), {x, 0, 50}], x]
LinearRecurrence[{2, -1, 1, -2, 1}, {1, 7, 19, 43, 73}, 50]; (* Vincenzo Librandi, Jul 14 2015 *)

G.f.: (1+5*x+6*x^2+11*x^3+x^4)/((1-x)^3*(1+x+x^2)).
a(n) = 2*a(n-1)-a(n-2)+a(n-3)-2*a(n-4)+a(n-5), n>5.
a(n) = A003215(n+1) + 6*A130518(n+1).
From Robert Israel, Jun 29 2015: (Start)
a(n) = 4*n^2 - 6*n + 1 if 3 divides n, 4*n^2 - 6*n + 3 otherwise.
a(n) = 1 + 6 * A000969(n-2) for n >= 2. (End)
a(n) = 4*n^2 - 6*n + 3^sign(n mod 3). - Wesley Ivan Hurt, Jul 13 2015

cp's OEIS Frontend

A085046 a(n) = n^2 - (1 + (-1)^n)/2.

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A056640 At stage 1, start with a unit square. At each successive stage add 4*(n-1) new squares around outside with edge-to-edge contacts. Sequence gives number of squares (regardless of size) at n-th stage.

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A255876 a(n) = (4*n^2 + 4*n - 3 - 3*(-1)^n)/2.

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Magma

Maple

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A259486 a(n) = 3*n^2 - 3*n + 1 + 6*floor((n-1)*(n-2)/6).

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Examples

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Crossrefs

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Magma

Magma

Maple

Mathematica

Formula

A255876 a(n) = (4n^2 + 4n - 3 - 3*(-1)^n)/2.

A259486 a(n) = 3n^2 - 3n + 1 + 6floor((n-1)(n-2)/6).