A085046 -id:A085046 - cp's OEIS frontend

A141222 Expansion of -1/(2x) + (2x-1)^2/(2x(1-4x)^(3/2)).

1, 5, 22, 95, 406, 1722, 7260, 30459, 127270, 529958, 2200276, 9111830, 37650172, 155266100, 639191160, 2627302995, 10784089350, 44208873390, 181025067300, 740483276610, 3026059513620, 12355464845100

Offset: 0

Paul Barry, Jun 14 2008

Apply Riordan array (1/sqrt(1-4x), xc(x)) to A131056, c(x) the g.f. of A000108.
Apply Riordan array (c(x)/sqrt(1-4*x), x*c(x)^2) to A131055.
Hankel transform appears to be (-1)^n*A085046(n).
Coefficients T(2*n+1,n) of triangle A103450. [Emanuele Munarini, Jun 01 2012, corrected by Werner Schulte, Nov 27 2021]

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Table[((1+3*n+n^2)*Binomial[2*n, n])/(n+1),{n,0,20}] (* Vaclav Kotesovec, Feb 13 2014 *)
CoefficientList[Series[-1/(2*x)+(2*x-1)^2/(2*x*(1-4x)^(3/2)),{x,0,20}],x] (* Vaclav Kotesovec, Feb 13 2014 *)
a[n_] := (1 + 3 n + n^2) CatalanNumber[n];
Table[a[n], {n, 0, 21}] (* Peter Luschny, Nov 28 2021 *)

a(n):=sum(binomial(2*n,k)*binomial(n+1,2*n-k),k,0,n); makelist(a(n),n,0,40); /* Emanuele Munarini, Jun 01 2012 */

a(n) = Sum_{k=0..n} (1 + (k+1)*2^(k-1) - 0^k/2)*C(2n-k,n-k); a(n) = Sum_{k=0..n} C(2n,k)*C(n+1,2n-k).
Equals the Narayana transform (A001263) of integer squares. - Gary W. Adamson, Jul 29 2011
Conjecture: (n+1)*a(n) + 2*(-3*n-1)*a(n-1) + 4*(2*n-3)*a(n-2) = 0. - R. J. Mathar, Nov 24 2012
From Vaclav Kotesovec, Feb 13 2014: (Start)
G.f.: -1/(2*x) + (2*x-1)^2/(2*x*(1-4x)^(3/2)).
a(n) = (1 + 3*n + n^2) * C(2*n,n) / (n+1).
Recurrence: (n+1)*(n^2 + n - 1)*a(n) = 2*(2*n-1)*(n^2 + 3*n + 1)*a(n-1).
(End)

Name of the sequence corrected by Vaclav Kotesovec, Feb 13 2014

A255840 a(n) = (4n^2 - 4n + 1 - (-1)^n)/2.

Original entry on oeis.org

0, 1, 4, 13, 24, 41, 60, 85, 112, 145, 180, 221, 264, 313, 364, 421, 480, 545, 612, 685, 760, 841, 924, 1013, 1104, 1201, 1300, 1405, 1512, 1625, 1740, 1861, 1984, 2113, 2244, 2381, 2520, 2665, 2812, 2965, 3120, 3281, 3444, 3613, 3784, 3961, 4140, 4325, 4512

Offset: 0

Wesley Ivan Hurt, Mar 07 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) is the total area of each figure. The perimeter, P, of each figure is given by P(n) = 4*A042963(n), n>0 (see example).

For n>0, partial sums of a(n) are in A056640.

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

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

Cf. A000290 (squares), A002620 (quarter-squares), A042963.

Cf. A056640, A085046, A255876.

[(4*n^2 - 4*n + 1 - (-1)^n)/2 : n in [0..100]];

A255840:=n->(4*n^2 - 4*n + 1 - (-1)^n)/2: seq(A255840(n), n=0..100);

CoefficientList[Series[x (1 + 2 x + 5 x^2)/((1 + x) (1 - x)^3), {x, 0, 50}], x]

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

G.f.: x*(1+2*x+5*x^2)/((1+x)*(1-x)^3).
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4).
a(n) = A000290(n) + 4*A002620(n).
a(n) - a(n-1) = A047471(n). - Wesley Ivan Hurt, Apr 28 2017

A302488 Total domination number of the n X n grid graph.

Original entry on oeis.org

0, 1, 2, 3, 6, 9, 12, 15, 20, 25, 30, 35, 42, 49, 56, 63, 72, 81, 90, 99, 110, 121, 132, 143, 156, 169, 182, 195, 210, 225, 240, 255, 272, 289, 306, 323, 342, 361, 380, 399, 420, 441, 462, 483, 506, 529, 552, 575, 600, 625, 650, 675, 702, 729, 756, 783, 812, 841, 870, 899, 930

Offset: 0

Eric W. Weisstein, Apr 08 2018

Extended to a(0) and a(1) using the formula/recurrence. The total domination number of the 1 X 1 grid graph is undefined.

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Grid Graph.
Eric Weisstein's World of Mathematics, Total Domination Number.
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,1,-2,1).

Main diagonal of A300358.
The four quadrasections are A002943, A016754, A002939(n+1), A000466(n+1).
Bisections are A002378 and A085046.
Cf. A303142.

R:=RealField(); [Round(((-1)^n + 2*n*(n + 2) + 4*Sin(n*Pi(R)/2) - 1)/8): n in [0..30]]; // G. C. Greubel, Apr 09 2018

Table[(-1 + (-1)^n + 2 n (2 + n) + 4 Sin[n Pi/2])/8, {n, 0, 20}]
LinearRecurrence[{2, -1, 0, 1, -2, 1}, {0, 1, 2, 3, 6, 9}, 20]
CoefficientList[Series[x (-1 - 2 x^3 + x^4)/((-1 + x)^3 (1 + x + x^2 + x^3)), {x, 0, 20}], x]

for(n=0,30, print1(round(((-1)^n + 2*n*(n + 2) + 4*sin(n*Pi/2) - 1)/8), ", ")) \\ G. C. Greubel, Apr 09 2018

a(n)=my(m=n\4); (2*m+1)*(2*m + n%4) \\ Andrew Howroyd, Aug 17 2025

a(n) = ((-1)^n + 2*n*(n + 2) + 4*sin(n*Pi/2) - 1)/8.
a(n) = 2*a(n-1) - a(n-2) + a(n-4) - 2*a(n-5) + a(n-6).
G.f.: x*(1 + 2*x^3 - x^4)/((1 - x)^3*(1 + x + x^2 + x^3)).
a(4*m + r) = (2*m + 1)*(2*m + r) for 0 <= r < 4. - Charles Kusniec, Aug 16 2025
From Amiram Eldar, Aug 26 2025: (Start)
Sum_{n>=1} 1/a(n) = Pi^2/8 + 3/2.
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/8 - 1/2. (End)

a(0)=0 prepended and offset corrected by Andrew Howroyd, Aug 17 2025

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).

A304487 a(n) = (3 + 2n - 3n^2 + 4n^3 - 3((-1 + n) mod 2))/6.

Original entry on oeis.org

1, 4, 15, 36, 73, 128, 207, 312, 449, 620, 831, 1084, 1385, 1736, 2143, 2608, 3137, 3732, 4399, 5140, 5961, 6864, 7855, 8936, 10113, 11388, 12767, 14252, 15849, 17560, 19391, 21344, 23425, 25636, 27983, 30468, 33097, 35872, 38799, 41880, 45121, 48524, 52095

Offset: 1

Stefano Spezia, Aug 17 2018

a(n) is the trace of an n X n matrix A in which the entries are 1 through n^2, spiraling inward starting with 1 in the (1,1)-entry (proved).

The first three terms of a(n) coincide with those of A317614.

			For n = 1 the matrix A is
   1
with trace Tr(A) = a(1) = 1.
For n = 2 the matrix A is
   1, 2
   4, 3
with Tr(A) = a(2) = 4.
For n = 3 the matrix A is
   1, 2, 3
   8, 9, 4
   7, 6, 5
with Tr(A) = a(3) = 15.
For n = 4 the matrix A is
   1,  2,  3, 4
  12, 13, 14, 5
  11, 16, 15, 6
  10,  9,  8, 7
with Tr(A) = a(4) = 36.

Stefano Spezia, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (3,-2,-2,3,-1).

Cf. A317614, A228958, A045991, A085046.

Cf. A126224 (determinant of the matrix A), A317298 (first differences).

a_n:=List([1..43], n->(3 + 2*n - 3*n^2 + 4*n^3 - 3*RemInt(-1 + n, 2))/6);

List([1..43],n->(3+2*n-3*n^2+4*n^3-3*((-1+n) mod 2))/6); # Muniru A Asiru, Sep 17 2018

I:=[1,4,15,36,73]; [n le 5 select I[n] else 3*Self(n-1)-2*Self(n-2)-2*Self(n-3)+3*Self(n-4)-Self(n-5): n in [1..43]]; // Vincenzo Librandi, Aug 26 2018

seq((3+2*n-3*n^2+4*n^3-3*modp((-1+n),2))/6,n=1..43); # Muniru A Asiru, Sep 17 2018

Table[1/6 (3 + 2 n - 3 n^2 + 4 n^3 - 3 Mod[-1 + n, 2]), {n, 1, 43}] (* or *)
CoefficientList[ Series[x*(1 + x + 5 x^2 + x^3)/((-1 + x)^4 (1 + x)), {x, 0, 43}], x] (* or *)
LinearRecurrence[{3, -2, -2, 3, -1}, {1, 4, 15, 36, 73}, 43]

a(n):=(3 + 2*n - 3*n^2 + 4*n^3 - 3*mod(-1 + n, 2))/6$ makelist(a(n), n, 1, 43);

Vec(x*(1 + x + 5*x^2 + x^3)/((-1 + x)^4*(1 + x)) + O(x^44))

a(n) = (3 + 2*n - 3*n^2 + 4*n^3 - 3*((-1 + n)%2))/6

a(n) = A045991(n) - Sum_{k=2..n-1} A085046(k) for n > 2 (proved).
G.f.: x*(1 + x + 5 x^2 + x^3)/((-1 + x)^4 (1 + x)).
a(n) + a(n + 1) = A228958(2*n + 1).
From Colin Barker, Aug 17 2018: (Start)
a(n) = (2*n - 3*n^2 + 4*n^3) / 6 for n even.
a(n) = (3 + 2*n - 3*n^2 + 4*n^3) / 6 for n odd.
a(n) = 3*a(n - 1) - 2*a(n - 2) - 2*a(n - 3) + 3*a(n - 4) - a(n - 5) for n > 5.
(End)
E.g.f.: (1/12)*exp(-x)*(-3 + exp(2*x)*(3 + 6*x + 18*x^2 + 8*x^3)). - Stefano Spezia, Feb 10 2019

A308896 Walk a rook along the square spiral numbered 0, 1, 2, ... (cf. A274641); a(n) = mex of earlier values the rook can move to.

Original entry on oeis.org

0, 1, 0, 1, 2, 3, 0, 2, 3, 1, 2, 3, 0, 2, 3, 1, 4, 5, 6, 7, 0, 4, 5, 6, 7, 1, 4, 5, 6, 7, 0, 4, 5, 6, 7, 1, 2, 5, 4, 7, 6, 3, 0, 2, 5, 4, 7, 6, 3, 1, 2, 5, 4, 7, 6, 3, 0, 2, 5, 4, 7, 6, 3, 1, 8, 9, 10, 11, 12, 13, 14, 15, 0, 8, 9, 10, 11, 12, 13, 14, 15, 1, 8

Offset: 0

N. J. A. Sloane, Jul 02 2019

Analog of A308884 but using a rook rather than a knight.

The array of values - see the illustration in the link - appears to have a number of interesting symmetries.

			The central 21 X 21 portion of the plane:
[ 4  1  3 30 31 28 29 26 27 24 25 22 23 20 21 18 19 16 17  2  0]
[ 5  2  1 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16  0  3]
[18 17 16  1  3  6  7 12 13 14 15  8  9 10 11  4  5  2  0 31 30]
[19 16 17  2  1  7  6 13 12 15 14  9  8 11 10  5  4  0  3 30 31]
[16 19 18  5  4  1  3 14 15 12 13 10 11  8  9  2  0  7  6 29 28]
[17 18 19  4  5  2  1 15 14 13 12 11 10  9  8  0  3  6  7 28 29]
[22 21 20 11 10  9  8  1  3  6  7  4  5  2  0 15 14 13 12 27 26]
[23 20 21 10 11  8  9  2  1  7  6  5  4  0  3 14 15 12 13 26 27]
[20 23 22  9  8 11 10  5  4  1  3  2  0  7  6 13 12 15 14 25 24]
[21 22 23  8  9 10 11  4  5  2  1  0  3  6  7 12 13 14 15 24 25]
*26 25 24 15 14 13 12  7  6  3 *0* 1  2  5  4 11 10  9  8 23 22]
[27 24 25 14 15 12 13  6  7  0  2  3  1  4  5 10 11  8  9 22 23]
[24 27 26 13 12 15 14  3  0  4  5  6  7  1  2  9  8 11 10 21 20]
[25 26 27 12 13 14 15  0  2  5  4  7  6  3  1  8  9 10 11 20 21]
[30 29 28  7  6  3  0  8  9 10 11 12 13 14 15  1  2  5  4 19 18]
[31 28 29  6  7  0  2  9  8 11 10 13 12 15 14  3  1  4  5 18 19]
[28 31 30  3  0  4  5 10 11  8  9 14 15 12 13  6  7  1  2 17 16]
[29 30 31  0  2  5  4 11 10  9  8 15 14 13 12  7  6  3  1 16 17]
[ 6  3  0 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31  1  2]
[ 7  0  2 17 16 19 18 21 20 23 22 25 24 27 26 29 28 31 30  3  1]
[ 0  4  5 18 19 16 17 22 23 20 21 26 27 24 25 30 31 28 29  6  7]
===============================**===============================

Rémy Sigrist, Table of n, a(n) for n = 0..16128
F. Michel Dekking, Jeffrey Shallit, and N. J. A. Sloane, Queens in exile: non-attacking queens on infinite chess boards, Electronic J. Combin., 27:1 (2020), #P1.52.
Rémy Sigrist, Colored representation of the spiral for -511 <= x, y <= 511 (where dark pixels correspond to higher values and red pixels correspond to 0's)
Rémy Sigrist, Scatterplot of (x,y) such that A(x,y) has bit b set to one for b = 0..6 and -63 <= x <= 64 and -63 <= y <= 64
Rémy Sigrist, PARI program for A308896
N. J. A. Sloane, Initial terms of spiral
N. J. A. Sloane, Explicit formulas for the array in A308896, Jul 02 2019
N. J. A. Sloane, The two kinds of sectors. (Rows y=1 and above form a sector of the first type, rows y=0 and below form the second type.)

Cf. A002378, A003987, A085046, A274641, A308884, A308897.

PARI
```
See Links section.
```

a(n) = 0, 1 iff n belongs to A002378, A085046, respectively. - Rémy Sigrist, Jul 02 2019

For formulas for the terms in the array, see the "Explicit formulas" link.

More terms from Rémy Sigrist, Jul 02 2019

A290561 a(n) = n + cos(n*Pi/2).

Original entry on oeis.org

1, 1, 1, 3, 5, 5, 5, 7, 9, 9, 9, 11, 13, 13, 13, 15, 17, 17, 17, 19, 21, 21, 21, 23, 25, 25, 25, 27, 29, 29, 29, 31, 33, 33, 33, 35, 37, 37, 37, 39, 41, 41, 41, 43, 45, 45, 45, 47, 49, 49, 49, 51, 53, 53, 53, 55, 57, 57, 57, 59, 61, 61, 61, 63, 65, 65, 65

Offset: 0

Jean-François Alcover and Paul Curtz, Aug 06 2017

a(n) divides A289296(n).

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

Cf. A004524, A085046, A289296, A290562.

A290561:=n->n+cos(n*Pi/2): seq(A290561(n), n=0..150); # Wesley Ivan Hurt, Aug 06 2017

a[n_] := n + Cos[n*Pi/2]; Table[a[n], {n, 0, 60}]

a(n) = n + round(cos(n*Pi/2)); \\ Michel Marcus, Aug 06 2017

Vec((x^3 + x^2 - x + 1)/((x - 1)^2*(x^2 + 1)) + O(x^100)) \\ Colin Barker, Aug 06 2017

G.f.: (x^3 + x^2 - x + 1)/((x - 1)^2*(x^2 + 1)).
a(n) = n if n == 3 (mod 4), and a(n) = a(n-4) + 4 otherwise, for n>2.
a(n) = a(n+20) - 20.
a(n) = 2*A004524(n) + 1.
a(n) + A290562(n) = 2*n.
a(n) * A290562(n) = n^2 - cos(n*Pi/2)^2 = A085046(n) for n>0.
A290562(n) = -a(-n).
From Colin Barker, Aug 06 2017: (Start)
a(n) = ((-i)^n + i^n)/2 + n where i=sqrt(-1).
a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - a(n-4) for n>3. (End)

A290562 a(n) = n - cos(n*Pi/2).

Original entry on oeis.org

-1, 1, 3, 3, 3, 5, 7, 7, 7, 9, 11, 11, 11, 13, 15, 15, 15, 17, 19, 19, 19, 21, 23, 23, 23, 25, 27, 27, 27, 29, 31, 31, 31, 33, 35, 35, 35, 37, 39, 39, 39, 41, 43, 43, 43, 45, 47, 47, 47, 49, 51, 51, 51, 53, 55, 55, 55, 57, 59, 59, 59, 61, 63, 63, 63, 65, 67

Offset: 0

Jean-François Alcover and Paul Curtz, Aug 06 2017

a(n) divides A289870(n).

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

Cf. A085046, A289870, A290561.

a[n_] := n - Cos[n*Pi/2]; Table[a[n], {n, 0, 60}]

a(n) = n - round(cos(n*Pi/2)); \\ Michel Marcus, Aug 06 2017

Vec((x^3 - x^2 + 3*x - 1)/((x - 1)^2*(x^2 + 1)) + O(x^100)) \\ Colin Barker, Aug 08 2017

G.f.: (x^3 - x^2 + 3 x - 1)/((x - 1)^2*(x^2 + 1)).
a(n) = n if n == 1 (mod 4), and a(n) = a(n-4) + 4 otherwise, for n>4.
a(n) = a(n+20) - 20.
a(n) = -A290561(-n).
a(n) + A290561(n) = 2*n.
a(n) * A290561(n) = n^2 - cos(n*Pi/2)^2 = A085046(n) for n>0.
From Colin Barker, Aug 08 2017: (Start)
a(n) = n - (-i)^n/2 - i^n/2 where i=sqrt(-1).
a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - a(n-4) for n>3.
(End)

A120413 Largest even number strictly less than n^2.

Original entry on oeis.org

0, 2, 8, 14, 24, 34, 48, 62, 80, 98, 120, 142, 168, 194, 224, 254, 288, 322, 360, 398, 440, 482, 528, 574, 624, 674, 728, 782, 840, 898, 960, 1022, 1088, 1154, 1224, 1294, 1368, 1442, 1520, 1598, 1680, 1762, 1848, 1934, 2024, 2114, 2208, 2302, 2400, 2498, 2600

Offset: 1

Henry Bottomley, Jul 06 2006

Longest non-intersecting route from (0, 0) to (n - 1, n - 1) staying in an (n - 1) X (n - 1) box (shortest route is length 2n A005843).

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

Maple
```
seq(2*ceil(n^2/2)-2,n=1..50);
```

Flatten[Table[{(2n - 1)^2 - 1, 4n^2 - 2}, {n, 25}]] (* Alonso del Arte, Apr 15 2016 *)

lista(nn) = for(n=0, nn, print1((-1+(-1)^n+4*n+2*n^2)/2, ", ")); \\ Altug Alkan, Apr 15 2016

a(n) = 2*ceiling[n^2/2] - 2 = 2*A074148(n) = A085046(n) - 1.
From Colin Barker, Jul 29 2012: (Start)
a(n) = (-1 + (-1)^n + 4*n + 2*n^2)/2.
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4).
G.f.: 2*x*(1 + 2*x - x^2)/((1-x)^3*(1+x)). (End)
a(n) = n^2 - 2 for even n; a(n) = n^2 - 1 for odd n. -Dennis P. Walsh, Apr 15 2016

Offset corrected by N. J. A. Sloane, Apr 15 2016

A264798 Irregular triangle read by rows: odd-valued terms of A094728(n+1).

Original entry on oeis.org

1, 3, 9, 5, 15, 7, 25, 21, 9, 35, 27, 11, 49, 45, 33, 13, 63, 55, 39, 15, 81, 77, 65, 45, 17, 99, 91, 75, 51, 19, 121, 117, 105, 85, 57, 21, 143, 135, 119, 95, 63, 23, 169, 165, 153, 133, 105, 69, 25, 195, 187, 171, 147, 115, 75, 27, 225, 221, 209, 189, 161, 125, 81, 29, 255, 247

Offset: 0

Paul Curtz, Nov 25 2015

A094728(n+1) comes from A120070(n+2). a(n) approximates frequencies of the spectral lines of the hydrogen atom.
Row sums: 1, 3, 14, 22, ... = A024598(n+1).
First column: A085046(n+1).
Row sums of A261046(n) = 1, 3, 8, 12, ... = A014255(n). See the formula.

			Irregular triangle begins:
1,
3,
9,  5,
15, 7,
25, 21,  9,
35, 27, 11,
49, 45, 33, 13,
63, 55, 39, 15,
...

Cf. A014255, A024598, A085046, A094728, A120070, A167268, A249947, A261046.

Table[n^2 - k^2, {n, 14}, {k, 0, n - 1}] /. n_ /; EvenQ@ n -> Nothing // Flatten (* Michael De Vlieger, Nov 25 2015 *)

for(n=1,20,for(k=0,n-1,s=n^2-k^2;if(s%2,print1(s,", ")))) \\ Derek Orr, Dec 24 2015

a(n) = A261046(n)*A167268(n+1)/2, where A167268 is Janet's sequence.

cp's OEIS Frontend

A141222 Expansion of -1/(2*x) + (2*x-1)^2/(2*x*(1-4x)^(3/2)).

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A255840 a(n) = (4*n^2 - 4*n + 1 - (-1)^n)/2.

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A302488 Total domination number of the n X n grid graph.

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

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A304487 a(n) = (3 + 2*n - 3*n^2 + 4*n^3 - 3*((-1 + n) mod 2))/6.

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A308896 Walk a rook along the square spiral numbered 0, 1, 2, ... (cf. A274641); a(n) = mex of earlier values the rook can move to.

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A141222 Expansion of -1/(2x) + (2x-1)^2/(2x(1-4x)^(3/2)).

A255840 a(n) = (4n^2 - 4n + 1 - (-1)^n)/2.

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

A304487 a(n) = (3 + 2n - 3n^2 + 4n^3 - 3((-1 + n) mod 2))/6.