Nicolay Avilov - cp's OEIS frontend

A382973 a(n) = 4n^3 - 6n^2 + 6*n - 2 + (-1)^n.

1, 19, 69, 183, 377, 683, 1117, 1711, 2481, 3459, 4661, 6119, 7849, 9883, 12237, 14943, 18017, 21491, 25381, 29719, 34521, 39819, 45629, 51983, 58897, 66403, 74517, 83271, 92681, 102779, 113581, 125119, 137409, 150483, 164357, 179063, 194617, 211051, 228381, 246639

Offset: 1

Nicolay Avilov, Jun 02 2025

a(n) is the number of 1 X 1 X 1 black cubes in a cube (2*n - 1) X (2*n - 1) X (2*n - 1), which is made up of 1 X 1 X 1 black cubes and 1 X 1 X 1 white cubes. In this case, any 1 X 1 X 1 cube is either completely black or completely white. The black and white cubes are arranged as follows: if the central cube has coordinates (0, 0, 0), then all cubes with coordinates (0, y, z), (x, 0, z) and (x, y, 0) are black. Then the cubes adjacent to the black ones are painted white so that a triangular layer with all white cubes is obtained. There will be eight such layers. In the next step, all cubes adjacent to the white ones are painted black so that a triangular layer of black cubes is formed, and so on, alternating layers of black cubes and layers of white cubes until all the cubes are painted (see the link "Illustration of coloring a cube").

			a(2) = 3^3 - 8*1 = 19;
a(3) = 5^3 - 8*7 = 69.

Nicolay Avilov, Illustration of a cube coloring.
Index entries for linear recurrences with constant coefficients, signature (3,-2,-2,3,-1).

Cf. A000578, A011934.

LinearRecurrence[{3, -2, -2, 3, -1}, {1, 19, 69, 183, 377}, 20] (* Hugo Pfoertner, Jun 12 2025 *)

a(n) = (2n - 1)^3 - 8*A011934(n-1).

G.f.: x*(1 + 16*x + 14*x^2 + 16*x^3 + x^4)/((1 - x)^4*(1 + x)). - Stefano Spezia, Jun 12 2025

A383585 Number of vertices of even degree in a cubic lattice n X n X n.

Original entry on oeis.org

0, 0, 13, 32, 63, 112, 185, 288, 427, 608, 837, 1120, 1463, 1872, 2353, 2912, 3555, 4288, 5117, 6048, 7087, 8240, 9513, 10912, 12443, 14112, 15925, 17888, 20007, 22288, 24737, 27360, 30163, 33152, 36333, 39712, 43295, 47088, 51097, 55328, 59787, 64480, 69413, 74592, 80023, 85712, 91665, 97888

Offset: 1

Nicolay Avilov, May 01 2025

An n X n X n cubic lattice is a graph with n^3 vertices. All interior vertices of the n X n X n cubic lattice are vertices of even degree, and all interior edge vertices of the cube are also vertices of even degree, so the number of vertices of even degree in an n X n X n lattice is (n - 2)^3 + 12*(n - 2).

			a(4) = 2^3 + 12*2 = 32;
a(5) = 3^3 + 12*3 = 63.

Nicolay Avilov, A technique for counting
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000578.

a[n_] := n^3 - 6*n^2 + 24*n - 32; a[1] = 0; Array[a, 48] (* Amiram Eldar, May 13 2025 *)

a(n) = n^3 - 6*n^2 + 24*n - 32 for n >= 2.
From Chai Wah Wu, May 19 2025: (Start)
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n > 5.
G.f.: x^3*(13*x^2 - 20*x + 13)/(x - 1)^4. (End)

A381380 Decimal expansion of the area of a ruled surface formed by moving a segment of length sqrt(6), the ends of which lie on the diagonals of opposite faces of a unit cube oriented at right angles to each other.

Original entry on oeis.org

2, 7, 2, 7, 0, 5, 4, 7, 7, 3, 8, 1, 2, 0, 4, 8, 9, 8, 8, 4, 3, 5, 1, 5, 5, 6, 7, 9, 0, 2, 0, 2, 5, 9, 8, 4, 2, 8, 3, 4, 6, 4, 7, 7, 1, 9, 9, 0, 3, 1, 3, 8, 7, 4, 0, 0, 3, 1, 0, 7, 1, 1, 8, 9, 3, 9, 5, 3, 9, 5, 1, 4, 0, 1, 3, 6, 7, 1, 4, 8, 4, 8, 4, 4, 9, 4, 0, 4, 0, 1, 1

Offset: 1

Nicolay Avilov, Feb 22 2025

A segment of constant length continuously sliding its endpoints along two intersecting straight lines defines a ruled surface -- a linoid. Here we consider a linoid defined by a segment of length sqrt(6)/2 sliding along two intersecting diagonals of opposite faces of a cube with edge 1. The surface of a linoid is given by the equation 2*x^2/(z - 1/2)^2 + 2*y^2/(z + 1/2)^2 = 1.
The surface of a linoid consists of four congruent surfaces. The area of one of them is calculated using integrals and multiplied by 4.
The name of the figure "linoid" was introduced by the author in the related article, see link.

			2.72705477381204898843515567902...

Nicolay Avilov, Construction of a linoid
Nicolay Avilov, Volume of a "linoid" (in Russian).

Equals sqrt(2)*Integral_{t=0..Pi/2} Integral_{z=0..1/2} sqrt(5 + 24*z^2 + 24*z*cos(2*t) + cos(4*t)) dz dt.

Terms corrected by Jinyuan Wang, Feb 23 2025

A380371 a(n) is the integer part of the area of a rhombus with side n and angle n degrees.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 3, 5, 8, 12, 17, 23, 29, 38, 47, 58, 70, 84, 100, 117, 136, 158, 181, 206, 234, 264, 296, 330, 368, 407, 450, 494, 542, 593, 646, 702, 761, 823, 889, 957, 1028, 1102, 1180, 1261, 1344, 1431, 1522, 1615, 1712, 1812, 1915, 2021, 2130, 2243, 2359

Offset: 0

Nicolay Avilov, Jan 23 2025

The angle n ranges 0 to 180 inclusive.

Terms increase to a maximum 12951 at n = 131 and then decrease.

			a(30) = 30*30*sin(30 degrees) = 450.
a(90) = 8100 is the area of a square with a side of 90.

Nicolay Avilov, Table of n, a(n) for n = 0..180

a(n) = floor(n^2*sin(Pi*n/180)).

A379747 The number of Hamiltonian cycles with rotational symmetry of order 3 on the triangular grid, n vertices on each side.

Original entry on oeis.org

0, 1, 1, 0, 2, 6, 0, 52, 270, 0, 17130, 189154, 0, 51417622

Offset: 1

Nicolay Avilov, Jan 01 2025

If n = 3*k + 1, then the triangular grid has a central node, so for such n there are no Hamiltonian cycles and a(n) = 0.

			a(3) = 1, the only Hamiltonian cycle being the obvious one running around the edge of the triangle;
a(4) = 0 because there are no Hamiltonian cycles;
a(5) = 2 because there are exactly two Hamiltonian cycles.

Nicolay Avilov, Examples of cycles counted by a(1) - a(8).

Cf. A112676.

a(9)-a(10) from Andrey Zabolotskiy, Jan 02 2025
a(11)-a(13) from Talmon Silver, Jan 04 2025
a(14) from Talmon Silver, Jan 06 2025

A375209 Number of simple symmetric Hamiltonian paths connecting opposite corners of a 2n+1 X 2n+1 grid.

Original entry on oeis.org

1, 2, 16, 564, 93866, 72054120, 260324223938, 4400423201461008, 349815282628284276844, 130501147375292529852604266, 228964256366276749773274186140858

Offset: 0

Nicolay Avilov, Oct 16 2024

			a(2) = 16, see link "Illustration".

Nicolay Avilov, Problem 2718. Broken Routes (in Russian).
Nicolay Avilov, Illustration for term a(2)

Cf. A001184, A331001.

a(n) = 2*A331001(2*n + 1) for n > 0. - Andrew Howroyd, Oct 16 2024

Thanks to Suleiman Makarenko.

a(5)-a(10) from Andrew Howroyd, Oct 16 2024

A374296 a(n) is the integer part of the area of a regular n-gon whose side lengths are n.

Original entry on oeis.org

3, 16, 43, 93, 178, 309, 500, 769, 1133, 1612, 2228, 3005, 3969, 5147, 6570, 8268, 10275, 12627, 15360, 18514, 22130, 26250, 30921, 36187, 42099, 48707, 56063, 64221, 73239, 83174, 94087, 106039, 119095, 133320, 148782, 165551, 183699, 203299

Offset: 3

Nicolay Avilov, Jul 03 2024

nonn

			Areas of polygons (starting from n=3):
  3.897... (equilateral triangle), so a(3) = 3,
 16.000... (square), so a(4) = 16,
 43.011... (pentagon), so a(5) = 43,
 93.530... (hexagon), so a(6) = 93,
178.061... (heptagon), so a(7) = 178.

Nicolay Avilov, Illustration of a(3)-a(7)

Cf. A064313.

a(n) = floor(n^3/(4*tan(Pi/n))).

a(n) = n^4/(4*Pi) - (Pi/12)*n^2 + O(1). - Charles R Greathouse IV, Jul 03 2024

A373584 a(n) is equal to the number of shaded cells in a regular hexagon with side n drawn on a hexagonal grid.

Original entry on oeis.org

1, 7, 13, 19, 31, 49, 67, 85, 109, 139, 169, 199, 235, 277, 319, 361, 409, 463, 517, 571, 631, 697, 763, 829, 901, 979, 1057, 1135, 1219, 1309, 1399, 1489, 1585, 1687, 1789, 1891, 1999, 2113, 2227, 2341, 2461, 2587, 2713, 2839, 2971, 3109, 3247, 3385, 3529

Offset: 1

Nicolay Avilov, Jun 10 2024

On a hexagonal grid, cells are colored as follows: one cell and all those located along three straight lines passing through the center of the original cell and forming six 60° angles between each other are painted. In each of these corners, cells are painted over so that a V-shaped arrangement of cells repeats ad infinitum. The number of shaded cells in regular hexagons centered on the starting cell determines the sequence a(n).

			a(3) = 19 - 6*1 = 13;
a(4) = 37 - 6*3 = 19.
                                                   o . o . o
                                 o . . o          . o . . o .
                   o . o        . o . o .        o . o . o . o
         o o      . o o .      . . o o . .      . . . o o . . .
   o    o o o    o o o o o    o o o o o o o    o o o o o o o o o
         o o      . o o .      . . o o . .      . . . o o . . .
                   o . o        . o . o .        o . o . o . o
                                 o . . o          . o . . o .
                                                   o . o . o
   1      7         13             19                 31

Paolo Xausa, Table of n, a(n) for n = 1..10000
Nicolay Avilov, Members of the sequence a(1) - a(7).
Nicolay Avilov, Problem 2663. Snowflakes (in Russian).
Nicolay Avilov, Illustration a(13) and a(16)
Index entries for linear recurrences with constant coefficients, signature (3,-4,4,-3,1).

Cf. A003215, A011848, A054925.

Table[6*Ceiling[n*(n - 1)/4] + 1, {n, 100}] (* Paolo Xausa, Jul 01 2024 *)

a(n+4) = a(n) + 12*n + 18.
a(n) = 6*ceiling(n*(n - 1)/4) + 1.
a(n) = A003215(n) - 6*A011848(n+1).
a(n) = 6*A054925(n) + 1.
G.f.: (1 + 4*x - 4*x^2 + 4*x^3 + x^4)/((1 - x)^3*(1 + x^2)). - Stefano Spezia, Jun 11 2024
E.g.f.: (exp(x)*(5 + 6*x + 3*x^2) - 3*cos(x) + 3*sin(x))/2. - Stefano Spezia, Aug 31 2025

A372630 Numbers k with property that there exists an m>k such that the sum of the natural numbers from k^2 to m^2 inclusive is a square number.

Original entry on oeis.org

1, 3, 8, 11, 12, 14, 17, 23, 30, 33, 35, 37, 41, 48, 59, 60, 65, 68, 77, 79, 82, 84, 89, 93, 94, 99

Offset: 1

Nicolay Avilov, May 07 2024

			The number 3 is a member of the sequence because the sum of all natural numbers from 3^2 to 4^2 inclusive is 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16 = 100, with 100 = 10^2.

Nicolay Avilov, Problem 1876. Segment of a natural series (in Russian).

Cf. A000290, A372631, A373330.

check(k, mm=100) = my(d=2*k^2-1, v=List([]), x, y, z); for(t=d+1, 17*d, if(issquare((t^2-d^2)/2), listput(v, t))); if(v[#v\2] != 3*d, return(-1)); for(i=1, #v\2, x=v[i]; y=v[i+#v\2]; for(j=1, mm, if(issquare((x-1)/2) && x>d+2, return(1)); z=6*y-x; x=y; y=z)); 0; \\ Jinyuan Wang, Jul 06 2024; just for checking

A372631 Numbers m for which there exists some k < m where the sum of the natural numbers from k^2 to m^2 inclusive is a square.

Original entry on oeis.org

4, 5, 7, 12, 15, 19, 29, 34, 41, 47, 55, 56, 65, 71, 73, 80, 84, 98, 111, 119, 124, 126, 141, 158, 165, 169, 175, 191, 209, 231, 239, 253, 260, 265, 287, 322, 335, 345, 352, 359, 369, 376, 403, 408, 425, 436, 444, 463, 465, 491, 505, 532, 542, 548, 587, 620

Offset: 1

Nicolay Avilov, May 07 2024

nonn

			4 is a term because the sum of all natural numbers from 3^2 to 4^2 inclusive is 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16 = 100 = 10^2.

Chai Wah Wu, Table of n, a(n) for n = 1..1301
Nicolay Avilov, Problem 1876. Segment of a natural series (in Russian).

Cf. A000290, A372630.

a={}; For[m=1, m<=620, m++, flag=0; tot=m^2*(m^2+1)/2; For[k=1, kStefano Spezia, May 11 2024  after Michael S. Branicky, May 10 2024 *)

from math import isqrt
def ok(m):
    tot = m**2*(m**2+1)//2
    for k in range(1, m):
        skm = tot - k**2*(k**2-1)//2
        if isqrt(skm)**2 == skm:
            return True
    return False
print([m for m in range(621) if ok(m)]) # Michael S. Branicky, May 10 2024

from itertools import count, islice
from sympy.abc import x,y
from sympy.ntheory.primetest import is_square
from sympy.solvers.diophantine.diophantine import diop_quadratic
def A372631_gen(startvalue=2): # generator of terms >= startvalue
    for m in count(max(startvalue,2)):
        m2 = m**2
        for k in diop_quadratic(m2*(m2+1)-x*(x-1)-2*y**2):
            if (r:=int(k[0]))A372631_list = list(islice(A372631_gen(),56)) # Chai Wah Wu, May 13 2024

cp's OEIS Frontend

User: Nicolay Avilov

A382973 a(n) = 4*n^3 - 6*n^2 + 6*n - 2 + (-1)^n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A383585 Number of vertices of even degree in a cubic lattice n X n X n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A381380 Decimal expansion of the area of a ruled surface formed by moving a segment of length sqrt(6), the ends of which lie on the diagonals of opposite faces of a unit cube oriented at right angles to each other.

Views

Author

Keywords

Comments

Examples

Links

Formula

Extensions

A380371 a(n) is the integer part of the area of a rhombus with side n and angle n degrees.

Views

Author

Keywords

Comments

Examples

Links

Formula

A379747 The number of Hamiltonian cycles with rotational symmetry of order 3 on the triangular grid, n vertices on each side.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A375209 Number of simple symmetric Hamiltonian paths connecting opposite corners of a 2n+1 X 2n+1 grid.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

Extensions

A374296 a(n) is the integer part of the area of a regular n-gon whose side lengths are n.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A373584 a(n) is equal to the number of shaded cells in a regular hexagon with side n drawn on a hexagonal grid.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A372630 Numbers k with property that there exists an m>k such that the sum of the natural numbers from k^2 to m^2 inclusive is a square number.

Views

A382973 a(n) = 4n^3 - 6n^2 + 6*n - 2 + (-1)^n.