A140064 -id:A140064 - cp's OEIS frontend

A386478 Array read by upward antidiagonals: T(k,n) = 1 (k = 0, n >= 0), T(k,n) = k^2n^2/2 - (3k-4)*n/2 + 1 (k >= 1, n >= 0).

1, 1, 1, 1, 2, 1, 1, 2, 4, 1, 1, 3, 7, 7, 1, 1, 5, 14, 16, 11, 1, 1, 8, 25, 34, 29, 16, 1, 1, 12, 40, 61, 63, 46, 22, 1, 1, 17, 59, 97, 113, 101, 67, 29, 1, 1, 23, 82, 142, 179, 181, 148, 92, 37, 1, 1, 30, 109, 196, 261, 286, 265, 204, 121, 46, 1, 1, 38, 140, 259, 359, 416, 418, 365, 269, 154, 56, 1, 1, 47, 175, 331, 473, 571, 607, 575, 481, 343, 191, 67, 1

Offset: 0

N. J. A. Sloane, Jul 24 2025

A k-chain is a planar graph consisting of a continuous path made up of k straight segments. T(k,n) is the maximum number of regions the plane can be divided into by drawing n k-chains.

The array is almost symmetric: the difference between T(k,n) and T(n,k) is 2*|k-n| (which is exactly the difference between the numbers of infinite regions). All the rows and columns satisfy the recurrence u(n) = 3*u(n-1) - 3*u(n-2) + u(n-3).

			Array begins (the rows are T(0,n>=0),, T(1,n>=0), T(2,n>=0), ...):
  1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
  1, 2, 4, 7, 11, 16, 22, 29, 37, ...
  1, 2, 7, 16, 29, 46, 67, 92, 121, ...
  1, 3, 14, 34, 63, 101, 148, 204, 269, ...
  1, 5, 25, 61, 113, 181, 265, 365, 481, ...
  1, 8, 40, 97, 179, 286, 418, 575, 757, ...
  1, 12, 59, 142, 261, 416, 607, 834, 1097, ...
  1, 17, 82, 196, 359, 571, 832, 1142, 1501, ...
  1, 23, 109, 259, 473, 751, 1093, 1499, 1969, ...
  ...
The first few antidiagonals are:
  1,
  1, 1,
  1, 2, 1,
  1, 2, 4, 1,
  1, 3, 7, 7, 1,
  1, 5, 14, 16, 11, 1,
  1, 8, 25, 34, 29, 16, 1,
  1, 12, 40, 61, 63, 46, 22, 1,
  ...

David O. H. Cutler and N. J. A. Sloane, paper in preparation, August 1 2025.

Paolo Xausa, Table of n, a(n) for n = 0..11324 (first 150 antidiagonals, flattened).

The first few rows are A000124, A130883, A140064, A080856, A383465.

The n=1 and 2 columns are A152948 and A386479.

A386478[k_, n_] := If[k == 0, 1, ((k*n - 3)*k + 4)*n/2 + 1];
Table[A386478[k - n, n], {k, 0, 12}, {n, 0, k}] (* Paolo Xausa, Jul 26 2025 *)

Row 0 added by N. J. A. Sloane, Jul 26 2025

A214230 Sum of the eight nearest neighbors of n in a right triangular type-1 spiral with positive integers.

Original entry on oeis.org

53, 88, 78, 125, 85, 84, 125, 97, 108, 143, 223, 168, 158, 169, 201, 284, 208, 183, 179, 187, 210, 281, 226, 219, 227, 235, 261, 314, 430, 339, 311, 310, 318, 326, 346, 396, 515, 403, 360, 347, 355, 363, 371, 379, 411, 509, 427, 411, 419, 427, 435, 443, 451, 486, 557

Offset: 1

Alex Ratushnyak, Jul 08 2012

Right triangular type-1 spiral implements the sequence Up, Right-down, Left.
Right triangular type-2 spiral (A214251): Left, Up, Right-down.
Right triangular type-3 spiral (A214252): Right-down, Left, Up.
A140064 -- rightwards from 1: 3,14,34...
A064225 -- leftwards from 1: 8,24,49...
A117625 -- upwards from 1: 2,12,31...
A006137 -- downwards from 1: 6,20,43...
A038764 -- left-down from 1: 7,22,46...
A081267 -- left-up from 1: 9,26,52...
A081589 -- right-up from 1: 13, 61, 145...
9*x^2/2 - 19*x/2 + 6 -- right-down from 1: 5,18,40...

			Right triangular spiral begins:
56
55  57
54  29  58
53  28  30  59
52  27  11  31  60
51  26  10  12  32  61
50  25   9   2  13  33  62
49  24   8   1   3  14  34  63
48  23   7   6   5   4  15  35  64
47  22  21  20  19  18  17  16  36  65
46  45  44  43  42  41  40  39  38  37  66
78  77  76  75  74  73  72  71  70  69  68  67
The eight nearest neighbors of 3 are 1, 2, 13, 33, 14, 4, 5, 6. Their sum is a(3)=78.

David Radcliffe, Table of n, a(n) for n = 1..10000

Cf. A214251, A214252.

Cf. A214226, A214231.

SIZE=29  # must be odd
grid = [0] * (SIZE*SIZE)
saveX = [0]* (SIZE*SIZE)
saveY = [0]* (SIZE*SIZE)
saveX[1] = saveY[1] = posX = posY = SIZE//2
grid[posY*SIZE+posX]=1
n = 2
def walk(stepX,stepY,chkX,chkY):
  global posX, posY, n
  while 1:
    posX+=stepX
    posY+=stepY
    grid[posY*SIZE+posX]=n
    saveX[n]=posX
    saveY[n]=posY
    n+=1
    if posY==0 or grid[(posY+chkY)*SIZE+posX+chkX]==0:
        return
while 1:
    walk(0, -1,  1,  1)    # up
    if posY==0:
        break
    walk( 1, 1, -1,  0)    # right-down
    walk(-1, 0,  0, -1)    # left
for n in range(1,92):
    posX = saveX[n]
    posY = saveY[n]
    k = grid[(posY-1)*SIZE+posX] + grid[(posY+1)*SIZE+posX]
    k+= grid[(posY-1)*SIZE+posX-1] + grid[(posY-1)*SIZE+posX+1]
    k+= grid[(posY+1)*SIZE+posX-1] + grid[(posY+1)*SIZE+posX+1]
    k+= grid[posY*SIZE+posX-1] + grid[posY*SIZE+posX+1]
    print(k, end=', ')

A192136 a(n) = (5n^2 - 3n + 2)/2.

Original entry on oeis.org

1, 2, 8, 19, 35, 56, 82, 113, 149, 190, 236, 287, 343, 404, 470, 541, 617, 698, 784, 875, 971, 1072, 1178, 1289, 1405, 1526, 1652, 1783, 1919, 2060, 2206, 2357, 2513, 2674, 2840, 3011, 3187, 3368, 3554, 3745, 3941, 4142, 4348, 4559, 4775, 4996, 5222, 5453, 5689

Offset: 0

Eric Werley, Jun 24 2011

Binomial transform of [1, 1, 5, 0, 0, 0, 0, 0, ...]. - Johannes W. Meijer, Jul 07 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000124, A000217, A002522, A006137, A130883, A140063, A140064, A143689.

A192136:=func< n | (5*n^2-3*n+2)/2 >; [ A192136(n): n in [0..50] ]; // Klaus Brockhaus, Jun 27 2011

Table[(5n^2-3n+2)/2,{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{1,2,8},50] (* Harvey P. Dale, Aug 08 2016 *)

a(n)=n*(5*n-3)/2+1 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = (5*n^2 - 3*n + 2)/2.
a(n) = 2*a(n-1) - a(n-2) + 5.
a(n) = a(n-1) + 5*n - 4.
a(n) = 5*binomial(n+2,2) - 9*n - 4.
a(n) = A000217(n+1) - A000217(n) + 5*A000217(n-1); triangular numbers. - Johannes W. Meijer, Jul 07 2011
O.g.f.: (1-x+5*x^2)/(1-x)^3.
From Elmo R. Oliveira, Nov 16 2024: (Start)
E.g.f.: exp(x)*(2 + 2*x + 5*x^2)/2.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

A276819 a(n) = (9*n^2 - n)/2 + 1.

Original entry on oeis.org

1, 5, 18, 40, 71, 111, 160, 218, 285, 361, 446, 540, 643, 755, 876, 1006, 1145, 1293, 1450, 1616, 1791, 1975, 2168, 2370, 2581, 2801, 3030, 3268, 3515, 3771, 4036, 4310, 4593, 4885, 5186, 5496, 5815, 6143, 6480, 6826, 7181, 7545, 7918, 8300, 8691, 9091, 9500, 9918, 10345, 10781, 11226, 11680, 12143, 12615

Offset: 0

Yuriy Sibirmovsky, Sep 18 2016

Diagonal of triangular spiral in A051682. The other 5 diagonals are given by A140064, A117625, A081267, A064225, A006137. See the link as well.
First differences are given by A017209.
72*a(n) - 71 is a perfect square. - Klaus Purath, Jan 14 2022

Colin Barker, Table of n, a(n) for n = 0..1000
Yuriy Sibirmovsky, Six diagonals of the triangular spiral.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A002378, A003215, A006137, A017209, A051682, A064225, A081267, A117625, A140064.

Mathematica
```
Table[(9*n^2-n)/2+1, {n,0,100}]
```

Vec((1+2*x+6*x^2)/(1-x)^3 + O(x^60)) \\ Colin Barker, Sep 18 2016

a(n) = (9*n^2 - n)/2 + 1; \\ Altug Alkan, Sep 18 2016

a(n) = (9*n^2 - n)/2 + 1.
a(n) = a(n-1) + 9*n - 5 with a(0) = 1.
From Colin Barker, Sep 18 2016: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2.
G.f.: (1 + 2*x + 6*x^2)/(1 - x)^3. (End)
From Klaus Purath, Jan 14 2022: (Start)
a(n) = A006137(n) - n.
A003215(a(n)) - A003215(a(n)-3) = A002378(9*n-1). (End)
E.g.f.: exp(x)*(2 + 8*x + 9*x^2)/2. - Stefano Spezia, Dec 25 2022

A383464 a(n) = 8n^2 - 5n + 1.

Original entry on oeis.org

1, 4, 23, 58, 109, 176, 259, 358, 473, 604, 751, 914, 1093, 1288, 1499, 1726, 1969, 2228, 2503, 2794, 3101, 3424, 3763, 4118, 4489, 4876, 5279, 5698, 6133, 6584, 7051, 7534, 8033, 8548, 9079, 9626, 10189, 10768, 11363, 11974, 12601, 13244, 13903, 14578, 15269

Offset: 0

N. J. A. Sloane, Jun 26 2025

This is equal to A139272(n) + 1, but has its own entry because of an important geometrical interpretation.
Definition: A k-legged Wu is a pencil of k semi-infinite lines originating from a common point.
A 2-legged Wu is a long-legged V (see A130883), and a 3-legged Wu is a long-legged Wu as in A140064.
Theorem (David Cutler, Jonathan Pei, and Edward Xiong, Jun 24 2025): a(n) is the maximum number of regions in the plane that can be formed from n copies of a 4-legged Wu.
Proof: [To be added]

Vincenzo Librandi, Table of n, a(n) for n = 0..3499
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A130083, A139272, A140064.

I:=[1, 4, 23]; [n le 3 select I[n] else 3*Self(n-1)-3* Self(n-2)+Self(n-3): n in [1..40]]; // Vincenzo Librandi, Jun 27 2025

LinearRecurrence[{3,-3,1},{1,4,23},50] (* Vincenzo Librandi, Jun 27 2025 *)

G.f.: (1 + x + 14*x^2)/(1 - x)^3.

E.g.f.: exp(x)*(1 + 3*x + 8*x^2). - Stefano Spezia, Jun 30 2025

A140065 a(n) = (7n^2 - 17n + 12)/2.

Original entry on oeis.org

1, 3, 12, 28, 51, 81, 118, 162, 213, 271, 336, 408, 487, 573, 666, 766, 873, 987, 1108, 1236, 1371, 1513, 1662, 1818, 1981, 2151, 2328, 2512, 2703, 2901, 3106, 3318, 3537, 3763, 3996, 4236, 4483, 4737, 4998, 5266, 5541, 5823, 6112, 6408, 6711, 7021, 7338, 7662

Offset: 1

Gary W. Adamson, May 03 2008

Binomial transform of [1, 2, 7, 0, 0, 0, ...].

This sequence together with 1, 6, 18, 37, 63, 96, ... with signature (3,-3,1) [not yet in OEIS] contain all numbers k such that 56*k - 47 is a square. - Klaus Purath, Oct 21 2021

			a(4) = 28 = (1, 3, 3, 1) * (1, 2, 7, 0) = (1 + 6 + 21 + 0).

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000217, A007318, A140064, A140066.

[(7*n^2 - 17*n + 12)/2 : n in [1..60]]; // Wesley Ivan Hurt, Oct 10 2021

seq((12-17*n+7*n^2)*1/2, n=1..40); # Emeric Deutsch, May 07 2008

Table[(7 n^2 - 17 n + 12)/2, {n, 1, 50}] (* Bruno Berselli, Mar 12 2015 *)
LinearRecurrence[{3,-3,1},{1,3,12},50] (* Harvey P. Dale, May 28 2017 *)

x = 'x + O('x^50); Vec(x*(1+6*x^2)/(1-x)^3) \\ G. C. Greubel, Feb 23 2017

A007318 * [1, 2, 7, 0, 0, 0, ...].
a(n) = A000217(n) + 6*A000217(n-2) = (A140064(n) + A140066(n))/2. - R. J. Mathar, May 06 2008
O.g.f.: x*(1+6*x^2)/(1-x)^3. - Alexander R. Povolotsky, May 06 2008
a(n) = 7*n + a(n-1) - 12 for n > 1, a(1)=1. - Vincenzo Librandi, Jul 08 2010
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3), n >= 4. - Klaus Purath, Oct 21 2021
E.g.f.: exp(x)*(6 - 5*x + 7*x^2/2) - 6. - Elmo R. Oliveira, Oct 31 2024

More terms from R. J. Mathar and Emeric Deutsch, May 06 2008

More terms from Vladimir Joseph Stephan Orlovsky, Oct 25 2008

A386481 Array read by upward antidiagonals: T(k,n) = 1 (k = 0, n >= 0), T(k,n) = binomial(n,2)k^2 + n(k-1) + 1 (k >= 1, n >= 0).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 3, 7, 4, 1, 1, 4, 14, 16, 7, 1, 1, 5, 23, 34, 29, 11, 1, 1, 6, 34, 58, 63, 46, 16, 1, 1, 7, 47, 88, 109, 101, 67, 22, 1, 1, 8, 62, 124, 167, 176, 148, 92, 29, 1, 1, 9, 79, 166, 237, 271, 259, 204, 121, 37, 1, 1, 10, 98, 214, 319, 386, 400, 358, 269, 154, 46, 1, 1, 11, 119, 268, 413, 521, 571, 554, 473, 343, 191, 56, 1

Offset: 0

N. J. A. Sloane, Aug 11 2025

T(k,n) is the maximum number of regions the plane can be divided into by drawing n k-armed long-legged V's.

			Array begins (the rows are T(0,n>=0),, T(1,n>=0), T(2,n>=0), ...):
   1, 1, 1, 1, 1, 1, 1, 1, 1, ...
   1, 1, 2, 4, 7, 11, 16, 22, 29, ...
   1, 2, 7, 16, 29, 46, 67, 92, 121, ...
   1, 3, 14, 34, 63, 101, 148, 204, 269, ...
   1, 4, 23, 58, 109, 176, 259, 358, 473, ...
   1, 5, 34, 88, 167, 271, 400, 554, 733, ...
   1, 6, 47, 124, 237, 386, 571, 792, 1049, ...
   1, 7, 62, 166, 319, 521, 772, 1072, 1421, ...
    ...
The first few antidiagonals are:
    1,
    1, 1,
    1, 1, 1,
    1, 2, 2, 1,
    1, 3, 7, 4, 1,
    1, 4, 14, 16, 7, 1,
    1, 5, 23, 34, 29, 11, 1,
    1, 6, 34, 58, 63, 46, 16, 1,
    1, 7, 47, 88, 109, 101, 67, 22, 1,
     ...

David O. H. Cutler and N. J. A. Sloane, paper in preparation, August 1 2025.

N. J. A. Sloane, Illustration for T(3,2) = 14.
N. J. A. Sloane, Illustration for T(3,3) = 34 (a 3-armed V is also known as a Wu).

This is a companion to the array A386478.

The rows and columns include A000124, A130883, A140064, A383464, and A008865.

Under construction, please do not touch.

cp's OEIS Frontend

A386478 Array read by upward antidiagonals: T(k,n) = 1 (k = 0, n >= 0), T(k,n) = k^2*n^2/2 - (3*k-4)*n/2 + 1 (k >= 1, n >= 0).

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A214230 Sum of the eight nearest neighbors of n in a right triangular type-1 spiral with positive integers.

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A192136 a(n) = (5*n^2 - 3*n + 2)/2.

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A276819 a(n) = (9*n^2 - n)/2 + 1.

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A383464 a(n) = 8*n^2 - 5*n + 1.

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A140065 a(n) = (7*n^2 - 17*n + 12)/2.

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A386481 Array read by upward antidiagonals: T(k,n) = 1 (k = 0, n >= 0), T(k,n) = binomial(n,2)*k^2 + n*(k-1) + 1 (k >= 1, n >= 0).

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A386478 Array read by upward antidiagonals: T(k,n) = 1 (k = 0, n >= 0), T(k,n) = k^2n^2/2 - (3k-4)*n/2 + 1 (k >= 1, n >= 0).

A192136 a(n) = (5n^2 - 3n + 2)/2.

A383464 a(n) = 8n^2 - 5n + 1.

A140065 a(n) = (7n^2 - 17n + 12)/2.

A386481 Array read by upward antidiagonals: T(k,n) = 1 (k = 0, n >= 0), T(k,n) = binomial(n,2)k^2 + n(k-1) + 1 (k >= 1, n >= 0).