A006137 -id:A006137 - cp's OEIS frontend

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

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

A358994 The sum of the numbers that are inside the contour of an n-story Christmas tree drawn at the top of the numerical pyramid containing the positive integers in natural order.

Original entry on oeis.org

21, 151, 561, 1503, 3310, 6396, 11256, 18466, 28683, 42645, 61171, 85161, 115596, 153538, 200130, 256596, 324241, 404451, 498693, 608515, 735546, 881496, 1048156, 1237398, 1451175, 1691521, 1960551, 2260461, 2593528, 2962110, 3368646, 3815656, 4305741, 4841583, 5425945

Offset: 1

Nicolay Avilov, Dec 25 2022

The numbers of the natural series are written line by line in the form of a numerical pyramid: the first line contains the number 1, the second line contains the next two numbers 2 and 3, the third line contains the next three numbers 4, 5 and 6, etc.; that is, the line starting with the number k contains the k following numbers. In this numerical pyramid, the contour of a "multi-story Christmas tree" is distinguished, each floor of which occupies three lines. The numbers of the sequence are the sum of all the numbers that fall into the contour of the Christmas tree, which has n floors.

			a(1) = 1 + 2 + 3 + 4 + 5 + 6 = 21;
a(2) = a(1) + (8 + 9 + 12 + 13 + 14 + 17 +18 + 19 + 20) = 151.

Nicolay Avilov, Problem 2128 (in Russian).
Nicolay Avilov, Explanatory drawing
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A001844, A022266, A006137.

[n*(27*n^3 + 66*n^2 + 49*n + 26)/8 : n in [1..60]]; // Wesley Ivan Hurt, Jun 14 2025

def a(n): return n*(27*n**3 + 66*n**2 + 49*n + 26) // 8
print([a(n) for n in range(1, 36)]) # Michael S. Branicky, Dec 25 2022

a(n) = n*(27*n^3 + 66*n^2 + 49*n + 26) / 8.
G.f.: x*(21 + 46*x + 16*x^2 - 2*x^3)/(1 - x)^5. - Stefano Spezia, Dec 25 2022
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - Wesley Ivan Hurt, Jun 14 2025

cp's OEIS Frontend

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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A358994 The sum of the numbers that are inside the contour of an n-story Christmas tree drawn at the top of the numerical pyramid containing the positive integers in natural order.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Python

Formula

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