A081267 -id:A081267 - cp's OEIS frontend

A064225 a(n) = (9n^2 + 5n + 2)/2.

1, 8, 24, 49, 83, 126, 178, 239, 309, 388, 476, 573, 679, 794, 918, 1051, 1193, 1344, 1504, 1673, 1851, 2038, 2234, 2439, 2653, 2876, 3108, 3349, 3599, 3858, 4126, 4403, 4689, 4984, 5288, 5601, 5923, 6254, 6594, 6943, 7301, 7668, 8044, 8429, 8823, 9226

Offset: 0

N. J. A. Sloane, Sep 22 2001

Diagonal of triangular spiral in A051682. - Michael Somos, Jul 22 2006
Ehrhart polynomial of closed quadrilateral with vertices (0,2),(2,3),(3,1),(2,0). - Michael Somos, Jul 22 2006
In the natural number array A000027 this sequence is the first knight moves diagonal (A081267 is the second, A001844 is the main diagonal). It can be used to define this diagonal for any array: A007318(A064225-1)=A005809 (Subtraction by 1 because A007318 is defined with offset 0.) - Tilman Piesk, Mar 24 2012
Or positions of pentagonal numbers, such that p(a(n)) = p(a(n)-1) + p(3*n+1), where p=A000326. - Vladimir Shevelev, Jan 21 2014

			Illustration of initial terms:
.
.                                    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 o o o
.                      o       o o o o
.                                 o o
.                                    o
.
.   1     8        24           49
- _Aaron David Fairbanks_, Feb 23 2025

Harry J. Smith, Table of n, a(n) for n = 0..1000
National Security Agency, Intrigued? (advertisement), Notices of the Amer. Math. Soc., vol. 49 (2002), p. 216.
J. A. Siehler, Selections without adjacency on a rectangular grid, arXiv:1409.3869, Table 3, k=2 (different offset)
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000027, A000326, A001844, A005809, A007318, A051682, A064226, A081267, A235332.

Column w=2 of A371967.

Table[(9n^2+5n+2)/2,{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{1,8,24},51] (* Harvey P. Dale, Sep 13 2011 *)

{a(n) = 1 + n * (9*n + 5) / 2}; /* Michael Somos, Jul 22 2006 */

(define (A064225 n) (/ (+ (* 9 n n) (* 5 n) 2) 2))

a(n) = 9*n+a(n-1)-2, with n>0, a(0) = 1. - Vincenzo Librandi, Aug 07 2010
a(0)=1, a(1)=8, a(2)=24, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Sep 13 2011
G.f.: (1+5*x+3*x^2)/(1-x)^3. - Colin Barker, Feb 23 2012
A064226(n) = a(-1-n). - Michael Somos, Jul 22 2006 (While the sequence itself is only one-way infinite, this identity works, as the defining formula (in the Name-field) produces integers also for the negative values of n, -1, -2, -3, etc.) - Antti Karttunen, Mar 24 2012
E.g.f.: exp(x)*(2 + 14*x + 9*x^2)/2. - Stefano Spezia, Dec 25 2022

A064226 a(n) = (9n^2 + 13n + 6)/2.

Original entry on oeis.org

3, 14, 34, 63, 101, 148, 204, 269, 343, 426, 518, 619, 729, 848, 976, 1113, 1259, 1414, 1578, 1751, 1933, 2124, 2324, 2533, 2751, 2978, 3214, 3459, 3713, 3976, 4248, 4529, 4819, 5118, 5426, 5743, 6069, 6404, 6748, 7101, 7463, 7834, 8214, 8603, 9001, 9408, 9824

Offset: 0

N. J. A. Sloane, Sep 22 2001

Diagonal of triangular spiral in A051682. - Paul Barry, Mar 15 2003

Ehrhart polynomial of open quadrilateral with vertices (0,2),(2,3),(3,1),(2,0). - Michael Somos, Jul 22 2006

Harry J. Smith, Table of n, a(n) for n = 0..1000
Milan Janjic, Two Enumerative Functions.
National Security Agency, Intrigued? (advertisement), Notices of the American Mathematical Society, Vol. 49 (2002), p. 216.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A051682, A064225, A081267, A081268, A235332.

I:=[3,14,34]; [n le 3 select I[n] else 3*Self(n-1) - 3*Self(n-2) + Self(n-3): n in [1..50]]; // Vincenzo Librandi, Jul 19 2015

A064226:=n-> (9*n^2 + 13*n + 6) / 2; seq(A064226(n), n=0..50); # Wesley Ivan Hurt, May 08 2014

Table[(9 n^2 + 13 n + 6)/2, {n, 0, 50}] (* Wesley Ivan Hurt, May 08 2014 *)
LinearRecurrence[{3, -3, 1}, {3, 14, 34}, 50] (* Vincenzo Librandi, Jul 19 2015 *)

{a(n) = 3 + n * (9*n + 13) / 2}; /* Michael Somos, Jul 22 2006 */

From Paul Barry, Mar 15 2003: (Start)
a(n) = 3*C(n,0) + 11*C(n,1) + 9*C(n,2); binomial transform of (3, 11, 9, 0, 0, 0, ...).
G.f.: (3 + 5*x + x^2)/(1-x)^3.
a(n) = A081268(n) + 2. (End)
A064225(n) = a(-1-n). -  Michael Somos, Jul 22 2006
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Wesley Ivan Hurt, Apr 16 2023
E.g.f.: (3 + 11*x + 9*x^2/2)*exp(x). - Elmo R. Oliveira, Oct 21 2024

A220083 a(n) = (15n^2 + 9n + 2)/2.

Original entry on oeis.org

1, 13, 40, 82, 139, 211, 298, 400, 517, 649, 796, 958, 1135, 1327, 1534, 1756, 1993, 2245, 2512, 2794, 3091, 3403, 3730, 4072, 4429, 4801, 5188, 5590, 6007, 6439, 6886, 7348, 7825, 8317, 8824, 9346, 9883, 10435, 11002, 11584, 12181, 12793, 13420, 14062

Offset: 0

Bruno Berselli, Dec 10 2012

Sequence related to the heptagonal numbers (A000566) by a(n) = n*A000566(n)-(n-1)*A000566(n-1).
Other similar sequences:
A005408(m) = (m+1)*A001477(m+1)-m*A001477(m), A001477 = nonn. integers;
A000326(m) = m*A000217(n)-(m-1)*A000217(m-1), A000217 = triangular numbers;
A003215(m) = (m+1)*A000290(m+1)-n*A000290(m), A000290 = square numbers;
A081267(m) = (m+1)*A000326(m+1)-n*A000326(m), A000326 = pentagonal numbers;
A080859(m) = (m+1)*A000384(m+1)-n*A000384(m), A000384 = hexagonal numbers;
A214675(m) = m*A000567(m)-(m-1)*A000567(m-1), A000567 = octagonal numbers.

Bruno Berselli, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000326, A003215, A005408, A006597, A080859, A081267, A214675.

/* By first comment: */  A000566:=func; [n*A000566(n)-(n-1)*A000566(n-1): n in [1..45]];

[(15*n^2 + 9*n + 2)/2: n in [0..50]]; // Vincenzo Librandi, Aug 18 2013

A220083:=n->(15*n^2 + 9*n + 2)/2; seq(A220083(n), n=0..100); # Wesley Ivan Hurt, Nov 14 2013

Table[(15 n^2 + 9 n + 2)/2, {n, 0, 45}]
CoefficientList[Series[(1 + 10 x + 4 x^2) / (1 - x)^3, {x, 0, 50}], x] (* Vincenzo Librandi, Aug 18 2013 *)

A220083(n):=(15*n^2 + 9*n + 2)/2$ makelist(A220083(n),n,0,20); /* Martin Ettl, Dec 11 2012 */

a(n)=(15*n^2+9*n+2)/2 \\ Charles R Greathouse IV, Oct 07 2015

G.f.: (1+10*x+4*x^2)/(1-x)^3.
Sum( a(i), i=0..n ) = A006597(n+1).
a(n) + a(-n) = A010005(n) for n>0.

A235332 a(n) = n(9n + 25)/2 + 6.

Original entry on oeis.org

6, 23, 49, 84, 128, 181, 243, 314, 394, 483, 581, 688, 804, 929, 1063, 1206, 1358, 1519, 1689, 1868, 2056, 2253, 2459, 2674, 2898, 3131, 3373, 3624, 3884, 4153, 4431, 4718, 5014, 5319, 5633, 5956, 6288, 6629, 6979, 7338, 7706, 8083, 8469, 8864, 9268, 9681, 10103

Offset: 0

Bruno Berselli, Jan 22 2014

This is the case d=6 of n*(9*n + 4*d + 1)/2 + d. Other similar sequences are:
d=0, A022267;
d=1, A064225;
d=2, A062123;
d=3, A064226;
d=4, A022266 (with initial 0);
d=5, A178977.
First bisection of A235537.

Bruno Berselli, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A017257 (first differences), A022266, A022267, A062123, A064225, A064226, A081267, A178977, A235537.

Magma
```
[n*(9*n+25)/2+6: n in [0..50]];
```

Table[n (9 n + 25)/2 + 6, {n, 0, 50}]
LinearRecurrence[{3,-3,1},{6,23,49},50] (* Harvey P. Dale, Feb 12 2022 *)

a(n)=n*(9*n+25)/2+6 \\ Charles R Greathouse IV, Oct 07 2015

G.f.: (6 + 5*x - 2*x^2)/(1 - x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
2*a(n) - a(n+1) + 12 = A081267(n).
E.g.f.: exp(x)*(12 + 34*x + 9*x^2)/2. - Elmo R. Oliveira, Nov 13 2024

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=', ')

A081268 Diagonal of triangular spiral in A051682.

Original entry on oeis.org

1, 12, 32, 61, 99, 146, 202, 267, 341, 424, 516, 617, 727, 846, 974, 1111, 1257, 1412, 1576, 1749, 1931, 2122, 2322, 2531, 2749, 2976, 3212, 3457, 3711, 3974, 4246, 4527, 4817, 5116, 5424, 5741, 6067, 6402, 6746, 7099, 7461, 7832, 8212, 8601, 8999, 9406

Offset: 0

Paul Barry, Mar 15 2003

			a(1) = 9*1 +  1 + 2 = 12.
a(2) = 9*2 + 12 + 2 = 32.
a(3) = 9*3 + 32 + 2 = 61.

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

Cf. A064226, A081267.

a(n)=(9*n^2+13*n+2)/2 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = C(n,0) + 11*C(n,1) + 9*C(n,2); binomial transform of (1, 11, 9, 0, 0, 0, ...).
a(n) = (9*n^2 + 13*n + 2)/2.
G.f.: (1 + 9*x - x^2)/(1-x)^3.
a(n) = 9*n + a(n-1) + 2 (with a(0)=1). - Vincenzo Librandi, Aug 08 2010
From Elmo R. Oliveira, Nov 16 2024: (Start)
E.g.f.: exp(x)*(9*x^2 + 22*x + 2)/2.
a(n) = A064226(n) - 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

A239304 Triangle of permutations corresponding to the compressed square roots of Gray code * bit-reversal permutation (A239303).

Original entry on oeis.org

1, 1, 2, 3, 1, 2, 4, 2, 1, 3, 2, 5, 4, 1, 3, 2, 5, 6, 3, 1, 4, 6, 2, 3, 7, 5, 1, 4, 7, 3, 2, 6, 8, 4, 1, 5, 3, 8, 7, 2, 4, 9, 6, 1, 5, 3, 8, 9, 4, 2, 7, 10, 5, 1, 6, 9, 3, 4, 10, 8, 2, 5, 11, 7, 1, 6, 10, 4, 3, 9

Offset: 1

Tilman Piesk, Mar 14 2014

The symmetrical binary matrices corresponding to the rows of A239303 can be interpreted as adjacency matrices of undirected graphs. These graphs are chains where one end is connected to itself, so they can be interpreted as permutations. The end connected to itself is always the first element of the permutation, i.e., on the left side of the triangle.
Columns of the square array:
T(m,1) = A008619(m) = 1,2,2,3,3...
T(m,2) = 1,1,1...
T(m,3) = A028242(m+3) = 3,2,4,3,5,4,6,5,7,6,8,7,9,8,10,9,11,10,12...
T(m,4) = m+3 = 4,5,6...
T(m,5) = A084964(m+4) = 2,5,3,6,4,7,5,8,6,9,7,10,8,11,9,12,10,13...
T(m,6) = 2,2,2...
T(m,7) = A168230(m+5) = 6,3,7,4,8,5,9,6,10,7,11,8,12,9,13,10,14...
T(m,8) = m+6 = 7,8,9...
T(m,9) = A152832(m+9) = 3,8,4,9,5,10,6,11,7,12,8,13,9,14,10,15...
T(m,10) = 3,3,3...
Diagonals of the square array:
T(n,n) = a(A001844(n)) = 1,1,4,7,4,2,9,14,7,3,14,21,10,4,19,28,13,5,24...
T(n,2n-1) = a(A064225(n)) = 1,2,3...
T(2n-1,n) = a(A081267(n)) = 1,1,5,10,6,2,12,21,11,3,19,32,16,4,26,43,21...

			Triangular array begins:
  1
  1 2
  3 1 2
  4 2 1 3
  2 5 4 1 3
  2 5 6 3 1 4
Square array begins:
  1 1 3 4 2 2
  2 1 2 5 5 2
  2 1 4 6 3 2
  3 1 3 7 6 2
  3 1 5 8 4 2
  4 1 4 9 7 2
Row 5 of A239303 is the vector (12,18,1,17,10), which corresponds to the following binary matrix:
  0 0 1 1 0
  0 1 0 0 1
  1 0 0 0 0
  1 0 0 0 1
  0 1 0 1 0
Interpreted as an adjacency matrix it describes the following graph, where each number is connected to its neighbors, and only the 2 is connected to itself:
  2 5 4 1 3
This is row 5 of the triangular array.

Tilman Piesk, First 140 rows of the triangle, flattened
Tilman Piesk, Sequency ordered Walsh matrix (Wikiversity)
Tilman Piesk, Calculation in MATLAB

Cf. A239303, A028242, A084964, A168230, A152832.

cp's OEIS Frontend

A064225 a(n) = (9*n^2 + 5*n + 2)/2.

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A064226 a(n) = (9*n^2 + 13*n + 6)/2.

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A220083 a(n) = (15*n^2 + 9*n + 2)/2.

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A235332 a(n) = n*(9*n + 25)/2 + 6.

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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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A081268 Diagonal of triangular spiral in A051682.

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

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A064225 a(n) = (9n^2 + 5n + 2)/2.

A064226 a(n) = (9n^2 + 13n + 6)/2.

A220083 a(n) = (15n^2 + 9n + 2)/2.

A235332 a(n) = n(9n + 25)/2 + 6.