A117625 -id:A117625 - cp's OEIS frontend

A130883 a(n) = 2*n^2 - n + 1.

1, 2, 7, 16, 29, 46, 67, 92, 121, 154, 191, 232, 277, 326, 379, 436, 497, 562, 631, 704, 781, 862, 947, 1036, 1129, 1226, 1327, 1432, 1541, 1654, 1771, 1892, 2017, 2146, 2279, 2416, 2557, 2702, 2851, 3004, 3161, 3322, 3487, 3656, 3829, 4006, 4187, 4372, 4561

Offset: 0

Mohammad K. Azarian, Jul 26 2007

Maximum number of regions determined by n bent lines (or angular sectors). See Concrete Mathematics reference.
A "bent line" may also be regarded as a "long-legged letter V", meaning a letter V with both line segments extended to infinity.  See A117625 for the analogous sequence for a long-legged Z. - N. J. A. Sloane, Jun 18 2025
a(n)*Pi is the total length of half circle spiral after n rotations. It is formed as irregular spiral with two center points. At the 2nd stage, there are two alternatives: (1) select 2nd half circle radius, r2  = 2, the sequence will be A014105 or (2) select r2 = 0, the sequence will be A130883. See illustration in links. - Kival Ngaokrajang, Jan 19 2014
A128218(a(n)) = 2*n+1 and A128218(m) != 2*n+1 for m < a(n). - Reinhard Zumkeller, Jun 20 2015

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, 2nd ed., Addison-Wesley, Reading, MA, 1994, pp. 7-8, and Problem 1.18, pages 19 and 500.

G. C. Greubel, Table of n, a(n) for n = 0..5000
Dmitry Efimov, Hafnian of two-parameter matrices, arXiv:2101.09722 [math.CO], 2021.
Guo-Niu Han, Enumeration of Standard Puzzles
Guo-Niu Han, Enumeration of Standard Puzzles [Cached copy]
Ângela Mestre and José Agapito, Square Matrices Generated by Sequences of Riordan Arrays, J. Int. Seq., Vol. 22 (2019), Article 19.8.4.
Kival Ngaokrajang, Illustration of irregular spirals (center points: 1, 2)
Franck Ramaharo, Statistics on some classes of knot shadows, arXiv:1802.07701 [math.CO], 2018.
Franck Ramaharo, A generating polynomial for the pretzel knot, arXiv:1805.10680 [math.CO], 2018.
N. J. A. Sloane, Illustration for a(3) = 16
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000124, A000384, A014105, A084849, A128218, A128918, A152947, A270109.
See also A117625.
A row of the array in A386478.

a130883 = a128918 . (* 2)  -- Reinhard Zumkeller, Oct 27 2013

[2*n^2 - n + 1 : n in [0..50]]; // Wesley Ivan Hurt, Mar 25 2020

a[n_]:=2*n^2-n+1; (* or *) Array[ -#*(1-#*2)+1&,5!,0] (* Vladimir Joseph Stephan Orlovsky, Dec 21 2008 *)
LinearRecurrence[{3,-3,1},{1,2,7},50] (* Harvey P. Dale, Jul 20 2011 *)

a(n)=2*n^2-n+1 \\ Charles R Greathouse IV, Sep 24 2015

def A130883(n): return n*(2*n - 1) + 1 # Chai Wah Wu, May 24 2022

a(n) = a(n-1) + 4*n - 3 for n > 0, a(0)=1. - Vincenzo Librandi, Nov 23 2010
a(n) = A000124(2*n) - 2*n. - Geoffrey Critzer, Mar 30 2011
O.g.f.: (4*x^2-x+1)/(1-x)^3. - Geoffrey Critzer, Mar 30 2011
a(n) = 2*a(n-1) - a(n-2) + 4. - Eric Werley, Jun 27 2011
a(0)=1, a(1)=2, a(2)=7; for n > 2, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Jul 20 2011
a(n) = A128918(2*n). - Reinhard Zumkeller, Oct 27 2013
a(n) = 1 + A000384(n). - Omar E. Pol, Apr 27 2017
E.g.f.: (2*x^2 + x + 1)*exp(x). - G. C. Greubel, Jul 14 2017
a(n) = A152947(2*n+1). - Franck Maminirina Ramaharo, Jan 10 2018

A140064 a(n) = (9n^2 - 5n + 2)/2.

Original entry on oeis.org

1, 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, 10249, 10683, 11126, 11578, 12039, 12509, 12988, 13476, 13973

Offset: 0

Gary W. Adamson, May 03 2008

Originally this entry was defined by a(n) = (9*n^2 - 23*n + 16)/2 and had offset 1. The current, simpler definition seems preferable, since it matches the following two geometrical applications. This change will also require several changes to the rest of the entry. - N. J. A. Sloane, Jun 26 2025
The letter Wu, ᗐ, is like a V but with three arms instead of two. Wu is included in the Unified Canadian Aboriginal Syllabics section of Unicode. The Unicode symbol for Wu is 0x2a5b. Wu is also called a "Boolean OR with middle stem", and is also the alchemical symbol Dissolve-2.
The long-legged Wu is a pencil of three semi-infinite lines originating from a point (the "tip"). The angles between the three lines are arbitrary.
Theorem 1 (Edward Xiong, Jonathan Pei, and David Cutler, Jun 24 2025): a(n) is the maximum number of regions in the plane that can be formed from n copies of a long-legged Wu.
Theorem 2: a(n) is also the maximum number of regions in the plane that can be formed from n copies of a long-legged letter A.
For proofs of Theorems 1 and 2 see "The Pancake, Hatpin, and Wu Planar Graphs".
For analogous sequences for long-legged letters V and Z see A130883 and A117625.

N. J. A. Sloane, Table of n, a(n) for n = 0..5000 [First 1000 terms from G. C. Greubel]
N. J. A. Sloane, The long-legged letters A, I, V, X, and Z. The long-legged A is a long-legged V with a crossbar. The distances from the tip of the A to the end-points of the crossbar need not be equal.
N. J. A. Sloane, 14 regions using 2 copies of the Wu graph
N. J. A. Sloane, 34 regions using 3 copies of the Wu graph
N. J. A. Sloane, Transforming a Hatpin graph to a Wu graph.
N. J. A. Sloane, 14 regions using 2 long-legged A's [Crossbars are shown in red]
N. J. A. Sloane, 34 regions using 3 long-legged A's [Crossbars are shown in red]
N. J. A. Sloane, Transforming a long-legged A_n graph to a Wu_n graph, and vice-versa
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000217, A007318, A064226, A130883, A117625.

A row of the array in A386478.

[ n eq 1 select 1 else Self(n-1)+9*n-16: n in [1..50] ];

seq((16-23*n+9*n^2)*1/2,n=1..40); # Emeric Deutsch, May 07 2008

Table[(9n^2-23n+16)/2,{n,40}] (* or *) LinearRecurrence[{3,-3,1},{1,3,14},40] (* Harvey P. Dale, Oct 01 2011 *)

x='x+O('x^50); Vec(x*(1+8*x^2)/(1-x)^3) \\ G. C. Greubel, Feb 18 2017

Binomial transform of [1, 2, 9, 0, 0, 0, ...].
a(n) = A000217(n) + 8*A000217(n-2). - R. J. Mathar, May 06 2008
O.g.f.: x*(1+8*x^2)/(1-x)^3. - R. J. Mathar, May 06 2008
a(n) = A064226(n-2), n>1. - R. J. Mathar, Jul 31 2008
a(n) = a(n-1) + 9*n - 16, a(1)=1. - Vincenzo Librandi, Nov 24 2010
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(1)=1, a(2)=3, a(3)=14. - Harvey P. Dale, Oct 01 2011
E.g.f.: exp(x)*(16 - 14*x + 9*x^2)/2. - Stefano Spezia, Dec 25 2022

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

Edited by N. J. A. Sloane, Jun 21 2025 and Jun 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=', ')

A245300 Triangle T(n,k) = (n+k)*(n+k+1)/2 + k, 0 <= k <= n, read by rows.

Original entry on oeis.org

0, 1, 4, 3, 7, 12, 6, 11, 17, 24, 10, 16, 23, 31, 40, 15, 22, 30, 39, 49, 60, 21, 29, 38, 48, 59, 71, 84, 28, 37, 47, 58, 70, 83, 97, 112, 36, 46, 57, 69, 82, 96, 111, 127, 144, 45, 56, 68, 81, 95, 110, 126, 143, 161, 180, 55, 67, 80, 94, 109, 125, 142, 160, 179, 199, 220

Offset: 0

Reinhard Zumkeller, Jul 17 2014

			First rows and their row sums (A245301):
   0                                                                  0;
   1,  4                                                              5;
   3,  7,  12                                                        22;
   6, 11,  17,  24                                                   58;
  10, 16,  23,  31,  40                                             120;
  15, 22,  30,  39,  49,  60                                        215;
  21, 29,  38,  48,  59,  71,  84                                   350;
  28, 37,  47,  58,  70,  83,  97, 112                              532;
  36, 46,  57,  69,  82,  96, 111, 127, 144                         768;
  45, 56,  68,  81,  95, 110, 126, 143, 161, 180                   1065;
  55, 67,  80,  94, 109, 125, 142, 160, 179, 199, 220              1430;
  66, 79,  93, 108, 124, 141, 159, 178, 198, 219, 241, 264         1870;
  78, 92, 107, 123, 140, 158, 177, 197, 218, 240, 263, 287, 312    2392.

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened

Cf. A000217, A046092, A062725, A245301.

a245300 n k = (n + k) * (n + k + 1) `div` 2 + k
a245300_row n = map (a245300 n) [0..n]
a245300_tabl = map a245300_row [0..]
a245300_list = concat a245300_tabl

[k + Binomial(n+k+1,2): k in [0..n], n in [0..15]]; // G. C. Greubel, Apr 01 2021

Table[k + Binomial[n+k+1,2], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Apr 01 2021 *)

flatten([[k + binomial(n+k+1,2) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Apr 01 2021

T(n, 0) = A000217(n).
T(n, n) = A046092(n).
T(2*n, n) = A062725(n) (central terms).
Sum_{k=0..n} T(n, k) = A245301(n).
From G. C. Greubel, Apr 01 2021: (Start)
T(n, 1) = A000124(n+1) = A134869(n+1), n >= 1.
T(n, 2) = A152948(n+4), n >= 2.
T(n, 3) = A152950(n+4), n >= 3.
T(n, 4) = A145018(n+5), n >= 4.
T(n, 5) = A167499(n+4), n >= 5.
T(n, 6) = A166136(n+5), n >= 6.
T(n, 7) = A167487(n+6), n >= 7.
T(n, n-1) = A056220(n), n >= 1.
T(n, n-2) = A142463(n-1), n >= 2.
T(n, n-3) = A054000(n-1), n >= 3.
T(n, n-4) = A090288(n-3), n >= 4.
T(n, n-5) = A268581(n-4), n >= 5.
T(n, n-6) = A059993(n-4), n >= 6.
T(n, n-7) = (-1)*A147973(n), n >= 7.
T(n, n-8) = A139570(n-5), n >= 8.
T(n, n-9) = A271625(n-5), n >= 9.
T(n, n-10) = A222182(n-4), n >= 10.
T(2*n, n-1) = A081270(n-1), n >= 1.
T(2*n, n+1) = A117625(n+1), n >= 1. (End)

A198392 a(n) = (6n(3n+7)+(2n+13)*(-1)^n+3)/16 + 1.

Original entry on oeis.org

2, 4, 12, 18, 31, 41, 59, 73, 96, 114, 142, 164, 197, 223, 261, 291, 334, 368, 416, 454, 507, 549, 607, 653, 716, 766, 834, 888, 961, 1019, 1097, 1159, 1242, 1308, 1396, 1466, 1559, 1633, 1731, 1809, 1912, 1994, 2102, 2188, 2301, 2391, 2509, 2603, 2726, 2824, 2952

Offset: 0

Bruno Berselli, Oct 25 2011

For an origin of this sequence, see the triangular spiral illustrated in the Links section.

First bisection gives A117625 (without the initial term).

Bruno Berselli, Table of n, a(n) for n = 0..1000
Bruno Berselli, Illustration of initial terms.
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A152832 (by Superseeker).

Cf. sequences related to the triangular spiral: A022266, A022267, A027468, A038764, A045946, A051682, A062708, A062725, A062728, A062741, A064225, A064226, A081266-A081268, A081270-A081272, A081275 [incomplete list].

[(6*n*(3*n+7)+(2*n+13)*(-1)^n+3)/16+1: n in [0..50]];

LinearRecurrence[{1,2,-2,-1,1},{2,4,12,18,31},60] (* Harvey P. Dale, Jun 15 2022 *)

for(n=0, 50, print1((6*n*(3*n+7)+(2*n+13)*(-1)^n+3)/16+1", "));

G.f.: (2+2*x+4*x^2+2*x^3-x^4)/((1+x)^2*(1-x)^3).
a(n) = a(n-1)+2*a(n-2)-2*a(n-3)-a(n-4)+a(n-5).
a(n)-a(-n-1) = A168329(n+1).
a(n)+a(n-1) = A102214(n).
a(2n)-a(2n-1) = A016885(n).
a(2n+1)-a(2n) = A016825(n).

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

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A130883 a(n) = 2*n^2 - n + 1.

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A140064 a(n) = (9*n^2 - 5*n + 2)/2.

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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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A245300 Triangle T(n,k) = (n+k)*(n+k+1)/2 + k, 0 <= k <= n, read by rows.

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A198392 a(n) = (6*n*(3*n+7)+(2*n+13)*(-1)^n+3)/16 + 1.

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

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A140064 a(n) = (9n^2 - 5n + 2)/2.

A198392 a(n) = (6n(3n+7)+(2n+13)*(-1)^n+3)/16 + 1.