A157914 -id:A157914 - cp's OEIS frontend

A195605 a(n) = (4n(n+2)+(-1)^n+1)/2 + 1.

2, 7, 18, 31, 50, 71, 98, 127, 162, 199, 242, 287, 338, 391, 450, 511, 578, 647, 722, 799, 882, 967, 1058, 1151, 1250, 1351, 1458, 1567, 1682, 1799, 1922, 2047, 2178, 2311, 2450, 2591, 2738, 2887, 3042, 3199, 3362, 3527, 3698, 3871, 4050, 4231, 4418, 4607, 4802

Offset: 0

Bruno Berselli, Sep 21 2011 - based on remarks and sequences by Omar E. Pol

Sequence found by reading the numbers in increasing order on the vertical line containing 2 of the square spiral whose vertices are the triangular numbers (A000217) - see Pol's comments in other sequences visible in this numerical spiral.

Also A077591 (without first term) and A157914 interleaved.

Bruno Berselli, Table of n, a(n) for n = 0..1000
Bruno Berselli, Illustration of initial terms: An origin of A195605.
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A000217, A077591, A157914.
Cf. A047621 (contains first differences), A016754 (contains the sum of any two consecutive terms).
Cf. A033585, A069129, A077221, A102083, A139098, A139271-A139277, A139592, A139593, A188135, A194268, A194431, A195241 [incomplete list].

[(4*n*(n+2)+(-1)^n+3)/2: n in [0..48]];

CoefficientList[Series[(2 + 3 x + 4 x^2 - x^3) / ((1 + x) (1 - x)^3), {x, 0, 50}], x] (* Vincenzo Librandi, Aug 19 2013 *)
LinearRecurrence[{2,0,-2,1},{2,7,18,31},50] (* Harvey P. Dale, Jan 21 2017 *)

for(n=0, 48, print1((4*n*(n+2)+(-1)^n+3)/2", "));

G.f.: (2+3*x+4*x^2-x^3)/((1+x)*(1-x)^3).
a(n) = a(-n-2) = 2*a(n-1)-2*a(n-3)+a(n-4).
a(n) = A047524(A000982(n+1)).
Sum_{n>=0} 1/a(n) = 1/2 + Pi^2/16 - cot(Pi/(2*sqrt(2)))*Pi/(4*sqrt(2)). - Amiram Eldar, Mar 06 2023

A185869 (Odd,even)-polka dot array in the natural number array A000027; read by antidiagonals.

Original entry on oeis.org

2, 7, 9, 16, 18, 20, 29, 31, 33, 35, 46, 48, 50, 52, 54, 67, 69, 71, 73, 75, 77, 92, 94, 96, 98, 100, 102, 104, 121, 123, 125, 127, 129, 131, 133, 135, 154, 156, 158, 160, 162, 164, 166, 168, 170, 191, 193, 195, 197, 199, 201, 203, 205, 207, 209, 232, 234, 236, 238, 240, 242, 244, 246, 248, 250, 252, 277, 279, 281, 283, 285, 287, 289, 291, 293, 295, 297, 299, 326, 328, 330, 332, 334, 336, 338, 340, 342, 344, 346, 348, 350, 379, 381, 383, 385, 387, 389, 391, 393, 395, 397, 399, 401, 403, 405

Offset: 1

Clark Kimberling, Feb 05 2011

This is the second of four polka dot arrays; see A185868.
row 1: A130883;
row 2: A100037;
row 3: A100038;
row 4: A100039;
col 1: A014107;
col 2: A033537;
col 3: A100040;
col 4: A100041;
diag (2,18,...): A077591;
diag (7,31,...): A157914;
diag (16,48,...): A035008;
diag (29,69,...): A108928;
antidiagonal sums: A033431;
antidiagonal sums: 2*(1^3, 2^3, 3^3, 4^3,...) = 2*A000578.
A060432(n) + n is odd if and only if n is in this sequence. - Peter Kagey, Feb 03 2016

			Northwest corner:
  2....7....16...29...46
  9....18...31...48...69
  20...33...50...71...96
  35...52...73...98...127

Peter Kagey, Table of n, a(n) for n = 1..10000

Cf. A000027 (as an array), A060432, A185868, A185870, A185871.

a185869 n = a185869_list !! (n - 1)
a185869_list = scanl (+) 2 $ a' 1
  where  a' n = 2 * n + 3 : replicate n 2 ++ a' (n + 1)
-- Peter Kagey, Sep 02 2016

f[n_,k_]:=2n-1+(2n+2k-3)(n+k-1);
TableForm[Table[f[n,k],{n,1,10},{k,1,15}]]
Table[f[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten

from math import isqrt, comb
def A185869(n):
    a = (m:=isqrt(k:=n<<1))+(k>m*(m+1))
    x = n-comb(a,2)
    y = a-x+1
    return y*((y+(c:=x<<1)<<1)-5)+x*(c-3)+2 # Chai Wah Wu, Jun 18 2025

T(n,k) = 2n-1+(n+k-1)*(2n+2k-3), k>=1, n>=1.

A185438 a(n) = 8n^2 - 2n + 1.

Original entry on oeis.org

1, 7, 29, 67, 121, 191, 277, 379, 497, 631, 781, 947, 1129, 1327, 1541, 1771, 2017, 2279, 2557, 2851, 3161, 3487, 3829, 4187, 4561, 4951, 5357, 5779, 6217, 6671, 7141, 7627, 8129, 8647, 9181, 9731, 10297, 10879, 11477, 12091, 12721, 13367, 14029, 14707, 15401, 16111, 16837, 17579

Offset: 0

Paul Curtz, Feb 03 2011

Odd numbers (A005408) written clockwise as a square spiral:
.
  41--43--45--47--49--51
   |                   |
  39  13--15--17--19  53
   |   |           |   |
  37  11   1---3  21  55
   |   |       |   |   |
  35   9---7---5  23  57
   |               |   |
  33--31--29--27--25  59
                       |
  71--69--67--65--63--61
.
Walking in straight lines away from the center:
  1, 17, 49, ... = A069129(n+1) =  1 - 8*n + 8*n^2,
  1,  3, 21, ... = A033567(n)   =  1 - 6*n + 8*n^2,
  1, 15, 45, ... = A014634(n)   =  1 + 6*n + 8*n^2,
  1,  5, 25, ... = A080856(n)   =  1 - 4*n + 8*n^2,
  1, 13, 41, ... = A102083(n)   =  1 + 4*n + 8*n^2,
  1,  7, 29, ... = a(n)         =  1 - 2*n + 8*n^2,
  1, 11, 37, ... = A188135(n)   =  1 + 2*n + 8*n^2,
  1,  9, 33, ... = A081585(n)   =  1       + 8*n^2,
  5, 29, 69, ... = A108928(n+1) = -3       + 8*n^2,
  7, 31, 71, ... = A157914(n+1) = -1       + 8*n^2,
  9, 35, 77, ... = A033566(n+1) = -1 + 2*n + 8*n^2.
All are quadrisections of sequences in A181407(n) (example: A014634(n) and A033567(n) in A064038(n+1)) or of this family (?): a(n) is a quadrisection of f(n) = 1,1,1,1,2,7,11,8,11,29,37,23,28,67,79,46,... f(n) is just before A064038(n+1) (fifth vertical) in A181407(n). The companion to a(n) is A188135(n), another quadrisection of f(n). Two last quadrisections of f(n) are A054552(n) and A033951(n).
For n >= 1, bisection of A193867. - Omar E. Pol, Aug 16 2011
Also the sequence may be obtained by starting with the segment (1, 7) followed by the line from 7 in the direction 7, 29, ... in the square spiral whose vertices are the generalized hexagonal numbers (A000217). - Omar E. Pol, Aug 01 2016

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

Cf. A000217, A005408, A014634, A014635, A033566, A033567, A033951, A054552, A064038.

Cf. A069129, A080856, A081585, A102083, A108928, A157914, A181407, A188135, A193867.

[1-2*n+8*n^2: n in [0..50]]; // Vincenzo Librandi, Feb 03 2011

Table[1 - 2n + 8n^2, {n, 0, 39}] (* Alonso del Arte, Feb 03 2011 *)
CoefficientList[Series[(-1 - 4 x - 11 x^2)/(x - 1)^3, {x, 0, 47}], x] (* Michael De Vlieger, Aug 01 2016 *)

a(n)=8*n^2-2*n+1 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = a(n-1) + 16*n - 10 (n > 0).
a(n) = 2*a(n-1) - a(n-2) + 16 (n > 1).
a(n) = 3*(n-1) - 3*a(n-2) + a(n-3) (n > 2).
G.f.: (-1 - 4*x - 11*x^2)/(x-1)^3. - R. J. Mathar, Feb 03 2011
a(n) = A014635(n) + 1. - Bruno Berselli, Apr 09 2011
E.g.f.: exp(x)*(1 + 6*x + 8*x^2). - Elmo R. Oliveira, Nov 17 2024

A158443 a(n) = 16*n^2 - 4.

Original entry on oeis.org

12, 60, 140, 252, 396, 572, 780, 1020, 1292, 1596, 1932, 2300, 2700, 3132, 3596, 4092, 4620, 5180, 5772, 6396, 7052, 7740, 8460, 9212, 9996, 10812, 11660, 12540, 13452, 14396, 15372, 16380, 17420, 18492, 19596, 20732, 21900, 23100, 24332, 25596, 26892, 28220, 29580

Offset: 1

Vincenzo Librandi, Mar 19 2009

The identity (8*n^2 - 1)^2 - (16*n^2 - 4) *(2*n)^2 = 1 can be written as A157914(n)^2 - a(n)*A005843(n)^2 = 1.

Sequence found by reading the line from 12, in the direction 12, 60, ... in the square spiral whose vertices are the generalized decagonal numbers A074377. - Omar E. Pol, Nov 02 2012

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Vincenzo Librandi, X^2-AY^2=1, Math Forum, 2007. [Wayback Machine link]
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000466, A005843, A074377, A157914.

I:=[12, 60, 140]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..50]];

16Range[60]^2-4  (* Harvey P. Dale, Mar 18 2011 *)

PARI
```
a(n) = 16*n^2 - 4.
```

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f: 4*x*(3+6*x-x^2)/(1-x)^3.
From Amiram Eldar, Mar 05 2023: (Start)
Sum_{n>=1} 1/a(n) = 1/8.
Sum_{n>=1} (-1)^(n+1)/a(n) = (Pi-2)/16. (End)
From Elmo R. Oliveira, Jan 16 2025: (Start)
E.g.f.: 4*(exp(x)*(4*x^2 + 4*x - 1) + 1).
a(n) = 4*A000466(n). (End)

A157913 a(n) = 64*n^2 - 16.

Original entry on oeis.org

48, 240, 560, 1008, 1584, 2288, 3120, 4080, 5168, 6384, 7728, 9200, 10800, 12528, 14384, 16368, 18480, 20720, 23088, 25584, 28208, 30960, 33840, 36848, 39984, 43248, 46640, 50160, 53808, 57584, 61488, 65520, 69680, 73968, 78384, 82928, 87600, 92400, 97328, 102384

Offset: 1

Vincenzo Librandi, Mar 09 2009

The identity (8*n^2 - 1)^2 - (64*n^2 - 16)*n^2 = 1 can be written as A157914(n)^2 - a(n)*n^2 = 1. - Vincenzo Librandi, Feb 09 2012

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

Cf. A000466, A157914.

I:=[48, 240, 560]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..40]]; // Vincenzo Librandi, Feb 09 2012

LinearRecurrence[{3, -3, 1}, {48, 240, 560}, 50] (* Vincenzo Librandi, Feb 09 2012 *)
64*Range[40]^2-16 (* Harvey P. Dale, Jul 27 2012 *)

for(n=1, 40, print1(64*n^2 - 16", ")); \\ Vincenzo Librandi, Feb 09 2012

From  Vincenzo Librandi, Feb 09 2012: (Start)
G.f.: -16*x*(3 + 6*x - x^2)/(x - 1)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End)
From Amiram Eldar, Mar 07 2023: (Start)
Sum_{n>=1} 1/a(n) = 1/32.
Sum_{n>=1} (-1)^(n+1)/a(n) = (Pi-2)/64. (End)
From Elmo R. Oliveira, Jan 16 2025: (Start)
E.g.f.: 16*(exp(x)*(4*x^2 + 4*x - 1) + 1).
a(n) = 16*A000466(n). (End)

A158487 a(n) = 64*n^2 - 8.

Original entry on oeis.org

56, 248, 568, 1016, 1592, 2296, 3128, 4088, 5176, 6392, 7736, 9208, 10808, 12536, 14392, 16376, 18488, 20728, 23096, 25592, 28216, 30968, 33848, 36856, 39992, 43256, 46648, 50168, 53816, 57592, 61496, 65528, 69688, 73976, 78392, 82936, 87608, 92408, 97336, 102392

Offset: 1

Vincenzo Librandi, Mar 20 2009

The identity (16*n^2 - 1)^2 - (64*n^2 - 8)*(2*n)^2 = 1 can be written as A141759(n)^2 - a(n)*A005843(n)^2 = 1. - Vincenzo Librandi, Feb 09 2012

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

Cf. A005843, A141759, A157914.

I:=[56, 248, 568]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..40]]; // Vincenzo Librandi, Feb 09 2012

LinearRecurrence[{3, -3, 1}, {56, 248, 568}, 50] (* Vincenzo Librandi, Feb 09 2012 *)

for(n=1, 40, print1(64*n^2 - 8", ")); \\ Vincenzo Librandi, Feb 09 2012

From Vincenzo Librandi, Feb 09 2012: (Start)
G.f.: -8*x*(7 + 10*x - x^2)/(x - 1)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End)
From Amiram Eldar, Mar 05 2023: (Start)
Sum_{n>=1} 1/a(n) = (1 - cot(Pi/(2*sqrt(2)))*Pi/(2*sqrt(2)))/16.
Sum_{n>=1} (-1)^(n+1)/a(n) = (cosec(Pi/(2*sqrt(2)))*Pi/(2*sqrt(2)) - 1)/16. (End)
From Elmo R. Oliveira, Jan 16 2025: (Start)
E.g.f.: 8*(exp(x)*(8*x^2 + 8*x - 1) + 1).
a(n) = 8*A157914(n). (End)

A320431 The number of tiles inside a regular n-gon created by lines that run from each of the vertices of the n edges orthogonal to these edges.

Original entry on oeis.org

1, 1, 31, 13, 71, 25, 127, 41, 199, 61, 287, 85, 391, 113, 511, 145, 647, 181, 799, 221, 967, 265, 1151, 313, 1351, 365, 1567, 421, 1799, 481, 2047, 545, 2311, 613, 2591, 685, 2887, 761, 3199, 841, 3527, 925, 3871, 1013, 4231, 1105, 4607, 1201, 4999, 1301, 5407, 1405, 5831, 1513, 6271, 1625, 6727, 1741

Offset: 3

R. J. Mathar, Jan 08 2019

Sequence proposed by Thomas Young: draw the regular n-gon and construct 2*n lines that run from both ends of the n edges perpendicular into the n-gon until they hit an opposite edge. (For n even the lines actually hit another vertex, so there are only n additional lines). a(n) is the number of non-overlapping tiles inside the n-gon with edges that are sections of the lines or n-gon edges.

R. J. Mathar, OEIS A320431
Thomas Young, R. J. Mathar, The surfer and the hut: a polygon dissection (2019)
Index to sequences on drawing diagonals in regular polygons
Index entries for linear recurrences with constant coefficients, signature (0,3,0,-3,0,1).

Cf. A165217, A320422

a(2n) = 2*n^2+2*n+1 = A001844(n), n>1. a(2n+1) = 8*n^2-1 = A157914(n), n>1. - Thomas Young (tyoung(AT)district16.org), Nov 11 2017
G.f.: x^3 +x^4 -x^5*(31+13*x-22*x^2-14*x^3+7*x^4+5*x^5) / ( (x-1)^3*(1+x)^3 ). - R. J. Mathar, Jan 21 2019
a(n) = 1+n*A064680(n-2), n>=5. - R. J. Mathar, Jan 21 2019

A342873 Numbers whose distance to the nearest cube equals the distance to the nearest product of 3 consecutive integers (three-dimensional oblong).

Original entry on oeis.org

0, 7, 16, 62, 92, 213, 276, 508, 616, 995, 1160, 1722, 1956, 2737, 3052, 4088, 4496, 5823, 6336, 7990, 8620, 10637, 11396, 13812, 14712, 17563, 18616, 21938, 23156, 26985, 28380, 32752, 34336, 39287, 41072, 46638, 48636, 54853, 57076, 63980, 66440, 74067

Offset: 1

Lamine Ngom, Mar 28 2021

That is, numbers k such that A074989(k) = A342872(k).
They form 2 partitions:
7, 62, 213, ... = 8*k^3 - k = k*A157914(k).
0, 16, 92,  ... = 8*k^3 + 6*k^2 + 2*k = 2*k*A033951(k).

Cf. A074989, A342872.

Cf. A157914, A033951, A074378, A201053, A000578, A007531, A081134.

def aupto(limit):
  cubes = [k**3 for k in range(int((limit+1)**1/3)+2)]
  proms = [k*(k+1)*(k+2) for k in range(int((limit+1)**1/3)+1)]
  A074989 = [min(abs(n-c) for c in cubes) for n in range(limit+1)]
  A342872 = [min(abs(n-p) for p in proms) for n in range(limit+1)]
  return [m for m in range(limit+1) if A074989[m] == A342872[m]]
print(aupto(10**4)) # Michael S. Branicky, Mar 28 2021

cp's OEIS Frontend

A195605 a(n) = (4*n*(n+2)+(-1)^n+1)/2 + 1.

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A185869 (Odd,even)-polka dot array in the natural number array A000027; read by antidiagonals.

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A185438 a(n) = 8*n^2 - 2*n + 1.

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A158443 a(n) = 16*n^2 - 4.

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A157913 a(n) = 64*n^2 - 16.

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A158487 a(n) = 64*n^2 - 8.

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A320431 The number of tiles inside a regular n-gon created by lines that run from each of the vertices of the n edges orthogonal to these edges.

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A195605 a(n) = (4n(n+2)+(-1)^n+1)/2 + 1.

A185438 a(n) = 8n^2 - 2n + 1.