A069131 -id:A069131 - cp's OEIS frontend

A101321 Table T(n,m) = 1 + nm(m+1)/2 read by antidiagonals: centered polygonal numbers.

1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 7, 7, 4, 1, 1, 11, 13, 10, 5, 1, 1, 16, 21, 19, 13, 6, 1, 1, 22, 31, 31, 25, 16, 7, 1, 1, 29, 43, 46, 41, 31, 19, 8, 1, 1, 37, 57, 64, 61, 51, 37, 22, 9, 1, 1, 46, 73, 85, 85, 76, 61, 43, 25, 10, 1, 1, 56, 91, 109, 113, 106, 91, 71, 49, 28, 11, 1, 1, 67

Offset: 0

Eugene McDonnell (eemcd(AT)mac.com), Dec 24 2004

Row n gives the centered figurate numbers of the n-gon.

Antidiagonal sums are in A101338.

			The upper left corner of the infinite array T is
|0| 1   1   1   1   1   1   1   1   1   1 ... A000012
|1| 1   2   4   7  11  16  22  29  37  46 ... A000124
|2| 1   3   7  13  21  31  43  57  73  91 ... A002061
|3| 1   4  10  19  31  46  64  85 109 136 ... A005448
|4| 1   5  13  25  41  61  85 113 145 181 ... A001844
|5| 1   6  16  31  51  76 106 141 181 226 ... A005891
|6| 1   7  19  37  61  91 127 169 217 271 ... A003215
|7| 1   8  22  43  71 106 148 197 253 316 ... A069099
|8| 1   9  25  49  81 121 169 225 289 361 ... A016754
|9| 1  10  28  55  91 136 190 253 325 406 ... A060544

Cf. A000217, A001263, A101338.

A101321 := proc(n,k)
        n*k*(k+1)/2+1 ;
end proc: # R. J. Mathar, Jul 28 2016

T[n_, m_] := 1 + n m (m + 1)/2;
Table[T[n - m, m], {n, 0, 12}, {m, n, 0, -1}] // Flatten (* Jean-François Alcover, Mar 23 2020 *)

T(n,m) = 1 + n*m*(m+1)/2 \\ Charles R Greathouse IV, Jul 28 2016

T(n,2) = A016777(n). T(n,3) = A016921(n). T(n,4) = A017281(n).
T(10,m) = A062786(m+1).
T(11,m) = A069125(m+1).
T(12,m) = A003154(m+1).
T(13,m) = A069126(m+1).
T(14,m) = A069127(m+1).
T(15,m) = A069128(m+1).
T(16,m) = A069129(m+1).
T(17,m) = A069130(m+1).
T(18,m) = A069131(m+1).
T(19,m) = A069132(m+1).
T(20,m) = A069133(m+1).
T(n+1,m) = T(n,m) + m*(m+1)/2. - Gary W. Adamson and Michel Marcus, Oct 13 2015

Edited by R. J. Mathar, Oct 21 2009

A069190 Centered 24-gonal numbers.

Original entry on oeis.org

1, 25, 73, 145, 241, 361, 505, 673, 865, 1081, 1321, 1585, 1873, 2185, 2521, 2881, 3265, 3673, 4105, 4561, 5041, 5545, 6073, 6625, 7201, 7801, 8425, 9073, 9745, 10441, 11161, 11905, 12673, 13465, 14281, 15121, 15985, 16873, 17785, 18721, 19681, 20665, 21673

Offset: 1

Terrel Trotter, Jr., Apr 10 2002

Sequence found by reading the line from 1, in the direction 1, 25, ..., in the square spiral whose vertices are the generalized octagonal numbers A001082. Semi-axis opposite to A135453 in the same spiral. - Omar E. Pol, Sep 16 2011

			a(5) = 241 because 12*5^2 - 12*5 + 1 = 300 - 60 + 1 = 241.

Ivan Panchenko, Table of n, a(n) for n = 1..1000
John Elias, Illustration: Odd Ordered Star Perimeters.
Eric Weisstein's World of Mathematics, Centered Polygonal Numbers.
R. Yin, J. Mu, and T. Komatsu, The p-Frobenius Number for the Triple of the Generalized Star Numbers, Preprints 2024, 2024072280. See p. 2.
Index entries for sequences related to centered polygonal numbers.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. centered k-gonal numbers with k=3..25: A005448, A001844, A005891, A003215, A069099, A016754, A060544, A062786, A069125, A003154, A069126, A069127, A069128, A069129, A069130, A069131, A069132, A069133, A069178, A069173, A069174, A069190, A262221.

FoldList[#1 + #2 &, 1, 24 Range@ 45] (* Robert G. Wilson v *)
Table[12n^2-12n+1,{n,50}] (* or *) LinearRecurrence[{3,-3,1},{1,25,73},50] (* Harvey P. Dale, Jul 17 2011 *)

a(n)=12*n^2-12*n+1 \\ Charles R Greathouse IV, Sep 24 2015

a(n) = 12*n^2 - 12*n + 1.
a(n) = 24*n + a(n-1) - 24 with a(1)=1. - Vincenzo Librandi, Aug 08 2010
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(1)=1, a(2)=25, a(3)=73. - Harvey P. Dale, Jul 17 2011
G.f.: x*(1+22*x+x^2)/(1-x)^3. - Harvey P. Dale, Jul 17 2011
Binomial transform of [1, 24, 24, 0, 0, 0, ...] and Narayana transform (cf. A001263) of [1, 24, 0, 0, 0, ...]. - Gary W. Adamson, Jul 26 2011
From Amiram Eldar, Jun 21 2020: (Start)
Sum_{n>=1} 1/a(n) = Pi*tan(Pi/sqrt(6))/(4*sqrt(6)).
Sum_{n>=1} a(n)/n! = 13*e - 1.
Sum_{n>=1} (-1)^n * a(n)/n! = 13/e - 1. (End)
E.g.f.: exp(x)*(1 + 12*x^2) - 1. - Stefano Spezia, May 31 2022

More terms from Harvey P. Dale, Jul 17 2011

A082108 a(n) = 4n^2 + 6n + 1.

Original entry on oeis.org

1, 11, 29, 55, 89, 131, 181, 239, 305, 379, 461, 551, 649, 755, 869, 991, 1121, 1259, 1405, 1559, 1721, 1891, 2069, 2255, 2449, 2651, 2861, 3079, 3305, 3539, 3781, 4031, 4289, 4555, 4829, 5111, 5401, 5699, 6005, 6319, 6641, 6971, 7309, 7655, 8009, 8371

Offset: 0

Paul Barry, Apr 03 2003

a(n) is the sum of the numerator and denominator of (n+1)/(2*n) + (n+2)/(2*(n+1)); all fractions are reduced and n > 0. - J. M. Bergot, Jun 14 2017

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. A028387, A069131.

[4*n^2+6*n+1: n in [0..60]]; // G. C. Greubel, Dec 22 2022

(* Programs from Michael De Vlieger, Jun 15 2017 *)
Table[4n^2 +6n +1, {n,0,50}]
LinearRecurrence[{3,-3,1}, {1,11,29}, 51]
CoefficientList[Series[(1+8*x-x^2)/(1-x)^3, {x,0,50}], x] (* End *)

a(n)=4*n^2+6*n+1 \\ Charles R Greathouse IV, Oct 07 2015

[4*n^2+6*n+1 for n in range(61)] # G. C. Greubel, Dec 22 2022

a(n) = a(n-1) + 8*n + 2. - Vincenzo Librandi, Aug 08 2010
From Michael De Vlieger, Jun 15 2017: (Start)
G.f.: (1 + 8*x - x^2)/(1 - x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End)
a(n) = A016742(n) + A008588(n) + 1. - Felix Fröhlich, Jun 16 2017
Sum_{k=1..n} a(k-1)/(2*k)! = 1 - 1/(2*n)!. - Robert Israel, Jul 19 2017
E.g.f.: (1 + 10*x + 4*x^2)*exp(x). - G. C. Greubel, Dec 22 2022

Incorrect formula and useless examples deleted by R. J. Mathar, Aug 31 2010

A195042 Concentric 9-gonal numbers.

Original entry on oeis.org

0, 1, 9, 19, 36, 55, 81, 109, 144, 181, 225, 271, 324, 379, 441, 505, 576, 649, 729, 811, 900, 991, 1089, 1189, 1296, 1405, 1521, 1639, 1764, 1891, 2025, 2161, 2304, 2449, 2601, 2755, 2916, 3079, 3249, 3421, 3600, 3781, 3969, 4159, 4356, 4555, 4761, 4969, 5184, 5401, 5625

Offset: 0

Omar E. Pol, Sep 27 2011

Also concentric enneagonal numbers or concentric nonagonal numbers.
A016766 and A069131 interleaved.
Partial sums of A056020. - Reinhard Zumkeller, Jan 07 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Column 9 of A195040.

Cf. A016766, A069131, A077221, A195041, A195142, A195043.

a195042 n = a195042_list !! n
a195042_list = scanl (+) 0 a056020_list
-- Reinhard Zumkeller, Jan 07 2012

[(9*n^2+5/2*((-1)^n-1))/4: n in [0..50]]; // Vincenzo Librandi, Sep 29 2011

LinearRecurrence[{2,0,-2,1},{0,1,9,19},60] (* Harvey P. Dale, Nov 24 2019 *)

a(n)=(9*n^2+5/2*((-1)^n-1))/4 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = (9*n^2 + 5/2*((-1)^n - 1))/4.
From R. J. Mathar, Sep 28 2011: (Start)
G.f.: -x*(1+7*x+x^2) / ( (1+x)*(x-1)^3 ).
a(n) + a(n+1) = A060544(n+1). (End)
Sum_{n>=1} 1/a(n) = Pi^2/54 + tan(sqrt(5)*Pi/6)*Pi/(3*sqrt(5)). - Amiram Eldar, Jan 16 2023

A195316 Centered 36-gonal numbers.

Original entry on oeis.org

1, 37, 109, 217, 361, 541, 757, 1009, 1297, 1621, 1981, 2377, 2809, 3277, 3781, 4321, 4897, 5509, 6157, 6841, 7561, 8317, 9109, 9937, 10801, 11701, 12637, 13609, 14617, 15661, 16741, 17857, 19009, 20197, 21421, 22681, 23977, 25309, 26677, 28081, 29521, 30997, 32509

Offset: 1

Omar E. Pol, Sep 16 2011

Sequence found by reading the line from 1, in the direction 1, 37, ..., in the square spiral whose vertices are the generalized hendecagonal numbers A195160. Semi-axis opposite to A195321 in the same spiral.

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

Bisection of A195147.
Cf. A003154, A069129, A069133, A069190, A195314, A195315, A195317, A195318.
Cf. A069131, A195160, A195321.

[(18*n^2-18*n+1): n in [1..50]]; // Vincenzo Librandi, Sep 19 2011

Table[18*n^2 - 18*n + 1, {n, 1, 40}] (* Amiram Eldar, Feb 11 2022 *)

a(n)=18*n^2-18*n+1 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = 18*n^2 - 18*n + 1.
G.f.: -x*(1 + 34*x + x^2)/(x-1)^3. - R. J. Mathar, Sep 18 2011
Sum_{n>=1} 1/a(n) = Pi*tan(sqrt(7)*Pi/6)/(6*sqrt(7)). - Amiram Eldar, Feb 11 2022
From Elmo R. Oliveira, Nov 14 2024: (Start)
E.g.f.: exp(x)*(18*x^2 + 1) - 1.
a(n) = 2*A069131(n) - 1.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3. (End)

A082046 Square array, A(n, k) = (kn)^2 + 3k*n + 1, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 5, 1, 1, 11, 11, 1, 1, 19, 29, 19, 1, 1, 29, 55, 55, 29, 1, 1, 41, 89, 109, 89, 41, 1, 1, 55, 131, 181, 181, 131, 55, 1, 1, 71, 181, 271, 305, 271, 181, 71, 1, 1, 89, 239, 379, 461, 461, 379, 239, 89, 1, 1, 109, 305, 505, 649, 701, 649, 505, 305, 109, 1

Offset: 0

Paul Barry, Apr 03 2003

			Array, A(n, k), begins as:
  1,  1,   1,   1,   1,    1,    1,    1, ... A000012;
  1,  5,  11,  19,  29,   41,   55,   71, ... A028387;
  1, 11,  29,  55,  89,  131,  181,  239, ... A082108;
  1, 19,  55, 109, 181,  271,  379,  505, ... A069131;
  1, 29,  89, 181, 305,  461,  649,  869, ... ;
  1, 41, 131, 271, 461,  701,  991, 1331, ... ;
  1, 55, 181, 379, 649,  991, 1405, 1891, ... ;
  1, 71, 239, 505, 869, 1331, 1891, 2549, ... ;
Antidiagonals, T(n, k), begin as:
  1;
  1,  1;
  1,  5,   1;
  1, 11,  11,   1;
  1, 19,  29,  19,   1;
  1, 29,  55,  55,  29,   1;
  1, 41,  89, 109,  89,  41,   1;
  1, 55, 131, 181, 181, 131,  55,  1;
  1, 71, 181, 271, 305, 271, 181, 71,  1;

G. C. Greubel, Antidiagonals n = 0..50, flattened

Cf. A028387, A057721, A069131, A072025, A082039, A082043, A082047, A082105, A082108.

[(k*(n-k))^2 + 3*(k*(n-k)) + 1: k in [0..n], n in [0..13]]; // G. C. Greubel, Dec 22 2022

T[n_, k_]:= (k*(n-k))^2 + 3*(k*(n-k)) + 1;
Table[T[n,k], {n,0,13}, {k,0,n}]//Flatten (* G. C. Greubel, Dec 22 2022 *)

def A082046(n,k): return (k*(n-k))^2 + 3*(k*(n-k)) + 1
flatten([[A082046(n,k) for k in range(n+1)] for n in range(14)]) # G. C. Greubel, Dec 22 2022

A(n, k) = (k*n)^2 + 3*k*n + 1 (square array).
A(k, n) = A(n, k).
A(n, n) = T(2*n, n) = A057721(n).
A(n, n+1) = A072025(n).
T(n, k) = (k*(n-k))^2 + 3*k*(n-k) + 1 (antidiagonals).
Sum_{k=0..n} T(n, k) = A082047(n) (antidiagonal sums).
From G. C. Greubel, Dec 22 2022: (Start)
Sum_{k=0..n} (-1)^k*T(n, k) = (1/2)*(1 + (-1)^n)*(1 - 2*n).
T(2*n+1, n-1) = T(2*n-1, n-1) = A072025(n-1). (End)

A010008 a(0) = 1, a(n) = 18*n^2 + 2 for n>0.

Original entry on oeis.org

1, 20, 74, 164, 290, 452, 650, 884, 1154, 1460, 1802, 2180, 2594, 3044, 3530, 4052, 4610, 5204, 5834, 6500, 7202, 7940, 8714, 9524, 10370, 11252, 12170, 13124, 14114, 15140, 16202, 17300, 18434, 19604, 20810, 22052, 23330, 24644, 25994, 27380, 28802, 30260

Offset: 0

N. J. A. Sloane

The identity (18*n^2+2)^2-(9*n^2+2)*(6*n)^2=4 can be written as a(n+1)^2-A010002(n+1)*A008588(n+1)^2=4. - Vincenzo Librandi, Feb 07 2012

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

After 20, all terms are in A000408.

Cf. A206399.

[1] cat [18*n^2+2: n in [1..50]]; // Vincenzo Librandi, Aug 03 2015

Join[{1}, 18 Range[41]^2 + 2] (* Bruno Berselli, Feb 06 2012 *)
Join[{1}, LinearRecurrence[{3, -3, 1}, {20, 74, 164}, 50]] (* Vincenzo Librandi, Aug 03 2015 *)

G.f.: (1+x)*(1+16*x+x^2)/(1-x)^3. - Bruno Berselli, Feb 06 2012
a(n) = (3*n-1)^2+(3*n+1)^2 = (n-1)^2+(n+1)^2+(4*n)^2 for n>0. - Bruno Berselli, Feb 06 2012
E.g.f.: (x*(x+1)*18+2)*e^x-1. - Gopinath A. R., Feb 14 2012
Sum_{n>=0} 1/a(n) = 3/4+ (1/12)*Pi*coth(Pi/3) = 1.0853330948... - R. J. Mathar, May 07 2024
a(n) = 2*A247792(n), n>0. - R. J. Mathar, May 07 2024
a(n) = A069131(n)+A069131(n+1). - R. J. Mathar, May 07 2024

More terms from Bruno Berselli, Feb 06 2012

A194273 Concentric triangular numbers (see Comments lines for definition).

Original entry on oeis.org

0, 1, 3, 6, 9, 12, 15, 19, 24, 30, 36, 42, 48, 55, 63, 72, 81, 90, 99, 109, 120, 132, 144, 156, 168, 181, 195, 210, 225, 240, 255, 271, 288, 306, 324, 342, 360, 379, 399, 420, 441, 462, 483, 505, 528, 552, 576, 600, 624, 649, 675, 702, 729, 756, 783, 811

Offset: 0

Omar E. Pol, Aug 20 2011

This can be interpreted as a cellular automaton on the infinite hexagonal net. The sequence gives the number of cells "ON" in the structure after n-th stage. A194272 gives the first differences. For a definition without words see the illustration of initial terms in the example section. For other concentric polygonal numbers see A194274, A194275 and A032528.

Also, row sums of an infinite square array T(n,k) in which column k lists 6*k-1 zeros followed by the numbers A008486 (see example).

			Using the numbers A008486 we can write:
0, 1, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36,...
0, 0, 0, 0, 0,  0,  0,  1,  3,  6,  9, 12, 15, 18,...
0, 0, 0, 0, 0,  0,  0,  0,  0,  0,  0,  0,  0,  1,...
And so on.
=========================================================
The sums of the columns give this sequence:
0, 1, 3, 6, 9, 12, 15, 19, 24, 30, 36, 42, 48, 55,...
...
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
.
. 1    3      6        9          12           15
.
.                                           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
.
.       19               24                 30

Index entries for sequences related to cellular automata

Cf. A000217, A008486, A032528, A069131, A152751, A194272, A194274, A194275.

G.f.: x/(1-3*x+3*x^2-3*x^4+3*x^5-x^6) = x/((1-x)^3*(1+x)*(1-x+x^2)).

A214581 Triangle read by rows: T(n,k) is the number of unordered pairs of nodes at distance k in the circumcoronene H(n) (n=1,2,3,4,5; see definition in the Klavzar papers).

Original entry on oeis.org

6, 6, 3, 30, 48, 57, 54, 45, 30, 12, 72, 126, 165, 186, 195, 186, 168, 138, 102, 66, 27, 132, 240, 327, 390, 435, 456, 462, 444, 414, 366, 309, 246, 177, 114, 48, 210, 390, 543, 666, 765, 834, 882, 900, 900, 870, 825, 756, 675, 582, 480, 378, 270, 174, 75

Offset: 1

Emeric Deutsch, Aug 31 2012

The entries in row n are the coefficients of the Wiener polynomial of the corresponding graph.
Row n contains 4n-1 entries.
T(n,1) = 9n^2-3n = A152743(n).
T(n,2) = 6n(3n-2)= A153796(n).
T(n,3) = 3(9n^2-9n+1)= 3*A069131(n) (for n>5 this is a conjecture).
T(n,2n) = n(7n^2-1) = 6*A004126(n) (for n>5 this is a conjecture).
T(n,4n-2) = 6(n^2+n-1) = 6*A028387(n-1) (for n>5 this is a conjecture).
T(n,4n-1) = 3n^2 = A033428(n) (for n>5 this is a conjecture).
Sum(k*T(n,k), k>=1) = A143366(n).

S. Klavzar, A bird's eye view of the cut method and a survey of its applications in chemical graph theory, MATCH, Commun. Math. Comput. Chem. 60, 2008, 255-274.
Bo-Yin Yang and Yeong-Nan Yeh, A Crowning Moment for Wiener Indices, Studies in Appl. Math., 112 (2004), 333-340.
Bo-Yin Yang and Yeong-Nan Yeh, Wiener polynomials of some chemically interesting graphs, International Journal of Quantum Chemistry, 99 (2004), 80-91, 2004.
P. Zigert, S. Klavzar, and I. Gutman, Calculating the hyper-Wiener index of benzenoid hydrocarbons, ACH Models Chem., 137, 2000, 83-94.

Cf. A143366, A214580, A152743, A153796, A069131, A004126, A028387, A033428.

The entries have been obtained by using the Maple Graph Theory package for finding the distance matrix of each of the five graphs H(n) (n=1,2,3,4,5). The given Maple program yields the Wiener polynomial of H(2) (having as coefficients the entries in row 2).

A264825 Centered 18-gonal (or octadecagonal) primes.

Original entry on oeis.org

19, 109, 181, 271, 379, 811, 991, 2161, 3079, 4159, 4969, 5851, 7309, 8929, 10099, 10711, 13339, 17029, 21169, 22051, 23869, 25759, 26731, 28729, 32941, 34039, 37441, 38611, 39799, 48619, 58321, 59779, 67339, 70489, 72091, 89101, 90901, 102079, 109891, 117991, 122149

Offset: 1

Ilya Gutkovskiy, Nov 26 2015

nonn

Primes of the form 9*k^2 - 9*k + 1.

OEIS Wiki, Figurate numbers
Eric Weisstein's World of Mathematics, Centered Polygonal Number
Eric Weisstein's World of Mathematics, Prime Number

Cf. A069131, A000040.

for(n=1, 1e3, if(isprime(k=9*n^2-9*n+1), print1(k,", "))) \\ Altug Alkan, Nov 26 2015

cp's OEIS Frontend

A101321 Table T(n,m) = 1 + n*m*(m+1)/2 read by antidiagonals: centered polygonal numbers.

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A069190 Centered 24-gonal numbers.

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A082108 a(n) = 4*n^2 + 6*n + 1.

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A195042 Concentric 9-gonal numbers.

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A195316 Centered 36-gonal numbers.

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A082046 Square array, A(n, k) = (k*n)^2 + 3*k*n + 1, read by antidiagonals.

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A010008 a(0) = 1, a(n) = 18*n^2 + 2 for n>0.

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A101321 Table T(n,m) = 1 + nm(m+1)/2 read by antidiagonals: centered polygonal numbers.

A082108 a(n) = 4n^2 + 6n + 1.

A082046 Square array, A(n, k) = (kn)^2 + 3k*n + 1, read by antidiagonals.