A004120 -id:A004120 - cp's OEIS frontend

A081344 Natural numbers in square maze arrangement, read by antidiagonals.

1, 2, 4, 9, 3, 5, 10, 8, 6, 16, 25, 11, 7, 15, 17, 26, 24, 12, 14, 18, 36, 49, 27, 23, 13, 19, 35, 37, 50, 48, 28, 22, 20, 34, 38, 64, 81, 51, 47, 29, 21, 33, 39, 63, 65, 82, 80, 52, 46, 30, 32, 40, 62, 66, 100, 121, 83, 79, 53, 45, 31, 41, 61, 67, 99, 101, 122, 120, 84, 78, 54

Offset: 1

Paul Barry, Mar 19 2003

Arrange the natural numbers by taking clockwise and counterclockwise turns. Begin (LL) and then repeat (RRR)(LLL).
a(n) is a pairing function: a function that reversibly maps Z^{+} x Z^{+} onto Z^{+}, where Z^{+} is the set of integer positive numbers. - Boris Putievskiy, Dec 16 2012
For generalizations see A219159, A213928. - Boris Putievskiy, Mar 10 2013

			The start of the sequence as table T(i,j), i,j > 0:
   1   4    5    16 ...
   2   3    6    15 ...
   9   8    7    14 ...
  10  11   12    13 ...
  ....

Boris Putievskiy, Rows n = 1..100 of triangle, flattened
Boris Putievskiy, Transformations Integer Sequences And Pairing Functions, arXiv:1212.2732 [math.CO], 2012.
Eric Weisstein's World of Mathematics, Pairing functions
Index entries for sequences that are permutations of the natural numbers

Cf. A219159, A213928. The main diagonal is A002061. The following appear within interlaced sequences: A016754, A001844, A053755, A004120. The first row is A081345. The first column is A081346. The inverse permutation A194280, the first inverse function (numbers of rows) A220603, the second inverse function (numbers of columns) A220604.

T[n_, k_] := T[n, k] = Which[OddQ[n] && k==1, n^2, EvenQ[k] && n==1, k^2, EvenQ[n] && k==1, T[n-1, 1]+1, OddQ[k] && n==1, T[1, k-1]+1, k <= n, T[n, k-1]+1 - 2 Mod[n, 2], True, T[n-1, k]-1 + 2 Mod[k, 2]]; Table[T[n-k+1, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Feb 20 2019 *)

t=int((math.sqrt(8*n-7) - 1)/ 2)
i=n-t*(t+1)/2
j=(t*t+3*t+4)/2-n
if j >= i:
     m=(j-1)**2 + j + (j-i)*(-1)**(j-1)
else:
     m=(i-1)**2 + i - (i-j)*(-1)**(i-1)
# Boris Putievskiy, Dec 19 2012

from math import isqrt
def A081344(n):
    t = (k:=isqrt(m:=n<<1))+((m<<2)>(k<<2)*(k+1)+1)-1
    i, j = n-(t*(t+1)>>1), (t*(t+3)>>1)+2-n
    r = max(i,j)
    return (r-1)**2+r+(j-i if r&1 else i-j) # Chai Wah Wu, Nov 04 2024

From Boris Putievskiy, Dec 19 2012: (Start)
a(n) = (i-1)^2 + i + (i-j)*(-1)^(i-1) if i >= j,
a(n) = (j-1)^2 + j - (j-i)*(-1)^(j-1) if i <  j,
where
i = n - t*(t+1)/2,
j = (t*t + 3*t + 4)/2-n,
t = floor((-1 + sqrt(8*n-7))/2). (End)
Enumeration by boustrophedonic ("ox-plowing") method: If i >= j: T(i,j)=(i-1)^2+i + (i-j)*(-1)^(i-1), if i  < j: T(i,j)=(j-1)^2+j - (j-i)*(-1)^(j-1). - Boris Putievskiy, Dec 19 2012
T(i,j) = m^2 - m + 1 - (i - j)*(-1)^m where m = max(i,j). - Ziad Ahmed, Jun 09 2025

A005337 Number of ways in which n identical balls can be distributed among 4 boxes in a row such that each pair of adjacent boxes contains at least 4 balls.

Original entry on oeis.org

15, 40, 76, 124, 185, 260, 350, 456, 579, 720, 880, 1060, 1261, 1484, 1730, 2000, 2295, 2616, 2964, 3340, 3745, 4180, 4646, 5144, 5675, 6240, 6840, 7476, 8149, 8860, 9610, 10400, 11231, 12104, 13020, 13980, 14985

Offset: 8

N. J. A. Sloane, Simon Plouffe

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 8..1000
D. R. Breach, Letter to N. J. A. Sloane, Jun 1980
Philippe Flajolet, Balls and Urns, etc., A problem in submarine detection (solution to problem 68-16).
M. Hayes (proposer) and D. R. Breach (solver), A combinatorial problem, Problem 68-16, SIAM Rev. 12 (1970), 294-297.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A005338, A005339, A005340. A column of A259975.

A005337:=(15-20*z+6*z**2)/(z-1)**4; # Simon Plouffe in his 1992 dissertation

CoefficientList[Series[(15 - 20 x + 6 x^2)/(1 - x)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Oct 14 2013 *)
LinearRecurrence[{4,-6,4,-1},{15,40,76,124},50] (* Harvey P. Dale, May 11 2014 *)

G.f.: x^8*(15 - 20*x + 6*x^2)/(1 - x)^4.

a(n) = (546 - 169*n + 6*n^2 + n^3)/6. [Colin Barker, Jul 08 2012]

G.f. corrected by Colin Barker, Jul 08 2012

Name clarified by Alois P. Heinz, Oct 02 2017

A005338 Number of ways in which n identical balls can be distributed among 5 boxes in a row such that each pair of adjacent boxes contains at least 4 balls.

Original entry on oeis.org

1, 8, 31, 85, 190, 360, 610, 956, 1415, 2005, 2745, 3655, 4756, 6070, 7620, 9430, 11525, 13931, 16675, 19785, 23290, 27220, 31606, 36480, 41875, 47825, 54365, 61531, 69360, 77890, 87160, 97210, 108081, 119815, 132455, 146045, 160630

Offset: 8

N. J. A. Sloane

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 8..1000
D. R. Breach, Letter to N. J. A. Sloane, Jun 1980
Philippe Flajolet, Balls and Urns, etc., A problem in submarine detection (solution to problem 68-16).
M. Hayes (proposer) and D. R. Breach (solver), A combinatorial problem, Problem 68-16, SIAM Rev. 12 (1970), 294-297.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A005337, A005339, A005340.

I:=[1, 8, 31, 85, 190, 360, 610]; [n le 7 select I[n] else 5*Self(n-1)-10*Self(n-2)+10*Self(n-3)-5*Self(n-4)+Self(n-5): n in [1..40]]; // Vincenzo Librandi, May 11 2012

f[x_] := x^8*(1 + 3*x + x^2 - 11*x^5 + 7*x^6)/(1 - x)^5; Drop[ CoefficientList[ Series[f[x], {x, 0, 44}], x], 8] (* Jean-François Alcover, Oct 05 2011, after Vladeta Jovovic *)
LinearRecurrence[{5,-10,10,-5,1},{1,8,31,85,190,360,610},40] (* Harvey P. Dale, Aug 26 2019 *)

G.f.: x^8*(1 + 3*x + x^2 - 11*x^5 + 7*x^6)/(1 - x)^5. - Vladeta Jovovic, Apr 13 2008

a(n) = (n^4 + 10*n^3 - 445*n^2 + 2690*n - 1656)/24 for n > 9. - Colin Barker, May 10 2012

Corrected and extended by Vladeta Jovovic, Apr 13 2008

Name clarified by Alois P. Heinz, Oct 02 2017

A005340 Number of ways in which n identical balls can be distributed among 7 boxes in a row such that each pair of adjacent boxes contains at least 4 balls.

Original entry on oeis.org

1, 13, 76, 295, 889, 2188, 4652, 8891, 15686, 26011, 41056, 62251, 91291, 130162, 181168, 246959, 330560, 435401, 565348, 724735, 918397, 1151704, 1430596, 1761619, 2151962, 2609495, 3142808, 3761251, 4474975, 5294974, 6233128

Offset: 12

N. J. A. Sloane

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 12..1000
D. R. Breach, Letter to N. J. A. Sloane, Jun 1980
Philippe Flajolet, Balls and Urns, etc., A problem in submarine detection (solution to problem 68-16).
M. Hayes (proposer) and D. R. Breach (solver), A combinatorial problem, Problem 68-16, SIAM Rev. 12 (1970), 294-297.
Index entries for linear recurrences with constant coefficients, signature (7, -21, 35, -35, 21, -7, 1).

Cf. A005337, A005338, A005339.

f[x_] := x^12*(1 + 6*x + 6*x^2 + x^3 - 66*x^5 + 74*x^6 - 21*x^7)/(1-x)^7; Drop[ CoefficientList[ Series[f[x], {x, 0, 42}], x], 12] (* Jean-François Alcover, Oct 05 2011, after Vladeta Jovovic *)
LinearRecurrence[{7,-21,35,-35,21,-7,1},{1,13,76,295,889,2188,4652,8891},40] (* Harvey P. Dale, Apr 03 2025 *)

G.f.: x^12*(1 + 6*x + 6*x^2 + x^3 - 66*x^5 + 74*x^6 - 21*x^7)/(1-x)^7. - Vladeta Jovovic, Apr 13 2008

More terms from Vladeta Jovovic, Apr 13 2008

Name clarified by Alois P. Heinz, Oct 02 2017

A005339 Number of ways in which n identical balls can be distributed among 6 boxes in a row such that each pair of adjacent boxes contains at least 4 balls.

Original entry on oeis.org

35, 154, 424, 930, 1775, 3080, 4985, 7650, 11256, 16006, 22126, 29866, 39501, 51332, 65687, 82922, 103422, 127602, 155908, 188818, 226843, 270528, 320453, 377234, 441524, 514014, 595434, 686554, 788185, 901180, 1026435, 1164890, 1317530

Offset: 12

N. J. A. Sloane

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

D. R. Breach, Letter to N. J. A. Sloane, Jun 1980
Philippe Flajolet, Balls and Urns, etc., A problem in submarine detection (solution to problem 68-16).
M. Hayes (proposer) and D. R. Breach (solver), A combinatorial problem, Problem 68-16, SIAM Rev. 12 (1970), 294-297.
Index entries for linear recurrences with constant coefficients, signature (6, -15, 20, -15, 6, -1).

Cf. A005337, A005338, A005340.

Drop[CoefficientList[Series[x^12(35-56x+25x^2-4x^3+x^6)/(1-x)^6, {x,0, 60}], x],12] (* or *) Join[{35},LinearRecurrence[{6,-15,20,-15,6,-1},{154,424,930,1775,3080,4985},48]] (* Harvey P. Dale, Aug 12 2011 *)

G.f.: x^12*(35 - 56*x + 25*x^2 - 4*x^3 + x^6)/(1-x)^6. - Vladeta Jovovic, Apr 13 2008

a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6), a(12)=35, a(13)=154, a(14)=424, a(15)=930, a(16)=1775, a(17)=3080, a(18)=4985. - Harvey P. Dale, Aug 12 2011

More terms from Vladeta Jovovic, Apr 13 2008

Name clarified by Alois P. Heinz, Oct 02 2017

A081345 First row in maze arrangement of natural numbers A081344.

Original entry on oeis.org

1, 4, 5, 16, 17, 36, 37, 64, 65, 100, 101, 144, 145, 196, 197, 256, 257, 324, 325, 400, 401, 484, 485, 576, 577, 676, 677, 784, 785, 900, 901, 1024, 1025, 1156, 1157, 1296, 1297, 1444, 1445, 1600, 1601, 1764, 1765, 1936, 1937, 2116, 2117, 2304, 2305, 2500

Offset: 0

Paul Barry, Mar 19 2003

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A081346.

[n^2+n+1-n*(-1)^n: n in [0..50]]; // Vincenzo Librandi, Aug 08 2013

CoefficientList[Series[(5 x^3 - x^2 + 3 x + 1) / ((1 - x)^3 (1 + x)^2), {x, 0, 50}], x] (* Vincenzo Librandi, Aug 08 2013 *)

a(n) = n^2 + n + 1 - n*(-1)^n = n^2 + n + 1 + n*(-1)^(n+1).
a(2*n) = A053755(n), a(2*n+1) = 4 * A004120(n).
G.f.: (5*x^3-x^2+3*x+1)/((1-x)^3*(1+x)^2). [Colin Barker, Sep 03 2012]

A259975 Irregular triangle read by rows: T(n,k) = number of ways of placing n balls into k boxes in such a way that any two adjacent boxes contain at least 4 balls.

Original entry on oeis.org

1, 1, 1, 1, 1, 5, 1, 1, 6, 4, 1, 7, 9, 1, 8, 16, 1, 9, 25, 15, 1, 1, 10, 35, 40, 8, 1, 11, 46, 76, 31, 1, 12, 58, 124, 85, 1, 13, 71, 185, 190, 35, 1, 1, 14, 85, 260, 360, 154, 13, 1, 15, 100, 350, 610, 424, 76, 1, 16, 116, 456, 956, 930, 295

Offset: 0

N. J. A. Sloane, Jul 12 2015

			Triangle begins:
  1;
  1;
  1;
  1;
  1,  5,   1;
  1,  6,   4;
  1,  7,   9;
  1,  8,  16;
  1,  9,  25,  15,   1;
  1, 10,  35,  40,   8;
  1, 11,  46,  76,  31;
  1, 12,  58, 124,  85;
  1, 13,  71, 185, 190,  35,  1;
  1, 14,  85, 260, 360, 154, 13;
  1, 15, 100, 350, 610, 424, 76;
  ...

Alois P. Heinz, Rows n = 0..200, flattened
R. P. Boas & N. J. A. Sloane, Correspondence, 1974
D. R. Breach, Letter to N. J. A. Sloane, Jun 1980
M. Hayes (proposer) and D. R. Breach (solver), A combinatorial problem, Problem 68-16, SIAM Rev. 12 (1970), 294-297.

Columns: A004120, A005337, A005338, A005339, A005340.

Row sums give A257666.

b:= proc(n, v) option remember; expand(`if`(n=0,
      `if`(v=0, 1+x, 1), add(x*b(n-j, max(0, 4-j)), j=v..n)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(b(n, 0)):
seq(T(n), n=0..20);  # Alois P. Heinz, Jul 12 2015

b[n_, v_] := b[n, v] = Expand[If[n == 0, If[v == 0, 1+x, 1], Sum[x*b[n-j, Max[0, 4-j]], {j, v, n}]]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 1, Exponent[p, x]}]][b[n, 0]]; Table[T[n], {n, 0, 20}] // Flatten (* Jean-François Alcover, Feb 13 2016, after Alois P. Heinz *)

More terms from Alois P. Heinz, Jul 12 2015

A302537 a(n) = (n^2 + 13*n + 2)/2.

Original entry on oeis.org

1, 8, 16, 25, 35, 46, 58, 71, 85, 100, 116, 133, 151, 170, 190, 211, 233, 256, 280, 305, 331, 358, 386, 415, 445, 476, 508, 541, 575, 610, 646, 683, 721, 760, 800, 841, 883, 926, 970, 1015, 1061, 1108, 1156, 1205, 1255, 1306, 1358, 1411, 1465, 1520, 1576

Offset: 0

Franck Maminirina Ramaharo, Apr 09 2018

Binomial transform of [1, 7, 1, 0, 0, 0, ...].

Numbers m > 0 such that 8*m + 161 is a square.

			Illustration of initial terms (by the formula a(n) = A052905(n) + 3*n):
.                                                                    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   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 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      16        25          35            46              58

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics: A Foundation for Computer Science, Addison-Wesley, 1994.

G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Sequences whose n-th terms are of the form binomial(n, 2) + n*k + 1:
A152947 (k = 0); A000124 (k = 1); A000217 (k = 2); A034856 (k = 3);
A052905 (k = 4); A051936 (k = 5); A246172 (k = 6).
Cf. A000578, A002939, A004120, A008589, A056119, A060544, A162261.

A302537:= func< n | ((n+1)^2 +12*n +1)/2 >;
[A302537(n): n in [0..50]]; // G. C. Greubel, Jan 21 2025

a := n -> (n^2 + 13*n + 2)/2;
seq(a(n), n = 0 .. 100);

Table[(n^2 + 13 n + 2)/2, {n, 0, 100}]
CoefficientList[ Series[(5x^2 - 5x - 1)/(x - 1)^3, {x, 0, 50}], x] (* or *)
LinearRecurrence[{3, -3, 1}, {1, 8, 16}, 51] (* Robert G. Wilson v, May 19 2018 *)

makelist((n^2 + 13*n + 2)/2, n, 0, 100);

a(n) = (n^2 + 13*n + 2)/2; \\ Altug Alkan, Apr 12 2018

def A302537(n): return (n**2 + 13*n + 2)//2
print([A302537(n) for n in range(51)]) # G. C. Greubel, Jan 21 2025

a(n) = binomial(n + 1, 2) + 6*n + 1 = binomial(n, 2) + 7*n + 1.
a(n) = a(n-1) + n + 6.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n >= 3, where a(0) = 1, a(1) = 8 and a(2) = 16.
a(n) = 2*a(n-1) - a(n-2) + 1.
a(n) = A004120(n+1) for n > 1.
a(n) = A056119(n) + 1.
a(n) = A152947(n+1) + A008589(n).
a(n) = A060544(n+1) - A002939(n).
a(n) = A000578(n+1) - A162261(n) for n > 0.
G.f.: (1 + 5*x - 5*x^2)/(1 - x)^3.
E.g.f.: (1/2)*(2 + 14*x + x^2)*exp(x).
Sum_{n>=0} 1/a(n) = 24097/45220 + 2*Pi*tan(sqrt(161)*Pi/2) / sqrt(161) = 1.4630922534498496... - Vaclav Kotesovec, Apr 11 2018

A356754 Triangle read by rows: T(n,k) = ((n-1)(n+2))/2 + 2k.

Original entry on oeis.org

2, 4, 6, 7, 9, 11, 11, 13, 15, 17, 16, 18, 20, 22, 24, 22, 24, 26, 28, 30, 32, 29, 31, 33, 35, 37, 39, 41, 37, 39, 41, 43, 45, 47, 49, 51, 46, 48, 50, 52, 54, 56, 58, 60, 62, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87

Offset: 1

Torlach Rush, Aug 25 2022

The first column of the triangle is the Lazy Caterer's sequence A000124.
Each subsequent column starts with A000124(n) + (2 * (n-1)).
The first downward diagonal is A046691(n).
Columns and downward diagonals of the triangle identify many sequences (possibly shifted) in the database. Examples can be found in crossrefs below.
The sum of the n-th upward diagonal of the triangle is A356288(n).

			Triangle T(n,k) begins:
  n\k   1   2   3   4   5   6   7   8   9  10  11  ...
   1:   2
   2:   4   6
   3:   7   9  11
   4:  11  13  15  17
   5:  16  18  20  22  24
   6:  22  24  26  28  30  32
   7:  29  31  33  35  37  39  41
   8:  37  39  41  43  45  47  49  51
   9:  46  48  50  52  54  56  58  60  62
  10:  56  58  60  62  64  66  68  70  72  74
  11:  67  69  71  73  75  77  79  81  83  85  87
  ...

Cf. A000124, A004120, A046691, A051938, A055999, A056000, A155212, A167487, A167499, A167614, A246172, A334563, A356288.

Table[((n-1)(n+2))/2+2k,{n,20},{k,n}]//Flatten (* Harvey P. Dale, May 26 2023 *)

def T(n, k): return ((n-1) * (n+2))//2 + 2*k
for n in range(1, 12):
    for k in range(1,(n+1)): print(T(n,k), end = ', ')

# Indexed as a linear sequence.
def a000124(n): return n*(n+1)//2 + 1
def a(n):
    l = m = 0
    for k in range(1,n):
        lc = a000124(k - 1)
        if n >= lc:
            l = lc
            m = k
        else: break
    return n + m + (n - l)

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

T(n,k+1) = T(n,k) + 2, k < n.

cp's OEIS Frontend

A081344 Natural numbers in square maze arrangement, read by antidiagonals.

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A005337 Number of ways in which n identical balls can be distributed among 4 boxes in a row such that each pair of adjacent boxes contains at least 4 balls.

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A005338 Number of ways in which n identical balls can be distributed among 5 boxes in a row such that each pair of adjacent boxes contains at least 4 balls.

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A005340 Number of ways in which n identical balls can be distributed among 7 boxes in a row such that each pair of adjacent boxes contains at least 4 balls.

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A005339 Number of ways in which n identical balls can be distributed among 6 boxes in a row such that each pair of adjacent boxes contains at least 4 balls.

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A081345 First row in maze arrangement of natural numbers A081344.

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A259975 Irregular triangle read by rows: T(n,k) = number of ways of placing n balls into k boxes in such a way that any two adjacent boxes contain at least 4 balls.

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A356754 Triangle read by rows: T(n,k) = ((n-1)(n+2))/2 + 2k.