A014206 -id:A014206 - cp's OEIS frontend

A059173 Maximal number of regions into which 4-space can be divided by n hyperspheres.

1, 2, 4, 8, 16, 32, 62, 114, 198, 326, 512, 772, 1124, 1588, 2186, 2942, 3882, 5034, 6428, 8096, 10072, 12392, 15094, 18218, 21806, 25902, 30552, 35804, 41708, 48316, 55682, 63862, 72914, 82898, 93876, 105912, 119072, 133424, 149038

Offset: 0

N. J. A. Sloane, Feb 15 2001

n hyperspheres divide R^k into at most C(n-1, k) + Sum_{i=0..k} C(n, i) regions.
From Raphie Frank Nov 24 2012: (Start)
Define the gross polygonal sum, GPS(n), of an n-gon as the maximal number of combined points (p), intersections (i), connections (c = edges (e) + diagonals (d)) and areas (a) of a fully connected n-gon, plus the area outside the n-gon. The gross polygonal sum (p + i + c + a + 1) is equal to this sequence and, for all n > 0, then individual components of this sum can be calculated from the first 5 entries in row (n-1) of Pascal's triangle.
For example, the gross polygonal sum of a 7-gon (the heptagon):
Let row 6 of Pascal's triangle = {1, 6, 15, 20, 15, 6, 1} = A B C D E F G.
Points = 1 + 6 = A + B = 7 [A000027(n)].
Intersections = 20 + 15 = D + E = 35 [A000332(n+2)].
Connections = 6 + 15 = B + C = 21 [A000217(n)].
Areas inside = 15 + 20 + 15 = C + D + E = 50 [A006522(n+1)].
Areas outside = 1 = A = 1 [A000012(n)].
Then, GPS(7) = 7 + 35 + 21 + 50 + 1 = 2(A + B + C + D + E) = 114 = a(7). In general, a(n) = GPS(n). (End)

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 73, Problem 4.

Paolo Xausa, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A014206 (dim 2), A046127 (dim 3), A059173 (dim 4), A059174 (dim 5).
A row of A059250.
Cf. A000012, A000027, A000217, A000332, A006522.

LinearRecurrence[{5,-10,10,-5,1},2^Range[0,5],50] (* Paolo Xausa, Dec 29 2023 *)

a(0) = 1; a(n) = 2 * A000127(n), for n >= 1.
G.f.: -(x^5 + x^4 - 2*x^3 + 4*x^2 - 3*x + 1)/(x-1)^5. - Colin Barker, Oct 06 2012
E.g.f.: exp(x)*(2 + x^2 + x^4/12) - 1. - Stefano Spezia, May 19 2024

A059174 Maximal number of regions into which 5-space can be divided by n hyperspheres.

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 64, 126, 240, 438, 764, 1276, 2048, 3172, 4760, 6946, 9888, 13770, 18804, 25232, 33328, 43400, 55792, 70886, 89104, 110910, 136812, 167364, 203168, 244876, 293192, 348874, 412736, 485650, 568548, 662424, 768336, 887408, 1020832, 1169870

Offset: 0

N. J. A. Sloane, Feb 15 2001

n hyperspheres divide R^k into at most binomial(n-1, k) + Sum_{i=0..k} binomial(n, i) regions.

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 73, Problem 4.

Muniru A Asiru, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A014206 (dim 2), A046127 (dim 3), A059173 (dim 4), this sequence (dim 5).
Fifth row (k=5) of A059250.
Cf. A006261.

Concatenation([1], List([1..40], n-> Binomial(n-1,5) + Sum([0..5], i-> Binomial(n,i)))); # Muniru A Asiru, Dec 18 2018

[1] cat [(n^5-5*n^4+25*n^3+5*n^2+94*n+120)/60: n in [0..40]]; // Vincenzo Librandi, Dec 21 2018

seq(coeff(series((x^6+3*x^4-6*x^3+7*x^2-4*x+1)/(1-x)^6,x,n+1), x, n), n = 0 .. 40); # Muniru A Asiru, Dec 18 2018

Join[{1}, Table[((n^5 - 5 n^4 + 25 n^3 + 5 n^2 + 94 n + 120) / 60), {n, 0, 50}]] (* Vincenzo Librandi, Dec 21 2018 *)

a(n) = binomial(n-1, 5) + sum(i=0, 5, binomial(n, i)); \\ Michel Marcus, Jan 29 2016

a(n) = binomial(n-1, 5) + Sum_{i=0..5} binomial(n, i).
G.f.: (x^6 + 3*x^4 - 6*x^3 + 7*x^2 - 4*x + 1)/(x - 1)^6. - Colin Barker, Oct 06 2012
a(n) = 2*A006261(n-1), for n > 0. - Günter Rote, Dec 18 2018, by elementary manipulations.
E.g.f.: 1 + (1/60)*(120*x + 20*x^3 + x^5)*exp(x). - Franck Maminirina Ramaharo, Dec 21 2018

A241269 Denominator of c(n) = (n^2+n+2)/((n+1)(n+2)(n+3)).

Original entry on oeis.org

3, 6, 15, 60, 105, 21, 126, 360, 495, 330, 429, 1092, 1365, 420, 1020, 2448, 2907, 1710, 1995, 4620, 5313, 759, 3450, 7800, 8775, 4914, 5481, 12180, 13485, 3720, 8184, 17952, 19635, 10710, 11655, 25308, 27417, 3705, 15990, 34440, 37023, 19866, 21285, 45540

Offset: 0

Paul Curtz, Apr 18 2014

All terms are multiples of 3.
Difference table of c(n):
   1/3,     1/6,    2/15,    7/60,    2/21,...
  -1/6,    -1/30,  -1/60,   -1/84,   -1/105,...
   2/15,    1/60,   1/210,   1/420,   1/630,...
  -7/60,   -1/84,  -1/420,  -1/1260, -1/2520,... .
This is an autosequence of the second kind; the inverse binomial transform is the signed sequence. The main diagonal is the first upper diagonal multiplied by 2.
Denominators of the main diagonal: A051133(n+1).
Denominators of the first upper diagonal; A000911(n).
c(n) is a companion to A026741(n)/A045896(n).
Based on the Akiyama-Tanigawa transform applied to 1/(n+1) which yields the Bernoulli numbers A164555(n)/A027642(n).
Are the numerators of the main diagonal (-1)^n? If yes, what is the value of 1/3 - 1/30 + 1/210,... or 1 - 1/10 + 1/70 - 1/420, ... , from A002802(n)?
Is a(n+40) - a(n) divisible by 10?
No: a(5) = 21 but a(45) = 12972. # Robert Israel, Jul 17 2023
Are the common divisors to A014206(n) and A007531(n+3) of period 16: repeat 2, 4, 4, 2, 2, 16, 4, 2, 2, 4, 4, 2, 2, 8, 4, 2?
Reduce c(n) = f(n) = b(n)/a(n) = 1/3, 1/6, 2/15, 7/60, 11/105, 2/21, 11/126, 29/360, ... .
Consider the successively interleaved autosequences (also called eigensequences) of the second kind and of the first kind
  1,    1/2,  1/3,    1/4,     1/5,     1/6,   ...
  0,    1/6,  1/6,    3/20,    2/15,    5/42,  ...
  1/3,  1/6,  2/15,   7/60,   11/105,   2/21,  ...
  0,    1/10, 1/10,  13/140,   3/35,    5/63,  ...
  1/5,  1/10, 3/35,  11/140,  23/315,  43/630, ...
  0,    1/14, 1/14,  17/252,   4/63,   ...
This array is Au1(m,n). Au1(0,0)=1, Au1(0,1)=1/2.
Au1(m+1,n) = 2*Au1(m,n+1) - Au1(m,n).
First row: see A003506, Leibniz's Harmonic Triangle.
Second row: A026741/A045896.
a(n) is the denominator of the third row f(n).
The first column is 1, 0, 1/3, 0, 1/5, 0, 1/7, 0, ... . Numerators: A093178(n+1). This incites, considering tan(1), to introduce before the first row
Ta0(n) = 0, 1/2, 1/2, 5/12, 1/3, 4/15, 13/60, 151/840, ... .

Robert Israel, Table of n, a(n) for n = 0..10000

seq(denom((n^2+n+2)/((n+1)*(n+2)*(n+3))),n=0..1000);

Denominator[Table[(n^2+n+2)/Times@@(n+{1,2,3}),{n,0,50}]] (* Harvey P. Dale, Mar 27 2015 *)

for(n=0, 100, print1(denominator((n^2+n+2)/((n+1)*(n+2)*(n+3))), ", ")) \\ Colin Barker, Apr 18 2014

c(n) = A014206(n)/A007531(n+3).
The sum of the difference table main diagonal is 1/3 - 1/30 + 1/210 - ... = 10*A086466-4 = 4*(sqrt(5)*log(phi)-1) = 0.3040894... - Jean-François Alcover, Apr 22 2014
a(n) = (n+1)*(n+2)*(n+3)/gcd(4*n - 4, n^2 + n + 2), where gcd(4*n - 4, n^2 + n + 2) is periodic with period 16. - Robert Israel, Jul 17 2023

More terms from Colin Barker, Apr 18 2014

A059250 Square array read by antidiagonals: T(k,n) = binomial(n-1, k) + Sum_{i=0..k} binomial(n, i), k >= 1, n >= 0.

Original entry on oeis.org

1, 1, 2, 1, 2, 4, 1, 2, 4, 6, 1, 2, 4, 8, 8, 1, 2, 4, 8, 14, 10, 1, 2, 4, 8, 16, 22, 12, 1, 2, 4, 8, 16, 30, 32, 14, 1, 2, 4, 8, 16, 32, 52, 44, 16, 1, 2, 4, 8, 16, 32, 62, 84, 58, 18, 1, 2, 4, 8, 16, 32, 64, 114, 128, 74, 20, 1, 2, 4, 8, 16, 32, 64, 126, 198, 186, 92, 22, 1, 2, 4, 8, 16, 32, 64

Offset: 1

N. J. A. Sloane, Feb 15 2001

T(k,n) = maximal number of regions into which k-space can be divided by n hyperspheres (k >= 1, n >= 0).
For all fixed k, the sequences T(k,n) are complete. - Frank M Jackson, Jan 26 2012
T(k-1,n) is also the number of regions created by n generic hyperplanes through the origin in k-space (k >= 2). - Kent E. Morrison, Nov 11 2017

			Array begins
  1, 2, 4, 6,  8, 10, 12, ...
  1, 2, 4, 8, 14, 22, ...
  1, 2, 4, 8, 16, ...

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 73, Problem 4.

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
K. E. Morrison, From bocce to positivity: some probabilistic linear algebra, arXiv:1405.2994 [math.PR], 2014; Math. Mag., 86 (2013) 110-119.
L. Schläfli, Theorie der vielfachen Kontinuität, 1901. (See p. 41)
J. G. Wendel, A problem in geometric probability, Math. Scand., 11 (1962) 109-111.

Cf. A014206 (dim 2), A046127 (dim 3), A059173 (dim 4), A059174 (dim 5).

Apart from border, same as A059214. If the k=0 row is included, same as A178522.

getvalue[n_, k_] := If[n==0, 1, Binomial[n-1, k]+Sum[Binomial[n, i],{i, 0,k}]]; lexicographicLattice[{dim_, maxHeight_}] := Flatten[Array[Sort@Flatten[(Permutations[#1] &) /@     IntegerPartitions[#1 + dim - 1, {dim}], 1] &, maxHeight], 1]; pairs=lexicographicLattice[{2, 13}]-1; Table[getvalue[First[pairs[[j]]], Last[pairs[[j]]]+1], {j, 1, Length[pairs]}] (* Frank M Jackson, Mar 16 2013 *)

T(k,n) = 2 * Sum_{i=0..k-1} binomial(n-1, i), k >= 1, n >= 1. - Kent E. Morrison, Nov 11 2017

Corrected and edited by N. J. A. Sloane, Aug 31 2011, following a suggestion from Frank M Jackson

A011826 f-vectors for simplicial complexes of dimension at most 1 (graphs) on at most n-1 vertices.

Original entry on oeis.org

2, 3, 5, 9, 16, 27, 43, 65, 94, 131, 177, 233, 300, 379, 471, 577, 698, 835, 989, 1161, 1352, 1563, 1795, 2049, 2326, 2627, 2953, 3305, 3684, 4091, 4527, 4993, 5490, 6019, 6581, 7177, 7808, 8475, 9179, 9921, 10702, 11523, 12385, 13289

Offset: 1

Svante Linusson (linusson(AT)math.kth.se)

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Aviezri S. Fraenkel and Urban Larsson, Playability and arbitrarily large rat games, arXiv:1705.03061 [math.CO], 2017. See p. 27.
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.

Cf. A000124, A004006, A014206, A033547.

a=2;s=3;lst={0,1,s};Do[a+=n;s+=a;AppendTo[lst,s],{n,2,4!,1}];lst+2 (* Vladimir Joseph Stephan Orlovsky, May 24 2009 *)

a(n) = (n^3 - 3n^2 + 8n + 6)/6 fits all listed terms. - John W. Layman, Mar 13 1999

Empirical G.f.: -x*(x^3 - 5*x^2 + 5*x - 2) / (x - 1)^4. - Colin Barker, Sep 19 2012

A127410 Negative value of coefficient of x^(n-5) in the characteristic polynomial of a certain n X n integer circulant matrix.

Original entry on oeis.org

1875, 25920, 184877, 917504, 3582306, 11760000, 33820710, 87588864, 208295373, 461452992, 962836875, 1908408320, 3617795636, 6595852032, 11617856508, 19845120000, 32979115575, 53463778368, 84747328281, 131616866304, 200621093750, 300598812800, 443333396610

Offset: 5

Paul Max Payton, Jan 14 2007

The n X n circulant matrix used here has first row 1 through n and each successive row is a circular rotation of the previous row to the right by one element.

The coefficient of x^(n-5) exists only for n>4, so the sequence starts with a(5). In order to obtain a nonnegative sequence the coefficient (which is negative for all n>4) is multiplied by -1.

			The circulant matrix for n = 5 is
[1 2 3 4 5]
[5 1 2 3 4]
[4 5 1 2 3]
[3 4 5 1 2]
[2 3 4 5 1]
The characteristic polynomial of this matrix is x^5 - 5*x^4 -100*x^3 - 625*x^2 - 1750*x - 1875. The coefficient of x^(n-5) is -1875, hence a(5) = 1875.

Daniel Zwillinger, ed., "CRC Standard Mathematical Tables and Formulae", 31st Edition, ISBN 1-58488-291, Section 2.6.2.25 (page 141) and Section 2.6.11.3 (page 152).

T. D. Noe, Table of n, a(n) for n = 5..1000

Cf. A000142 (factorial numbers), A014206 (n^2+n+2), A127407, A127408, A127409, A127411, A127412.

[ -Coefficient(CharacteristicPolynomial(Matrix(IntegerRing(), n, n, [< i, j, 1 + (j-i) mod n > : i, j in [1..n] ] )), n-5) : n in [5..24] ]; // Klaus Brockhaus, Jan 27 2007

[ (n-4)*(n-3)*(n-2)*(n-1)*n^5*(4*n+16) / (2*Factorial(6)) : n in [5..24] ]; // Klaus Brockhaus, Jan 27 2007

n * (n+1) * (n+2) * (n+3) * (n+4)^5 * (4*n + 32) / (2 * factorial(6)); % Paul Max Payton, Jan 14 2007

a(n) = {-polcoef(charpoly(matrix(n,n,i,j,(j-i)%n+1),x),n-5)} \\ Klaus Brockhaus, Jan 27 2007

a(n) = {(4*n^10-24*n^9-20*n^8+360*n^7-704*n^6+384*n^5)/(2*6!)} \\ Klaus Brockhaus, Jan 27 2007

a(n+4) = n*(n+1)*(n+2)*(n+3)*(n+4)^5*(4*n+32)/(2*6!) for n>=1.
a(n) = (4*n^10-24*n^9-20*n^8+360*n^7-704*n^6+384*n^5)/(2*6!) for n>=5.
G.f.: x^5*(x^5+53*x^4-82*x^3-2882*x^2-5295*x-1875)/(x-1)^11. [Colin Barker, May 29 2012]

Edited by Klaus Brockhaus, Jan 27 2007

A127411 Negative value of coefficient of x^(n-6) in the characteristic polynomial of a certain n X n integer circulant matrix.

Original entry on oeis.org

27216, 453789, 3866624, 22674816, 103500000, 393286542, 1297410048, 3822832728, 10267329072, 25518796875, 59378761728, 130535973152, 273106821312, 547049504268, 1054272000000, 1962916959024, 3543150344976, 6218839661001, 10640820731904, 17789062500000

Offset: 6

Paul Max Payton, Jan 14 2007

The n X n circulant matrix used here has first row 1 through n and each successive row is a circular rotation of the previous row to the right by one element.

The coefficient of x^(n-6) exists only for n>5, so the sequence starts with a(6). In order to obtain a nonnegative sequence the coefficient (which is negative for all n>5) is multiplied by -1.

			The circulant matrix for n = 6 is
[1 2 3 4 5 6]
[6 1 2 3 4 5]
[5 6 1 2 3 4]
[4 5 6 1 2 3]
[3 4 5 6 1 2]
[2 3 4 5 6 1]
The characteristic polynomial of this matrix is x^6 - 6*x^5 -196*x^4 - 1980*x^3 - 10044*x^2 - 25920*x - 27216. The coefficient of x^(n-6) is -27216, hence a(6) = 27216.

Daniel Zwillinger, ed., "CRC Standard Mathematical Tables and Formulae", 31st Edition, ISBN 1-58488-291, Section 2.6.2.25 (page 141) and Section 2.6.11.3 (page 152).

T. D. Noe, Table of n, a(n) for n = 6..1000

Cf. A000142 (factorial numbers), A014206 (n^2+n+2), A127407, A127408, A127409, A127410, A127412.

[ -Coefficient(CharacteristicPolynomial(Matrix(IntegerRing(), n, n, [< i, j, 1 + (j-i) mod n > : i, j in [1..n] ] )), n-6) : n in [6..22] ]; // Klaus Brockhaus, Jan 27 2007

[ (n-5)*(n-4)*(n-3)*(n-2)*(n-1)*n^6*(5*n+19) / (2*Factorial(7)) : n in [6..22] ]; // Klaus Brockhaus, Jan 27 2007

n * (n+1) * (n+2) * (n+3) * (n+4) * (n+5)^6 * (5*n + 44) / (2*factorial(7)); % Paul Max Payton, Jan 14 2007

a(n) = {-polcoef(charpoly(matrix(n,n,i,j,(j-i)%n+1),x),n-6)} \\ Klaus Brockhaus, Jan 27 2007

a(n) = {(5*n^12-56*n^11+140*n^10+490*n^9-2905*n^8+4606*n^7-2280*n^6)/(2*7!)} \\ Klaus Brockhaus, Jan 27 2007

a(n+5) = n*(n+1)*(n+2)*(n+3)*(n+4)*(n+5)^6*(5*n+44)/(2*7!) for n>=1.
a(n) = (5*n^12 - 56*n^11 + 140*n^10 + 490*n^9 - 2905*n^8 + 4606*n^7 - 2280*n^6)/(2*7!) for n>=6.
G.f.: x^6*(x^6 + 131*x^5 + 150*x^4 - 20470*x^3 - 90215*x^2 - 99981*x - 27216)/(x-1)^13. - Colin Barker, May 29 2012

Edited by Klaus Brockhaus, Jan 27 2007

A347570 Table read by antidiagonals upward: the n-th row gives the lexicographically earliest infinite B_n sequence.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 2, 4, 4, 1, 2, 5, 8, 5, 1, 2, 6, 14, 13, 6, 1, 2, 7, 22, 33, 21, 7, 1, 2, 8, 32, 56, 72, 31, 8, 1, 2, 9, 44, 109, 154, 125, 45, 9, 1, 2, 10, 58, 155, 367, 369, 219, 66, 10, 1, 2, 11, 74, 257, 669, 927, 857, 376, 81, 11

Offset: 1

Peter Kagey, Sep 06 2021

A B_n sequence is a sequence such that all sums a(x_1) + a(x_2) + ... + a(x_n) are distinct for 1 <= x_1 <= x_2 <= ... <= x_n.

			Table begins:
n\k | 1  2   3   4    5     6     7      8
----+------------------------------------------
  1 | 1, 2,  3,  4,   5,    6,    7,     8, ...
  2 | 1, 2,  4,  8,  13,   21,   31,    45, ...
  3 | 1, 2,  5, 14,  33,   72,  125,   219, ...
  4 | 1, 2,  6, 22,  56,  154,  369,   857, ...
  5 | 1, 2,  7, 32, 109,  367,  927,  2287, ...
  6 | 1, 2,  8, 44, 155,  669, 2215,  6877, ...
  7 | 1, 2,  9, 58, 257, 1154, 4182, 14181, ...
  8 | 1, 2, 10, 74, 334, 1823, 8044, 28297, ...

Chai Wah Wu, Table of n, a(n) for n = 1..241
Eric Weisstein's World of Mathematics, B2 Sequence.

Cf. A000027 (n=1), A005282 (n=2), A096772 (n=3), A014206 (k=4), A370754 (k=5).

from itertools import count, islice, combinations_with_replacement
def A347570_gen(): # generator of terms
    asets, alists, klist = [set()], [[]], [1]
    while True:
        for i in range(len(klist)-1,-1,-1):
            kstart, alist, aset = klist[i], alists[i], asets[i]
            for k in count(kstart):
                bset = set()
                for d in combinations_with_replacement(alist+[k],i):
                    if (m:=sum(d)+k) in aset:
                        break
                    bset.add(m)
                else:
                    yield k
                    alists[i].append(k)
                    klist[i] = k+1
                    asets[i].update(bset)
                    break
        klist.append(1)
        asets.append(set())
        alists.append([])
A347570_list = list(islice(A347570_gen(),30)) # Chai Wah Wu, Sep 06 2023

A027713 Palindromes of form k^2 + k + 2.

Original entry on oeis.org

2, 4, 8, 22, 44, 212, 242, 464, 2552, 8558, 40604, 41414, 85558, 229922, 805508, 2029202, 2342432, 2737372, 4006004, 4437344, 4767674, 281585182, 400060004, 440727044, 805282508, 8059999508, 40000600004, 47997579974, 251476674152, 2626540456262, 2728292928272

Offset: 1

Patrick De Geest

Palindromes h such that 4*h - 7 is a square. - Bruno Berselli, Aug 29 2018

Giovanni Resta, Table of n, a(n) for n = 1..65
Patrick De Geest, Palindromic Quasi_Over_Squares of the form n^2+(n+X)

Cf. A014206, A027712, A028414, A027715.

[m: n in [0..10^7] | Intseq(m) eq Reverse(Intseq(m)) where m is n^2+n+2]; // Vincenzo Librandi, Mar 27 2011

palQ[n_] := Module[{idn = IntegerDigits[n]}, idn == Reverse[idn]]; Select[Table[n^2 + n + 2, {n, 0, 502000}], palQ] (* Harvey P. Dale, Sep 15 2011 *)

More terms from Giovanni Resta, Aug 28 2018

A127407 Negative value of coefficient of x^(n-2) in the characteristic polynomial of a certain n X n integer circulant matrix.

Original entry on oeis.org

3, 15, 44, 100, 195, 343, 560, 864, 1275, 1815, 2508, 3380, 4459, 5775, 7360, 9248, 11475, 14079, 17100, 20580, 24563, 29095, 34224, 40000, 46475, 53703, 61740, 70644, 80475, 91295, 103168, 116160, 130339, 145775, 162540, 180708, 200355

Offset: 2

Paul Max Payton, Jan 14 2007

The n X n circulant matrix used here has first row 1 through n and each successive row is a circular rotation of the previous row to the right by one element.

The coefficient of x^(n-2) exists only for n>1, so the sequence starts with a(2). In order to obtain a nonnegative sequence the coefficient (which is negative for all n>1) is multiplied by -1.

			The circulant matrix for n = 5 is
[1 2 3 4 5]
[5 1 2 3 4]
[4 5 1 2 3]
[3 4 5 1 2]
[2 3 4 5 1]
The characteristic polynomial of this matrix is x^5 - 5*x^4 -100*x^3 - 625*x^2 - 1750*x - 1875. The coefficient of x^(n-2) is -100, hence a(5) = 100.

Daniel Zwillinger, ed., "CRC Standard Mathematical Tables and Formulae", 31st Edition, ISBN 1-58488-291, Section 2.6.2.25 (page 141) and Section 2.6.11.3 (page 152).

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000142 (factorial numbers), A014206 (n^2+n+2), A127408, A127409, A127410, A127411, A127412.

[ -Coefficient(CharacteristicPolynomial(Matrix(IntegerRing(), n, n, [< i, j, 1 + (j-i) mod n > : i, j in [1..n] ] )), n-2) : n in [2..38] ]; // Klaus Brockhaus, Jan 27 2007

[ (n-1) * n^2 * (n+7) / (2 * Factorial(3)) : n in [2..38] ]; // Klaus Brockhaus, Jan 27 2007

n * (n+1)^2 * (n+8) / (2 * factorial(3)); % Paul Max Payton, Jan 14 2007

a(n) = {-polcoeff(charpoly(matrix(n,n,i,j,(j-i)%n+1),x),n-2)} \\ Klaus Brockhaus, Jan 27 2007

a(n) = {(n^4+6*n^3-7*n^2)/(2*3!)} \\ Klaus Brockhaus, Jan 27 2007

a(n+1) = n*(n+1)^2*(n+8)/(2*3!) for n>=1.
a(n) = ((n-1)^4+10*(n-1)^3+17*(n-1)^2+8*(n-1))/(2*3!) for n>=2.
a(n) = (n^2*(-7+6*n+n^2))/12. G.f.: x^2*(3-x^2)/(1-x)^5. - Colin Barker, May 13 2012

Edited by Klaus Brockhaus, Jan 27 2007

cp's OEIS Frontend

A059173 Maximal number of regions into which 4-space can be divided by n hyperspheres.

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A059174 Maximal number of regions into which 5-space can be divided by n hyperspheres.

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A241269 Denominator of c(n) = (n^2+n+2)/((n+1)*(n+2)*(n+3)).

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A059250 Square array read by antidiagonals: T(k,n) = binomial(n-1, k) + Sum_{i=0..k} binomial(n, i), k >= 1, n >= 0.

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A011826 f-vectors for simplicial complexes of dimension at most 1 (graphs) on at most n-1 vertices.

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A127410 Negative value of coefficient of x^(n-5) in the characteristic polynomial of a certain n X n integer circulant matrix.

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A127411 Negative value of coefficient of x^(n-6) in the characteristic polynomial of a certain n X n integer circulant matrix.

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A241269 Denominator of c(n) = (n^2+n+2)/((n+1)(n+2)(n+3)).