A091940 -id:A091940 - cp's OEIS frontend

A342128 Table read by antidiagonals upwards: T(n,k) is the number of n-colorings of the vertices of the k-dimensional hypercube such that no two adjacent vertices have the same color. n >= 0, k >=0.

0, 1, 0, 2, 0, 0, 3, 2, 0, 0, 4, 6, 2, 0, 0, 5, 12, 18, 2, 0, 0, 6, 20, 84, 114, 2, 0, 0, 7, 30, 260, 2652, 2970, 2, 0, 0, 8, 42, 630, 29660, 1321860, 1185282, 2, 0, 0, 9, 56, 1302, 198030, 187430900, 130253748108, 100301050602, 2, 0, 0, 10, 72, 2408, 932862, 10199069190, 2157531034816940

Offset: 0

Peter Kagey, Feb 28 2021

			Table begins:
  n\k|  0   1     2         3                4                              5
  ---+-----------------------------------------------------------------------
   0 |  0   0     0         0                0                              0
   1 |  1   0     0         0                0                              0
   2 |  2   2     2         2                2                              2
   3 |  3   6    18       114             2970                        1185282
   4 |  4  12    84      2652          1321860                   130253748108
   5 |  5  20   260     29660        187430900               2157531034816940
   6 |  6  30   630    198030      10199069190            7905235551766437150
   7 |  7  42  1302    932862     269591166222         7365707045872206479742
   8 |  8  56  2408   3440024    4221404762120      2337101560809838105414712
   9 |  9  72  4104  10599192   44876701584360    327425229254999498091796728
  10 | 10  90  6570  28478970  355148098691850  24489214732779742874109277530

Eric Weisstein's World of Mathematics, Chromatic Polynomial
Eric Weisstein's World of Mathematics, Hypercube Graph

Columns and rows: A002378 (k=1), A091940 (k=2), A140986 (k=3), A158348 (k=4), A380589 (k=5), A307334 (n=3).

Cf. A334278, A342088 (analogous for cross-polytope).

T(n,k) = Sum_{i=0..2^k} A334278(k,i)*n^i.

A100606 a(n) = n^4 + n^3 + n.

Original entry on oeis.org

0, 3, 26, 111, 324, 755, 1518, 2751, 4616, 7299, 11010, 15983, 22476, 30771, 41174, 54015, 69648, 88451, 110826, 137199, 168020, 203763, 244926, 292031, 345624, 406275, 474578, 551151, 636636, 731699, 837030, 953343, 1081376, 1221891, 1375674, 1543535, 1726308

Offset: 0

Douglas Winston (douglas.winston(AT)srupc.com), Nov 30 2004

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000583, A002523, A027445, A053699, A059826, A071253, A091940, A100019.

[n^4+n^3+n: n in [0..50]]; // Vincenzo Librandi, Jun 09 2011

Table[n^4+n^3+n,{n,0,40}] (* or *) LinearRecurrence[{5,-10,10,-5,1},{0,3,26,111,324},40] (* Harvey P. Dale, Apr 25 2015 *)

a(n)=n^4+n^3+n \\ Charles R Greathouse IV, Oct 21 2022

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5); a(0)=0, a(1)=3, a(2)=26, a(3)=111, a(4)=324. - Harvey P. Dale, Apr 25 2015
From Elmo R. Oliveira, Aug 29 2025: (Start)
G.f.: x*(3 + 11*x + 11*x^2 - x^3)/(1-x)^5.
E.g.f.: x*(3 + 10*x + 7*x^2 + x^3)*exp(x). (End)

A123658 a(n) = 1 + n^4 + n^6 + n^9 + n^10.

Original entry on oeis.org

5, 1617, 79543, 1315073, 11735001, 70591825, 322948907, 1208225793, 3874742893, 11001010001, 28297158495, 67080151297, 148467846593, 309923269713, 615105191251, 1168247947265, 2134605998037, 3768860634193, 6453801131783, 10752064160001, 17474246985385

Offset: 1

Jonathan Vos Post, Oct 04 2006

			a(40) = 1+40^(A001358(1))+40^(A001358(2))+40^(A001358(3))+40^(A001358(4)) = 1+40^4+40^6+40^9+40^10 = 10747908098560001.

T. D. Noe, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).

Cf. A001358, A002523, A091940, A123177, A123656-A123659.

[1+n^4+n^6+n^9+n^10: n in [0..50]]; // G. C. Greubel, Oct 17 2017

Table[1+n^4+n^6+n^9+n^10, {n,1,50}] (* G. C. Greubel, Oct 17 2017 *)
LinearRecurrence[{11,-55,165,-330,462,-462,330,-165,55,-11,1},{5,1617,79543,1315073,11735001,70591825,322948907,1208225793,3874742893,11001010001,28297158495},30] (* Harvey P. Dale, Jul 11 2025 *)

a(n)=1+n^4+n^6+n^9+n^10 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = 1 + n^4 + n^6 + n^9 + n^10.

G.f.: x*(x^10 -8*x^9 +615*x^8 +33654*x^7 +381288*x^6 +1242534*x^5 +1378908*x^4 +528210*x^3 +62031*x^2 +1562*x +5)/(1-x)^11. - Colin Barker, May 27 2012

A131473 a(n) = n^6 - n.

Original entry on oeis.org

0, 0, 62, 726, 4092, 15620, 46650, 117642, 262136, 531432, 999990, 1771550, 2985972, 4826796, 7529522, 11390610, 16777200, 24137552, 34012206, 47045862, 63999980, 85766100, 113379882, 148035866, 191102952, 244140600, 308915750

Offset: 0

Mohammad K. Azarian, Jul 27 2007

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (7, -21, 35, -35, 21, -7, 1).

Cf. A002378, A007531, A034262, A091940, A058895, A059826.

[n^6-n: n in [0..30]]; // Vincenzo Librandi, Aug 11 2011

A131473:=n->n^6-n; seq(A131473(n), n=0..30); # Wesley Ivan Hurt, Feb 25 2014

f[n_]:=n^6-n;f[Range[0,60]] (* Vladimir Joseph Stephan Orlovsky, Feb 10 2011 *)
Array[#^6-#&,60,0] (* Harvey P. Dale, Aug 10 2011 *)

[lucas_number1(3,n^3,n) for n in range(0, 27)] # Zerinvary Lajos, May 16 2009

A277443 Square array A(n,k) (n>=1, k>=1) read by antidiagonals: A(n,k) is the number of n-colorings of the prism graph Y_k on 2k vertices.

Original entry on oeis.org

0, 0, 0, 0, 2, 0, 0, 0, 18, 0, 0, 2, 12, 84, 0, 0, 0, 114, 264, 260, 0, 0, 2, 180, 2652, 1920, 630, 0, 0, 0, 858, 16080, 29660, 8520, 1302, 0, 0, 2, 1932, 119844, 367080, 198030, 28140, 2408, 0, 0, 0, 7074, 816984, 4843460, 4067280, 932862, 76272, 4104, 0, 0, 2, 18660, 5784492, 62682480, 85847910, 28576380, 3440024, 179424, 6570, 0

Offset: 1

Jeremy Tan, Oct 15 2016

Y_1 contains a loop, so has no colorings with any number of colors. Y_2 is the cycle graph C_4 with two double edges; these two graphs are therefore equivalent with respect to number of colorings.

			Square array A(n,k) begins:
  0,   0,    0,      0,       0,        0,          0, ...
  0,   2,    0,      2,       0,        2,          0, ...
  0,  18,   12,    114,     180,      858,       1932, ...
  0,  84,  264,   2652,   16080,   119844,     816984, ...
  0, 260, 1920,  29660,  367080,  4843460,   62682480, ...
  0, 630, 8520, 198030, 4067280, 85847910, 1800687000, ...

N. L. Biggs, R. M. Damerell and D. A. Sands, Recursive families of graphs, Journal of Combinatorial Theory Series B Volume 12 (1972), 123-131. MR0294172
Eric Weisstein's World of Mathematics, Prism Graph
Wikipedia, Chromatic polynomial

Cf. A277444 (colorings of Möbius ladders), A182406 (square grid graphs).
Columns k=1,2 are A000004 and A091940.
Rows n=1,2 are A000004 and A010673.

A(n,k) = (n^2-3n+3)^k+(n-1)((3-n)^k+(1-n)^k)+n^2-3n+1.

cp's OEIS Frontend

A342128 Table read by antidiagonals upwards: T(n,k) is the number of n-colorings of the vertices of the k-dimensional hypercube such that no two adjacent vertices have the same color. n >= 0, k >=0.

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A100606 a(n) = n^4 + n^3 + n.

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A123658 a(n) = 1 + n^4 + n^6 + n^9 + n^10.

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Magma

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A131473 a(n) = n^6 - n.

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A277443 Square array A(n,k) (n>=1, k>=1) read by antidiagonals: A(n,k) is the number of n-colorings of the prism graph Y_k on 2k vertices.

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