A002721 -id:A002721 - cp's OEIS frontend

A244878 Number of 6 X 6 traceless symmetric magic squares with magic sum n.

1, 15, 130, 760, 3355, 12043, 36935, 100135, 245870, 556580, 1177295, 2351165, 4469610, 8141210, 14284170, 24247962, 39970575, 64178685, 100639000, 154470030, 232524589, 343854445, 500269705, 717006745, 1013519780, 1414412506, 1950527645, 2660213675, 3590789540, 4800229700

Offset: 0

N. J. A. Sloane, Jul 08 2014

Colin Barker, Table of n, a(n) for n = 0..1000
R. P. Stanley, Examples of Magic Labelings, Unpublished Notes, 1973 [Cached copy, with permission]
Index entries for linear recurrences with constant coefficients, signature (9,-35,75,-90,42,42,-90,75,-35,9,-1).

Row n=6 of A333351.

LinearRecurrence[{9,-35,75,-90,42,42,-90,75,-35,9,-1},{1,15,130,760,3355,12043,36935,100135,245870,556580,1177295},30] (* Harvey P. Dale, Jul 18 2024 *)

Vec((1 + 6*x + 30*x^2 + 40*x^3 + 30*x^4 + 6*x^5 + x^6) / ((1 - x)^10*(1 + x)) + O(x^30)) \\ Colin Barker, Jan 12 2017

G.f.: (1 + 6*x + 30*x^2 + 40*x^3 + 30*x^4 + 6*x^5 + x^6) / ((1 - x)^10*(1 + x)).

a(n) = (945*(507+5*(-1)^n) + 1480896*n + 2062800*n^2 + 1747040*n^3 + 989100*n^4 + 383628*n^5 + 100800*n^6 + 17160*n^7 + 1710*n^8 + 76*n^9) / 483840. - Colin Barker, Jan 12 2017

A125196 Number of magic labelings of the Petersen graph with magic sum n.

Original entry on oeis.org

1, 6, 27, 87, 228, 513, 1034, 1914, 3315, 5440, 8541, 12921, 18942, 27027, 37668, 51428, 68949, 90954, 118255, 151755, 192456, 241461, 299982, 369342, 450983, 546468, 657489, 785869, 933570, 1102695, 1295496

Offset: 0

R. J. Mathar, Jan 25 2007

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
R. P. Stanley, Examples of Magic Labelings, Unpublished Notes, 1973 [Cached copy, with permission]
M. M. Ahmed, Algebraic Combinatorics of Magic Squares, math.CO/0405476.
Index entries for linear recurrences with constant coefficients, signature (5,-9,5,5,-9,5,-1).

a := proc(r) local r1 ; r1 := r^5/24+5*r^4/16+25*r^3/24+15*r^2/8+23*r/12 ; if r mod 2 = 0 then r1+1 ; else r1+13/16 ; fi ; end: for n from 0 to 30 do printf("%d ",a(n)) ; od;

CoefficientList[Series[(x^4 + x^3 + 6x^2 + x + 1)/(1 - x)^6/(1 + x), {x, 0, 40}], x] (* Vincenzo Librandi, Dec 12 2012 *)
LinearRecurrence[{5,-9,5,5,-9,5,-1},{1,6,27,87,228,513,1034},40] (* Harvey P. Dale, Sep 10 2024 *)

a(n) = (1/32)*(29*C(n+5,5) + 21*C(n+4,5) + 126*C(n+3,5) - 34*C(n+2,5) + 21*C(n+1,5) - 3*C(n,5) + 3*(-1)^n). [Stanley]. - N. J. A. Sloane, Jul 07 2014

G.f.: (x^4+x^3+6x^2+x+1)/((1-x)^6*(1+x)) [Stanley; Ahmed].

Stanley reference added by N. J. A. Sloane, Jul 07 2014

A125198 Number of magical labelings of the octahedral graph of magic sum n.

Original entry on oeis.org

1, 8, 40, 144, 417, 1032, 2272, 4568, 8545, 15072, 25320, 40824, 63553, 95984, 141184, 202896, 285633, 394776, 536680, 718784, 949729, 1239480, 1599456, 2042664, 2583841, 3239600, 4028584, 4971624, 6091905, 7415136, 8969728, 10786976, 12901249, 15350184

Offset: 0

R. J. Mathar, Jan 25 2007

Colin Barker, Table of n, a(n) for n = 0..1000
M. M. Ahmed, Algebraic Combinatorics of Magic Squares, math.CO/0405476, p73.
R. P. Stanley, Examples of Magic Labelings, Unpublished Notes, 1973 [Cached copy, with permission]
Index entries for linear recurrences with constant coefficients, signature (6,-14,14,0,-14,14,-6,1).

a := proc(r) local r2 ; r2 := r^6/120+r^5/10+25*r^4/48+3*r^3/2+38*r^2/15+12*r/5 ; if r mod 2 = 0 then r2+1 ; else r2+15/16 ; fi ; end: for n from 0 to 40 do printf("%d ",a(n)) ; od;

(1 + 2*x + 6*x^2 + 2*x^3 + x^4)/((1 - x)^7*(1 + x)) + O[x]^40 // CoefficientList[#, x]& (* Jean-François Alcover, Apr 01 2018 *)

Vec((1+2*x+6*x^2+2*x^3+x^4)/((1-x)^7*(1+x)) + O(x^40)) \\ Colin Barker, Jan 13 2017

G.f.: (1+2*x+6*x^2+2*x^3+x^4)/((1-x)^7*(1+x)). [Stanley] - N. J. A. Sloane, Jul 07 2014
From Colin Barker, Jan 13 2017: (Start)
a(n) = (15*(31+(-1)^n) + 1152*n + 1216*n^2 + 720*n^3 + 250*n^4 + 48*n^5 + 4*n^6) / 480.
a(n) = 6*a(n-1) - 14*a(n-2) + 14*a(n-3) - 14*a(n-5) + 14*a(n-6) - 6*a(n-7) + a(n-8) for n>7.
(End)

Stanley reference added by N. J. A. Sloane, Jul 07 2014

A244495 Number of 3 X 3 matrices of nonnegative integer entries with all row and column sums <= n.

Original entry on oeis.org

1, 34, 451, 3380, 17531, 70466, 235014, 679722, 1757085, 4147792, 9084361, 18683314, 36421463, 67798940, 121239308, 209285436, 350158809, 569759574, 904194895, 1402934104, 2132700691, 3182223374, 4667981330, 6741092150, 9595505205, 13477677876, 18697927509, 25643668006, 34794756655

Offset: 0

N. J. A. Sloane, Jul 06 2014

nonn

			a(1)=34:
0 1's: 1,
1 1: 9,
2 1's: 3*3*2 = 18,
3 1's: 6 (transversals),
total = 34.

Stanley, Richard P., Linear homogeneous Diophantine equations and magic labelings of graphs. Duke Math. J. 40 (1973), 607-632.
Stanley, Richard P., Magic labelings of graphs, symmetric magic squares, systems of parameters, and Cohen-Macaulay rings. Duke Math. J. 43 (1976), no. 3, 511-531.

Robert Israel, Table of n, a(n) for n = 0..10000
R. P. Stanley, Examples of Magic Labelings, Unpublished Notes, 1973 [Cached copy, with permission]

[1+(25/6)*k+(3337/420)*k^2+(13777/1512)*k^3+(3289/480)*k^4+(9983/2880)*k^5+(281/240)*k^6+(73/288)*k^7+(107/3360)*k^8+(107/60480)*k^9 : k in [0..30]]; // Wesley Ivan Hurt, Jul 06 2014

f:= k -> 1+(25/6)*k+(3337/420)*k^2+(13777/1512)*k^3+(3289/480)*k^4+(9983/2880)*k^5+(281/240)*k^6+(73/288)*k^7+(107/3360)*k^8+(107/60480)*k^9:
seq(f(k),k=0..1000); # Robert Israel, Jul 06 2014

CoefficientList[Series[(1 + 24*x + 156*x^2 + 280*x^3 + 156*x^4 + 24*x^5 + x^6)/(1 - x)^10, {x, 0, 30}], x] (* Wesley Ivan Hurt, Jul 06 2014 *)

G.f.: (1+24*x+156*x^2+280*x^3+156*x^4+24*x^5+x^6)/(1-x)^10.

a(k) = 1+(25/6)*k+(3337/420)*k^2+(13777/1512)*k^3+(3289/480)*k^4+(9983/2880)*k^5+(281/240)*k^6+(73/288)*k^7+(107/3360)*k^8+(107/60480)*k^9. - Robert Israel, Jul 06 2014

A244498 Number of magic labelings of the nodes of the 4 X 4 grid graph with magic sum n.

Original entry on oeis.org

1, 36, 446, 3172, 15891, 62408, 204828, 585672, 1501269, 3521452, 7674810, 15723500, 30556903, 56739216, 101252408, 174482832, 291507177, 473741364, 751024438, 1164218484, 1768415099, 2636848984, 3865629780, 5579414360, 7938153405, 11145058236, 15455946546, 21190138876, 28743091407

Offset: 0

N. J. A. Sloane, Jul 07 2014

The graph has 16 nodes and 24 edges.

The node labels are nonnegative integers, and the sum along any of the 4 rows or 4 columns is n.

Colin Barker, Table of n, a(n) for n = 0..1000
R. P. Stanley, Examples of Magic Labelings, Unpublished Notes, 1973 [Cached copy, with permission]
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Vec((1 + 26*x + 131*x^2 + 212*x^3 + 131*x^4 + 26*x^5 + x^6) / ((1 - x)^10) + O(x^40)) \\ Colin Barker, Jan 11 2017

G.f.: (1 + 26*x + 131*x^2 + 212*x^3 + 131*x^4 + 26*x^5 + x^6) / ((1 - x)^10).
From Colin Barker, Jan 11 2017: (Start)
a(n) = (7560 + 34164*n + 67044*n^2 + 75190*n^3 + 53382*n^4 + 25095*n^5 + 7896*n^6 + 1620*n^7 + 198*n^8 + 11*n^9) / 7560.
a(n) = 10*a(n-1) - 45*a(n-2) + 120*a(n-3) - 210*a(n-4) + 252*a(n-5) - 210*a(n-6) + 120*a(n-7) - 45*a(n-8) + 10*a(n-9) - a(n-10) for n>9.
(End)

A244866 Let G denote the 7-node, 12-edge graph formed from a hexagon with main diagonals drawn and a node at the center; a(n) = number of magic labelings of G with magic sum 2n.

Original entry on oeis.org

1, 18, 114, 438, 1263, 3024, 6356, 12132, 21501, 35926, 57222, 87594, 129675, 186564, 261864, 359720, 484857, 642618, 839002, 1080702, 1375143, 1730520, 2155836, 2660940, 3256565, 3954366, 4766958, 5707954, 6792003, 8034828, 9453264, 11065296, 12890097, 14948066, 17260866, 19851462, 22744159

Offset: 0

N. J. A. Sloane, Jul 07 2014

Colin Barker, Table of n, a(n) for n = 0..1000
R. P. Stanley, Examples of Magic Labelings, Unpublished Notes, 1973 [Cached copy, with permission]
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

LinearRecurrence[{6,-15,20,-15,6,-1},{1,18,114,438,1263,3024},40] (* Harvey P. Dale, Nov 09 2022 *)

Vec((1 + 12*x + 21*x^2 + 4*x^3) / (1 - x)^6 + O(x^40)) \\ Colin Barker, Jan 11 2017

G.f.: (1 + 12*x + 21*x^2 + 4*x^3) / (1 - x)^6.
From Colin Barker, Jan 11 2017: (Start)
a(n) = (n + 1)*(n + 2)*(19*n^3 + 63*n^2 + 68*n + 30) / 60.
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) for n>5.
(End)

A244867 Let G denote the 9-node, 16-edge graph formed from an octagon with main diagonals drawn and a node at the center; a(n) = number of magic labelings of G with magic sum 2n.

Original entry on oeis.org

1, 32, 320, 1784, 7040, 22104, 58980, 139320, 299343, 596200, 1115972, 1983488, 3374150, 5527952, 8765880, 13508880, 20299581, 29826960, 42954136, 60749480, 84521228, 115855784, 156659900, 209206920, 276187275, 360763416, 466629372, 598075120, 760055954, 958267040, 1199223344, 1490345120

Offset: 0

N. J. A. Sloane, Jul 07 2014

Colin Barker, Table of n, a(n) for n = 0..1000
R. P. Stanley, Examples of Magic Labelings, Unpublished Notes, 1973 [Cached copy, with permission]
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

LinearRecurrence[{8,-28,56,-70,56,-28,8,-1},{1,32,320,1784,7040,22104,58980,139320},40] (* Harvey P. Dale, Aug 17 2019 *)

Vec((1 + 24*x + 92*x^2 + 64*x^3 + 6*x^4) / (1 - x)^8 + O(x^40)) \\ Colin Barker, Jan 11 2017

G.f.: (1 + 24*x + 92*x^2 + 64*x^3 + 6*x^4) / (1 - x)^8.
From Colin Barker, Jan 11 2017: (Start)
a(n) = (5040 + 22164*n + 43092*n^2 + 46963*n^3 + 30240*n^4 + 11326*n^5 + 2268*n^6 + 187*n^7) / 5040.
a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8) for n>7.
(End)

A244877 Number of 4 X 4 traceless matrices of nonnegative integers with all row and column sums <= n.

Original entry on oeis.org

1, 108, 3396, 49852, 447777, 2862384, 14265040, 58788720, 208577633, 655618172, 1864806372, 4877412332, 11877770177, 27199536768, 59037366080, 122252641280, 242826555969, 464730264172, 860261412132, 1545252073596, 2700999547873, 4605325025136, 7675844020560, 12529356938800

Offset: 0

N. J. A. Sloane, Jul 08 2014

Colin Barker, Table of n, a(n) for n = 0..1000
R. P. Stanley, Examples of Magic Labelings, Unpublished Notes, 1973 [Cached copy, with permission]
Index entries for linear recurrences with constant coefficients, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).

Vec((1 + 95*x + 2070*x^2 + 13842*x^3 + 34416*x^4 + x^9 + 95*x^8 + 2070*x^7 + 13842*x^6 + 34416*x^5) / (1-x)^13 + O(x^30)) \\ Colin Barker, Jan 11 2017

G.f.: (1 + 95*x + 2070*x^2 + 13842*x^3 + 34416*x^4 + x^9 + 95*x^8 + 2070*x^7 + 13842*x^6 + 34416*x^5) / (1-x)^13.

a(n) = ((2+n)^2*(226800 + 1014480*n + 2137860*n^2 + 2775656*n^3 + 2422453*n^4 + 1468908*n^5 + 621705*n^6 + 180336*n^7 + 34191*n^8 + 3820*n^9 + 191*n^10)) / 907200. - Colin Barker, Jan 11 2017

A244882 Expansion of (1 + 2x + 2x^2) / (1 - x)^6.

Original entry on oeis.org

1, 8, 35, 110, 280, 616, 1218, 2220, 3795, 6160, 9581, 14378, 20930, 29680, 41140, 55896, 74613, 98040, 127015, 162470, 205436, 257048, 318550, 391300, 476775, 576576, 692433, 826210, 979910, 1155680, 1355816, 1582768, 1839145, 2127720, 2451435, 2813406

Offset: 0

N. J. A. Sloane, Jul 08 2014

Colin Barker, Table of n, a(n) for n = 0..1000
R. P. Stanley, Examples of Magic Labelings, Unpublished Notes, 1973 [Cached copy, with permission]See page 33.
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

CoefficientList[Series[(1+2x+2x^2)/(1-x)^6,{x,0,40}],x] (* or *) LinearRecurrence[{6,-15,20,-15,6,-1},{1,8,35,110,280,616},40] (* Harvey P. Dale, Dec 26 2016 *)

Vec((1 + 2*x + 2*x^2) / (1 - x)^6 + O(x^40)) \\ Colin Barker, Jan 12 2017

G.f.: (1 + 2*x + 2*x^2) / (1 - x)^6.
From Colin Barker, Jan 12 2017: (Start)
a(n) = (24 + 62*n + 63*n^2 + 33*n^3 + 9*n^4 + n^5) / 24.
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) for n>5.
(End)

A244883 Expansion of (1+6x+16x^2+8*x^3+x^4)/(1-x)^8.

Original entry on oeis.org

1, 14, 100, 472, 1691, 4988, 12744, 29160, 61149, 119482, 220220, 386464, 650455, 1056056, 1661648, 2543472, 3799449, 5553510, 7960468, 11211464, 15540019, 21228724, 28616600, 38107160, 50177205, 65386386, 84387564, 107938000, 136911407, 172310896, 215282848

Offset: 0

N. J. A. Sloane, Jul 08 2014

Todd Silvestri, Table of n, a(n) for n = 0..10000
R. P. Stanley, Examples of Magic Labelings, Unpublished Notes, 1973 [Cached copy, with permission]
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

[((n+1)*(n+2)*(n+3)*(n*(n+4)*(n*(16*n+57)+137)+420))/2520: n in [0..40]]; // Vincenzo Librandi, Nov 16 2014

a[n_Integer/;n>=0]:=((n+1) (n+2) (n+3) (n (n+4) (n (16 n+57)+137)+420))/2520 (* Todd Silvestri, Nov 16 2014 *)
CoefficientList[Series[(1 + 6 x + 16 x^2 + 8 x^3 + x^4) / (1 - x)^8, {x, 0, 100}], x] (* Vincenzo Librandi, Nov 16 2014 *)
LinearRecurrence[{8,-28,56,-70,56,-28,8,-1},{1,14,100,472,1691,4988,12744,29160},40] (* Harvey P. Dale, May 11 2020 *)

a(n) = ((n+1)*(n+2)*(n+3)*(n*(n+4)*(n*(16*n+57)+137)+420))/2520. - Todd Silvestri, Nov 16 2014

cp's OEIS Frontend

A244878 Number of 6 X 6 traceless symmetric magic squares with magic sum n.

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A125196 Number of magic labelings of the Petersen graph with magic sum n.

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Maple

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A125198 Number of magical labelings of the octahedral graph of magic sum n.

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A244495 Number of 3 X 3 matrices of nonnegative integer entries with all row and column sums <= n.

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A244498 Number of magic labelings of the nodes of the 4 X 4 grid graph with magic sum n.

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A244866 Let G denote the 7-node, 12-edge graph formed from a hexagon with main diagonals drawn and a node at the center; a(n) = number of magic labelings of G with magic sum 2n.

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A244867 Let G denote the 9-node, 16-edge graph formed from an octagon with main diagonals drawn and a node at the center; a(n) = number of magic labelings of G with magic sum 2n.

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A244877 Number of 4 X 4 traceless matrices of nonnegative integers with all row and column sums <= n.

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A244882 Expansion of (1 + 2*x + 2*x^2) / (1 - x)^6.

A244882 Expansion of (1 + 2x + 2x^2) / (1 - x)^6.

A244883 Expansion of (1+6x+16x^2+8*x^3+x^4)/(1-x)^8.