A002721 -id:A002721 - cp's OEIS frontend

A100182 Structured tetragonal anti-prism numbers.

1, 8, 28, 68, 135, 236, 378, 568, 813, 1120, 1496, 1948, 2483, 3108, 3830, 4656, 5593, 6648, 7828, 9140, 10591, 12188, 13938, 15848, 17925, 20176, 22608, 25228, 28043, 31060, 34286, 37728, 41393, 45288, 49420, 53796, 58423, 63308, 68458, 73880, 79581, 85568, 91848, 98428, 105315, 112516

Offset: 1

James A. Record (james.record(AT)gmail.com), Nov 07 2004

If offset is changed to 0, this is the number of magic labelings of the 5-node, 8-edge graph formed from a square with both diagonals drawn and a node at the center [Stanley]. - N. J. A. Sloane, Jul 07 2014

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
R. P. Stanley, Examples of Magic Labelings, Unpublished Notes, 1973 [Cached copy, with permission]
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A100185 - structured anti-prisms; A100145 for more on structured numbers.

[(1/6)*(7*n^3-3*n^2+2*n): n in [1..40]]; // Vincenzo Librandi, Aug 18 2011

Table[(7*n^3 - 3*n^2 + 2*n)/6, {n,1,40}] (* or *) LinearRecurrence[{4, -6, 4, -1}, {1, 8, 28, 68}, 40] (* G. C. Greubel, Nov 08 2018 *)

vector(40, n, (7*n^3 -3*n^2 +2*n)/6) \\ G. C. Greubel, Nov 08 2018

a(n) = (1/6)*(7*n^3 - 3*n^2 + 2*n). [Corrected by Luca Colucci, Mar 01 2011]
G.f.: x*(1 + 4*x + 2*x^2)/(1-x)^4. - Colin Barker, Jun 08 2012
E.g.f.: (6*x +18*x^2 +7*x^3)*exp(x)/6. - G. C. Greubel, Nov 08 2018
a(n) = binomial(n,3) + n^3. - Pedro Caceres, Jul 28 2019

A108675 a(n) = (n+1)(n+2)(61n^4 + 366n^3 + 845n^2 + 888n + 360)/720.

Original entry on oeis.org

1, 21, 157, 707, 2353, 6405, 15106, 31998, 62349, 113641, 196119, 323401, 513149, 787801, 1175364, 1710268, 2434281, 3397485, 4659313, 6289647, 8369977, 10994621, 14272006, 18326010, 23297365, 29345121, 36648171, 45406837

Offset: 0

Emeric Deutsch, Jun 17 2005

Kekulé numbers for certain benzenoids.

S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (p. 231, # 35).

R. P. Stanley, Examples of Magic Labelings, Unpublished Notes, 1973 [Cached copy, with permission] See p. 31.
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

a:=n->(n+1)*(n+2)*(61*n^4+366*n^3+845*n^2+888*n+360)/720: seq(a(n),n=0..32);

G.f.: (1 + 14*x + 31*x^2 + 14*x^3 + x^4)/(1-x)^7.

A244864 a(n) = binomial(n+5,5) + 4binomial(n+4,5) + 4binomial(n+3,5) + binomial(n+2,5).

Original entry on oeis.org

1, 10, 49, 165, 440, 1001, 2030, 3774, 6555, 10780, 16951, 25675, 37674, 53795, 75020, 102476, 137445, 181374, 235885, 302785, 384076, 481965, 598874, 737450, 900575, 1091376, 1313235, 1569799, 1864990, 2203015, 2588376, 3025880, 3520649, 4078130, 4704105, 5404701, 6186400, 7056049

Offset: 0

N. J. A. Sloane, Jul 07 2014

Alois P. Heinz, 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).

a:= n-> (2*n+3)*(n+2)*(n+1)*(n^2+3*n+4)/24:
seq(a(n), n=0..40);  # Alois P. Heinz, Jul 11 2014

Table[Binomial[n+5,5]+4*Binomial[n+4,5]+4*Binomial[n+3,5]+ Binomial[ n+2,5],{n,0,40}] (* or *) LinearRecurrence[{6,-15,20,-15,6,-1},{1,10,49,165,440,1001},40] (* Harvey P. Dale, Nov 13 2014 *)
a[n_] := (2 n^5 + 15 n^4 + 48 n^3 + 81 n^2 + 70 n + 24)/  24; Array[a, 40, 0] (* or *)
CoefficientList[Series[(x^3 + 4 x^2 + 4 x + 1)/(x - 1)^6, {x, 0, 40}], x] (* Robert G. Wilson v, Feb 26 2015 *)

a(n)=(2*n+3)*(n+2)*(n+1)*(n^2+3*n+4)/24 \\ Charles R Greathouse IV, Oct 21 2022

G.f.: (x+1)*(x^2+3*x+1)/(x-1)^6; a(n) = (2*n+3)*(n+2)*(n+1)*(n^2+3*n+4)/24. - Alois P. Heinz, Jul 11 2014

a(n) = Sum_{k=A000292(n)..A000292(n+1)} k. - J. M. Bergot, Feb 25 2015

A244865 Number of 3 X 3 symmetric matrices with row and column sums <= n.

Original entry on oeis.org

1, 14, 85, 336, 1023, 2610, 5860, 11942, 22555, 40068, 67677, 109578, 171157, 259196, 382096, 550116, 775629, 1073394, 1460845, 1958396, 2589763, 3382302, 4367364, 5580666, 7062679, 8859032, 11020933, 13605606, 16676745, 20304984, 24568384, 29552936, 35353081, 42072246, 49823397, 58729608

Offset: 0

N. J. A. Sloane, Jul 07 2014

a(n) is the number of perimeter-magic hollow triangles of order 4 (4 elements per edge, 9 elements in total) with magic edge sum n+4. The triangles have 9 elements >=1, not necessarily distinct. Triangles obtained by rotations and flips (D_6 symmetry) are counted as being distinct. Associated triangles of order 3 are counted in A019298. - R. J. Mathar, Mar 05 2025

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,-14,14,0,-14,14,-6,1).

Vec((1 + 8*x + 15*x^2 + 8*x^3 + x^4) / ((1 - x)^7*(1 + x)) + O(x^40)) \\ Colin Barker, Jan 11 2017

G.f.: (1 + 8*x + 15*x^2 + 8*x^3 + x^4) / ((1 - x)^7*(1 + x)).
From Colin Barker, Jan 11 2017: (Start)
a(n) = (15*(127+(-1)^n) + 6432*n + 8936*n^2 + 6480*n^3 + 2570*n^4 + 528*n^5 + 44*n^6) / 1920.
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)

A244868 Number of symmetric 5 X 5 matrices of nonnegative integers with zeros on the main diagonal and every row and column adding to n.

Original entry on oeis.org

1, 22, 158, 654, 1980, 4906, 10577, 20588, 37059, 62710, 100936, 155882, 232518, 336714, 475315, 656216, 888437, 1182198, 1548994, 2001670, 2554496, 3223242, 4025253, 4979524, 6106775, 7429526, 8972172, 10761058, 12824554, 15193130, 17899431, 20978352, 24467113, 28405334, 32835110, 37801086

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).

Even bisection of row n=5 of A333351.

Cf. A053494.

Vec((1 + 16*x + 41*x^2 + 16*x^3 + x^4) / (1 - x)^6 + O(x^40)) \\ Colin Barker, Jan 11 2017

G.f.: (1 + 16*x + 41*x^2 + 16*x^3 + x^4) / (1 - x)^6.
From Colin Barker, Jan 11 2017: (Start)
a(n) = (24 + 94*n + 165*n^2 + 155*n^3 + 75*n^4 + 15*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)

A244870 Number of magic labelings with magic sum n of 2nd graph shown in link.

Original entry on oeis.org

1, 8, 37, 121, 318, 717, 1446, 2678, 4639, 7614, 11955, 18087, 26516, 37835, 52732, 71996, 96525, 127332, 165553, 212453, 269434, 338041, 419970, 517074, 631371, 765050, 920479, 1100211, 1306992, 1543767, 1813688, 2120120, 2466649, 2857088, 3295485

Offset: 0

N. J. A. Sloane, Jul 08 2014

N. J. A. Sloane, Graphs for A244869-A244876.
R. P. Stanley, Examples of Magic Labelings, Unpublished Notes, 1973 [Cached copy, with permission]
Index entries for linear recurrences with constant coefficients, signature (5, -9, 5, 5, -9, 5, -1).

Cf. A244869-A244876.

Table[7 n^5/120 + 7 n^4/16 + 35 n^3/24 + 21 n^2/8 + 149 n/60 + (-1)^n/32 + 31/32, {n, 0, 40}] (* Bruno Berselli, Jul 08 2014 *)

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

a(n) = 7*n^5/120 + 7*n^4/16 + 35*n^3/24 + 21*n^2/8 + 149*n/60 + (-1)^n/32 + 31/32. [Bruno Berselli, Jul 08 2014]

A244871 Number of magic labelings with magic sum n of 3rd graph shown in link.

Original entry on oeis.org

1, 10, 55, 217, 672, 1755, 4030, 8386, 16135, 29140, 49941, 81915, 129430, 198037, 294652, 427780, 607725, 846846, 1159795, 1563805, 2078956, 2728495, 3539130, 4541382, 5769907, 7263880, 9067345, 11229631, 13805730, 16856745, 20450296, 24661000, 29570905

Offset: 0

N. J. A. Sloane, Jul 08 2014

N. J. A. Sloane, Graphs for A244869-A244876.
R. P. Stanley, Examples of Magic Labelings, Unpublished Notes, 1973 [Cached copy, with permission]
Index entries for linear recurrences with constant coefficients, signature (5, -8, 0, 14, -14, 0, 8, -5, 1).

Cf. A244869-A244876.

Table[n^6/48 + 3 n^5/16 + 77 n^4/96 + 2 n^3 + 37 n^2/12 - n (-1)^n/32 + 89 n/32 - 3 (-1)^n/64 + 67/64, {n,0,40}] (* Bruno Berselli, Jul 08 2014 *)

G.f.: (1+5*x+13*x^2+22*x^3+13*x^4+5*x^5+x^6)/((1-x)^7*(1+x)^2).

a(n) = n^6/48 + 3*n^5/16 + 77*n^4/96 + 2*n^3 + 37*n^2/12 - n*(-1)^n/32 + 89*n/32 - 3*(-1)^n/64 + 67/64. [Bruno Berselli, Jul 08 2014]

A244872 Number of magic labelings with magic sum n of 4th graph shown in link.

Original entry on oeis.org

1, 15, 114, 569, 2138, 6562, 17329, 40765, 87512, 174452, 327137, 582784, 993895, 1632561, 2595510, 4009958, 6040323, 8895861, 12839284, 18196419, 25366968, 34836428, 47189231, 63123163, 83465122, 109188274, 141430667, 181515362, 230972141, 291560851, 365296444

Offset: 0

N. J. A. Sloane, Jul 08 2014

N. J. A. Sloane, Graphs for A244869-A244876.
R. P. Stanley, Examples of Magic Labelings, Unpublished Notes, 1973 [Cached copy, with permission]
Index entries for linear recurrences with constant coefficients, signature (7, -20, 28, -14, -14, 28, -20, 7, -1).

Cf. A244869-A244876.

Table[17 n^7/1440 + 119 n^6/960 + 28 n^5/45 + 721 n^4/384 + 5243 n^3/1440 + 721 n^2/160 + 129 n/40 + (-1)^n/256 + 255/256, {n, 0, 30}]
LinearRecurrence[{7,-20,28,-14,-14,28,-20,7,-1},{1,15,114,569,2138,6562,17329,40765,87512},40] (* Harvey P. Dale, Nov 01 2021 *)

G.f.: (1+8*x+29*x^2+43*x^3+29*x^4+8*x^5+x^6)/((1-x)^8*(1+x)).

a(n) = 17*n^7/1440 + 119*n^6/960 + 28*n^5/45 + 721*n^4/384 + 5243*n^3/1440 + 721*n^2/160 + 129*n/40 + (-1)^n/256 + 255/256. [Bruno Berselli, Jul 08 2014]

A244874 Number of magic labelings with magic sum n of 6th graph shown in link.

Original entry on oeis.org

1, 17, 137, 707, 2709, 8417, 22408, 53008, 114251, 228431, 429325, 766167, 1308451, 2151643, 3423880, 5293736, 7979133, 11757477, 16977097, 24070067, 33566489, 46110317, 62476800, 83591624, 110551831, 144648595, 187391933, 240537431, 306115063, 386460183, 484246768

Offset: 0

N. J. A. Sloane, Jul 08 2014

Colin Barker, Table of n, a(n) for n = 0..1000
N. J. A. Sloane, Graphs for A244869-A244876.
R. P. Stanley, Examples of Magic Labelings, Unpublished Notes, 1973 [Cached copy, with permission]
Index entries for linear recurrences with constant coefficients, signature (7,-20,28,-14,-14,28,-20,7,-1).

Cf. A244869, A244870, A244871, A244872, A244873, A244875, A244876.

LinearRecurrence[{7,-20,28,-14,-14,28,-20,7,-1},{1,17,137,707,2709,8417,22408,53008,114251},40] (* Harvey P. Dale, Jun 30 2022 *)

Vec((1 + 10*x + 38*x^2 + 60*x^3 + 38*x^4 + 10*x^5 + x^6) / ((1 - x)^8*(1 + x)) + O(x^40)) \\ Colin Barker, Jan 11 2017

G.f.: (1 + 10*x + 38*x^2 + 60*x^3 + 38*x^4 + 10*x^5 + x^6) / ((1 - x)^8*(1 + x)).

a(n) = (-315*(-129+(-1)^n) + 138528*n + 202104*n^2 + 171248*n^3 + 93030*n^4 + 32312*n^5 + 6636*n^6 + 632*n^7) / 40320. - Colin Barker, Jan 11 2017

A244875 Number of magic labelings with magic sum n of 7th graph shown in link.

Original entry on oeis.org

1, 21, 179, 938, 3612, 11242, 29947, 70855, 152720, 305330, 573812, 1023939, 1748545, 2875153, 4574922, 7073018, 10660515, 15707931, 22680505, 32155320, 44840378, 61595732, 83456781, 111659833, 147670042, 193211824, 250301858, 321284777, 408871655, 516181395

Offset: 0

N. J. A. Sloane, Jul 08 2014

Colin Barker, Table of n, a(n) for n = 0..1000
N. J. A. Sloane, Graphs for A244869-A244876.
R. P. Stanley, Examples of Magic Labelings, Unpublished Notes, 1973 [Cached copy, with permission]
Index entries for linear recurrences with constant coefficients, signature (7,-20,28,-14,-14,28,-20,7,-1).

Cf. A244869, A244870, A244871, A244872, A244873, A244874, A244876.

Vec((1 + 14*x + 52*x^2 + 77*x^3 + 52*x^4 + 14*x^5 + x^6) / ((1 - x)^8*(1 + x)) + O(x^40)) \\ Colin Barker, Jan 11 2017

G.f.: (1 + 14*x + 52*x^2 + 77*x^3 + 52*x^4 + 14*x^5 + x^6) / ((1 - x)^8*(1 + x)).

a(n) = (315*(255+(-1)^n) + 306144*n + 498456*n^2 + 452312*n^3 + 250530*n^4 + 86576*n^5 + 17724*n^6 + 1688*n^7) / 80640. - Colin Barker, Jan 11 2017

cp's OEIS Frontend

A100182 Structured tetragonal anti-prism numbers.

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A108675 a(n) = (n+1)*(n+2)*(61*n^4 + 366*n^3 + 845*n^2 + 888*n + 360)/720.

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A244864 a(n) = binomial(n+5,5) + 4*binomial(n+4,5) + 4*binomial(n+3,5) + binomial(n+2,5).

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A244865 Number of 3 X 3 symmetric matrices with row and column sums <= n.

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A244868 Number of symmetric 5 X 5 matrices of nonnegative integers with zeros on the main diagonal and every row and column adding to n.

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A244870 Number of magic labelings with magic sum n of 2nd graph shown in link.

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A244871 Number of magic labelings with magic sum n of 3rd graph shown in link.

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A244872 Number of magic labelings with magic sum n of 4th graph shown in link.

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A244874 Number of magic labelings with magic sum n of 6th graph shown in link.

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A108675 a(n) = (n+1)(n+2)(61n^4 + 366n^3 + 845n^2 + 888n + 360)/720.

A244864 a(n) = binomial(n+5,5) + 4binomial(n+4,5) + 4binomial(n+3,5) + binomial(n+2,5).