A244873 -id:A244873 - cp's OEIS frontend

A006325 4-dimensional analog of centered polygonal numbers.

0, 0, 1, 7, 26, 70, 155, 301, 532, 876, 1365, 2035, 2926, 4082, 5551, 7385, 9640, 12376, 15657, 19551, 24130, 29470, 35651, 42757, 50876, 60100, 70525, 82251, 95382, 110026, 126295, 144305, 164176, 186032, 210001, 236215, 264810, 295926

Offset: 0

Albert Rich (Albert_Rich(AT)msn.com)

If X is an n-set and Y and Z disjoint 2-subsets of X then a(n-4) is equal to the number of 6-subsets of X intersecting both Y and Z. - Milan Janjic, Aug 26 2007
For n>0, a(n+1) is the n-th antidiagonal sum of A213751. - Clark Kimberling, Jun 20 2012
This sequence is the case m=n-1, k=n+3 of b(m,k) = m*(m+1)*((k-2)*m-(k-5))/6, which is the m-th k-gonal pyramidal number. - Luciano Ancora, Apr 11 2015
Starting (1, 7, 26, ...), this is the binomial transform of (1, 6, 13, 12, 4, 0, 0, 0, ...). - Gary W. Adamson, Jul 31 2015
Also starting (1, 7, 26, ...), this appears to be the number of magic labelings of the cycle-of-loops graph LOOP X C_4 having magic sum n, where LOOP is the 1-vertex, 1-loop-edge graph. - David J. Seal, Sep 13 2017
The conjecture by David J. Seal is true and easily proved using MacMahon's Omega operators via the "Omega" package for Mathematica authored by Axel Riese (obtaining (up to an offset) the generating function listed in the formula section below). See the second Mathematica program in which the edges of LOOP X C_4 are indexed as in the example below. The Omega package can be downloaded from the link provided in the article by G. E. Andrews et al. - L. Edson Jeffery, Oct 15 2017

			A representation of the LOOP X C_4 graph, with edges and loops indexed as shown, as used in the second Mathematica program below:
.             3         1
.              O_______O
.              |   2   |
.              |4     0|
.              |_______|
.              O   6   O
.             5         7

T. A. Gulliver, Sequences from Arrays of Integers, Int. Math. Journal, Vol. 1, No. 4, pp. 323-332, 2002.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
G. E. Andrews, P. Paule and A. Riese, MacMahon's partition analysis III. The Omega package.
David Galvin and Courtney Sharpe, Independent set sequence of linear hyperpaths, arXiv:2409.15555 [math.CO], 2024. See p. 7.
Milan Janjic, Two Enumerative Functions
Milan Janjić, On Restricted Ternary Words and Insets, arXiv:1905.04465 [math.CO], 2019.
Milan Janjic and B. Petkovic, A Counting Function, arXiv 1301.4550 [math.CO], 2013.
Milan Janjic and B. Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014) # 14.3.5.
J. K. Merikoski, R. Kumar and R. A. Rajput, Upper bounds for the largest eigenvalue of a bipartite graph, Electronic Journal of Linear Algebra ISSN 1081-3810, A publication of the International Linear Algebra Society, Volume 26, pp. 168-176, April 2013.
Richard P. Stanley, Examples of Magic Labelings, Unpublished Notes, 1973 [Cached copy, with permission]
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A005900, A169938, A213751, A288486, A019298, A061907, A244497, A244879, A244873, A244880.
Cf. A000027, A000217, A019298, A244497, A244879, A244873, A244880, A293310, A293309 (magic labelings of LOOP X C_k, for k = 1..3,5..10).
Cf. A293311, A293312.

[n*(n-1)*(n^2-n+1)/6: n in [0..40]]; // Vincenzo Librandi, May 22 2011

Table[n*(n-1)*(n^2-n+1)/6, {n,0,60}] (* Vladimir Joseph Stephan Orlovsky, Apr 22 2011 *)
<< Omega.m; n = 4; cond = {}; Do[AppendTo[cond, Sum[a[Mod[2*k - j, 2*n]], {j, 0, 2}] == a[2*n]], {k, 0, n - 1}]; f = OEqSum[Product[x[i]^a[i], {i, 0, 2*n}], cond, u][[1]]; Do[f = OEqR[f, Subscript[u, k]], {k, n}];
(* Generating function: *)
f = Factor[f /. {x[2*n] -> x} /. {x[_] -> 1}]
(* This sequence (with initial zeros dropped): *)
CoefficientList[Series[f, {x, 0, 35}], x] (* L. Edson Jeffery, Oct 15 2017 *)

a(n)=n*(n-1)*(n^2-n+1)/6 \\ Charles R Greathouse IV, Sep 24 2015

a(n) = n*(n-1)*(n^2-n+1)/6.
a(n) = ((n^5 - (n-1)^5) - (n^1 - (n-1)^1))/30 = (n^5 - (n-1)^5 - 1)/30. - Xavier Acloque, Jan 25 2003
The partial sums of the octahedral numbers: a(n+1) = Sum_{i=0..n} A005900(i). - Jonathan Vos Post, Mar 14 2006
G.f.: -x^2*(x+1)^2/(x-1)^5. - Maksym Voznyy (voznyy(AT)mail.ru), Aug 10 2009
a(n) = Sum_{i=1..n-1} Sum_{j=1..n-1} min(i,j)^2. - Enrique Pérez Herrero, Jan 15 2013 [Which is just rephrasing the partial sum formula with the Murthy formula in A005900. - R. J. Mathar, Jun 14 2014]
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5), n > 4. - Yosu Yurramendi, Sep 03 2013
From Daniel Poveda Parrilla, Sep 09 2017: (Start)
a(n) = A169938(n-1)/6.
a(n+1) = A288486(n)/24. (End)
Sum_{n>=2} 1/a(n) = 12 - 2 * sqrt(3) * tanh(sqrt(3)*Pi/2). - Amiram Eldar, Jun 28 2020
E.g.f.: exp(x)*x^2*(3 + 4*x + x^2)/6. - Stefano Spezia, Dec 12 2021

A019298 Number of balls in pyramid with base either a regular hexagon or a hexagon with alternate sides differing by 1 (balls in hexagonal pyramid of height n taken from hexagonal close-packing).

Original entry on oeis.org

0, 1, 4, 11, 23, 42, 69, 106, 154, 215, 290, 381, 489, 616, 763, 932, 1124, 1341, 1584, 1855, 2155, 2486, 2849, 3246, 3678, 4147, 4654, 5201, 5789, 6420, 7095, 7816, 8584, 9401, 10268, 11187, 12159, 13186

Offset: 0

Eric E Blom (eblom(AT)REM.re.uokhsc.edu)

Alternately add and subtract successively longer sets of integers: 0; 1 = 0+1; -4 = 1-2-3; 11 = -4+4+5+6; -23 = 11-7-8-9-10; 42 = -23+11+12+13+14+15; -69 = 42-16-17-18-19-20-21; ... then take absolute values. - Walter Carlini, Aug 28 2003
Number of 3 X 3 symmetric matrices with nonnegative integer entries, such that every row (and column) sum equals n-1.
Equals Sum_{0..n} of "three-quarter squares" sequence (A077043). - Philipp M. Buluschek (kitschen(AT)romandie.com), Aug 12 2007
a(n) is the sum of the n-th row in A220075, n > 0. - Reinhard Zumkeller, Dec 03 2012
Sum of all the smallest parts in the partitions of 3n into three parts (see example). - Wesley Ivan Hurt, Jan 23 2014
For n > 0, a(n) is the number of (nonnegative integer) magic labelings of the prism graph Y_3 with magic sum n - 1. - L. Edson Jeffery, Sep 09 2017
Or number of magic labelings of LOOP X C_3 with magic sum n - 1, where LOOP is the 1-vertex, 1-loop-edge graph, as Y_k = I X C_k and LOOP X C_k have the same numbers of magic labelings when k is odd. - David J. Seal, Sep 13 2017
a(n) is the number of triples of integers in [1,n]^3 such that each pair has sum larger than n. - Bob Zwetsloot, Jul 23 2020

			Add last column for a(n) (n > 0).
                                               13 + 1 + 1
                                               12 + 2 + 1
                                               11 + 3 + 1
                                               10 + 4 + 1
                                                9 + 5 + 1
                                                8 + 6 + 1
                                                7 + 7 + 1
                                   10 + 1 + 1  11 + 2 + 2
                                    9 + 2 + 1  10 + 3 + 2
                                    8 + 3 + 1   9 + 4 + 2
                                    7 + 4 + 1   8 + 5 + 2
                                    6 + 5 + 1   7 + 6 + 2
                        7 + 1 + 1   8 + 2 + 2   9 + 3 + 3
                        6 + 2 + 1   7 + 3 + 2   8 + 4 + 3
                        5 + 3 + 1   6 + 4 + 2   7 + 5 + 3
                        4 + 4 + 1   5 + 5 + 2   6 + 6 + 3
            4 + 1 + 1   5 + 2 + 2   6 + 3 + 3   7 + 4 + 4
            3 + 2 + 1   4 + 3 + 2   5 + 4 + 3   6 + 5 + 4
1 + 1 + 1   2 + 2 + 2   3 + 3 + 3   4 + 4 + 4   5 + 5 + 5
   3(1)        3(2)        3(3)        3(4)        3(5)     ..   3n
---------------------------------------------------------------------
    1           4           11          23          42      ..  a(n)

R. P. Stanley, Enumerative Combinatorics, Wadsworth, Vol. 1, 1986; see Prop. 4.6.21, p. 235, G_3(lambda).
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 7.14(a), p. 452.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
G. E. Andrews, P. Paule and A. Riese, MacMahon's partition analysis III. The Omega package, January 1999, p. 13.
L. Carlitz, Enumeration of symmetric arrays, Duke Math. J., Vol. 33 (1966), 771-782. MR0201332 (34 #1216).
R. J. Mathar, Illustrations of magic labelings (2025)
R. P. Stanley, Magic labelings of graphs, symmetric magic squares,..., Duke Math. J. 43 (3) (1976) 511-531, Section 5, F_3(x).
R. P. Stanley, Examples of Magic Labelings, Unpublished Notes, 1973 [Cached copy, with permission]
Index entries for linear recurrences with constant coefficients, signature (3,-2,-2,3,-1).

Cf. A053493, A077043 (first differences), A002717.
Cf. A061927, A244497, A292281, A244873, A289992 (# of magic labelings of prism graph Y_k = I X C_k, for k = 4,5,6,7,8, up to an offset).
Cf. A006325, A244879, A244880 (# of magic labelings of LOOP X C_k, for k = 4,6,8, up to an offset).

[Floor((n^2+1)*(2*n+3)/8): n in [0..80]]; // Vincenzo Librandi, Jul 28 2013

Maple
```
series(x*(x^2+x+1)/(x+1)/(x-1)^4,x,80);
```

Table[ Ceiling[3*n^2/4], {n, 0, 37}] // Accumulate (* Jean-François Alcover, Dec 20 2012, after Philipp M. Buluschek's comment *)
CoefficientList[Series[x (x^2 + x + 1) / ((x + 1) (x - 1)^4), {x, 0, 40}], x] (* Vincenzo Librandi, Jul 28 2013 *)
LinearRecurrence[{3, -2, -2, 3, -1}, {0, 1, 4, 11, 23}, 38] (* L. Edson Jeffery, Sep 09 2017 *)

a(n)=(n^2+1)*(2*n+3)\8 \\ Charles R Greathouse IV, Apr 04 2013

def A019298(n): return n*(n*(2*n+3)+2)+3>>3 # Chai Wah Wu, Jun 07 2025

a(n) = floor((n^2+1)(2n+3)/8).
G.f.: x*(x^2+x+1)/((x+1)*(x-1)^4).
a(n) = floor((2n^3 + 3n^2 + 2n)/8); also nearest integer to ((n+1)^4 - n^4)/16.
a(n) = (4n^3 + 6n^2 + 4n+1 - (-1)^n)/16. - Wesley Petty (Wesley.Petty(AT)mail.tamucc.edu), Mar 06 2004
a(n) = Sum_{i=1..n} i^2 - floor(i^2/4) = Sum_{i=1..n} i * (2n - 2i + 1 - floor((n - i + 1)/2) ). - Wesley Ivan Hurt, Jan 23 2014
E.g.f.: (1/16)*(-exp(-x) + exp(x)*(1 + 14*x + 18*x^2 + 4*x^3)). - Stefano Spezia, Nov 29 2019
a(2*n) = (1/2)*( n*(n + 1)^3 - (n - 1)*n^3 ); a(2*n-1) = (1/2)*( (n + 1)*n^3 - n*(n - 1)^3 ) (note: replacing the exponent 3 with 2 throughout gives the sequence of generalized pentagonal numbers A001318). - Peter Bala, Aug 11 2021
a(2n-1) = A213772(n). - R. J. Mathar, Mar 02 2025
(n-2)*a(n) -3*a(n-1) -(n+1)*a(n-2) +2*n-1 =0. - R. J. Mathar, Mar 09 2025

Error in n=8 term corrected May 15 1997

A244497 Number of magic labelings of the prism graph I X C_5 with magic sum n.

Original entry on oeis.org

1, 11, 57, 197, 533, 1223, 2494, 4654, 8105, 13355, 21031, 31891, 46837, 66927, 93388, 127628, 171249, 226059, 294085, 377585, 479061, 601271, 747242, 920282, 1123993, 1362283, 1639379, 1959839, 2328565, 2750815, 3232216, 3778776, 4396897, 5093387, 5875473, 6750813, 7727509, 8814119

Offset: 0

N. J. A. Sloane, Jul 07 2014

This sequence is also the number of magic labelings of the cycle-of-loops graph LOOP X C_5 with magic sum n, where LOOP is the 1-vertex, 1-loop-edge graph. A similar identity holds between the sequences for I X C_k and LOOP X C_k for all odd k. - David J. Seal, Sep 14 2017

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 (5,-9,5,5,-9,5,-1).

Cf. A019298, A061927, A292281, A244873, A289992 (analogs for prism graphs I X C_k, k = 3,4,6,7,8).

Cf. A006325, A244879, A244880 (analogs for LOOP X C_k, k = 4,6,8).

A244497:=n->(3*(63+(-1)^n) + 576*n + 720*n^2 + 460*n^3 + 150*n^4 + 20*n^5) / 192: seq(A244497(n), n=0..50); # Wesley Ivan Hurt, Sep 16 2017

CoefficientList[Series[(1 + 6 x + 11 x^2 + 6 x^3 + x^4)/((1 - x)^6*(1 + x)), {x, 0, 37}], x] (* Michael De Vlieger, Sep 15 2017 *)
LinearRecurrence[{5,-9,5,5,-9,5,-1},{1,11,57,197,533,1223,2494},40] (* Harvey P. Dale, Aug 04 2021 *)

Vec((1+6*x+11*x^2+6*x^3+x^4) / ((1-x)^6*(1+x)) + O(x^40)) \\ Colin Barker, Jan 13 2017

G.f.: (1 + 6*x + 11*x^2 + 6*x^3 + x^4) / ((1 - x)^6*(1 + x)).
From Colin Barker, Jan 13 2017: (Start)
a(n) = (3*(63+(-1)^n) + 576*n + 720*n^2 + 460*n^3 + 150*n^4 + 20*n^5) / 192.
a(n) = 5*a(n-1) - 9*a(n-2) + 5*a(n-3) + 5*a(n-4) - 9*a(n-5) + 5*a(n-6) - a(n-7) for n>6.
(End)

A244879 Number of magic labelings of the cycle-of-loops graph LOOP X C_6 having magic sum n, where LOOP is the 1-vertex, 1-loop-edge graph.

Original entry on oeis.org

1, 18, 129, 571, 1884, 5103, 11998, 25362, 49347, 89848, 154935, 255333, 404950, 621453, 926892, 1348372, 1918773, 2677518, 3671389, 4955391, 6593664, 8660443, 11241066, 14433030, 18347095, 23108436, 28857843, 35752969, 43969626, 53703129, 65169688, 78607848

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 (7,-21,35,-35,21,-7,1).

Cf. A019298, A006325, A244497, A292281, A244873, A244880.

CoefficientList[Series[(1 + 11 x + 24 x^2 + 11 x^3 + x^4)/(1 - x)^7, {x, 0, 31}], x] (* Michael De Vlieger, Sep 15 2017 *)
LinearRecurrence[{7,-21,35,-35,21,-7,1},{1,18,129,571,1884,5103,11998},40] (* Harvey P. Dale, Jul 30 2019 *)

Vec((1 + 11*x + 24*x^2 + 11*x^3 + x^4) / (1 - x)^7 + O(x^40)) \\ Colin Barker, Jan 11 2017

G.f.: (1 + 11*x + 24*x^2 + 11*x^3 + x^4) / (1 - x)^7.
From Colin Barker, Jan 11 2017: (Start)
a(n) = (120 + 438*n + 677*n^2 + 570*n^3 + 275*n^4 + 72*n^5 + 8*n^6) / 120.
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7) for n > 6.
(End)

Name corrected by David J. Seal, Sep 13 2017

A061927 a(n) = n(n+1)(2n+1)(n^2+n+3)/30.

Original entry on oeis.org

0, 1, 9, 42, 138, 363, 819, 1652, 3060, 5301, 8701, 13662, 20670, 30303, 43239, 60264, 82280, 110313, 145521, 189202, 242802, 307923, 386331, 479964, 590940, 721565, 874341, 1051974, 1257382, 1493703, 1764303, 2072784, 2422992, 2819025

Offset: 0

Henry Bottomley, May 17 2001

Also number of magic labelings of the cubical graph of magic sum n-1 [Ahmed]. - R. J. Mathar, Jan 25 2007
If Y_i (i=1,2,3) are 2-blocks of a (n+3)-set X then a(n-4) is the number of 8-subsets of X intersecting each Y_i (i=1,2,3). - Milan Janjic, Oct 28 2007
The cube graph is also the prism graph I X C_4, so this is related to the number of magic labelings of other prism & related graphs. - David J. Seal, Sep 13 2017

Harry J. Smith, Table of n, a(n) for n = 0..1000
Maya Mohsin Ahmed, Algebraic Combinatorics of Magic Squares, arXiv:math/0405476 [math.CO], 2004, p. 73.
Shalosh B. Ekhad and Doron Zeilberger, There are (1/30)(r+1)(r+2)(2r+3)(r^2+3r+5) Ways For the Four Teams of a World Cup Group to Each Have r Goals For and r Goals Against [Thanks to the Soccer Analog of Prop. 4.6.19 of Richard Stanley's (Classic!) EC1], arXiv:1407.1919 [math.CO], 2014.
David Galvin and Courtney Sharpe, Independent set sequence of linear hyperpaths, arXiv:2409.15555 [math.CO], 2024. See p. 7.
Yu-hong Guo, Some n-Color Compositions, J. Int. Seq. 15 (2012) 12.1.2, eq. (5), m=3.
Milan Janjic, Two Enumerative Functions
Milan Janjic and Boris Petkovic, A Counting Function, arXiv 1301.4550 [math.CO], 2013.
Milan Janjic and Boris Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014) # 14.3.5.
R. P. Stanley, Examples of Magic Labelings, Unpublished Notes, 1973. [Cached copy, with permission] See p. 32.
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A006325, A019298, A244497, A244873, A289992, A292281, partial sums of A014820, A006975 (binomial transform shifted left).

Table[n (n + 1) (2 n + 1) (n^2 + n + 3)/30, {n, 0, 33}] (* or *)
CoefficientList[Series[x (1 + x)^3/(-1 + x)^6, {x, 0, 33}], x] (* Michael De Vlieger, Sep 15 2017 *)
LinearRecurrence[{6,-15,20,-15,6,-1},{0,1,9,42,138,363},40] (* Harvey P. Dale, Apr 18 2018 *)

a(n) = { n*(n + 1)*(2*n + 1)*(n^2 + n + 3)/30 } \\ Harry J. Smith, Jul 29 2009

a(n) = a(n-1) + A014820(n) = A061926(9, n).

G.f.: x*(1+x)^3/(-1+x)^6 = 20/(-1+x)^5 + 1/(-1+x)^2 + 7/(-1+x)^3 + 18/(-1+x)^4 + 8/(-1+x)^6. - R. J. Mathar, Nov 18 2007

A244880 Number of magic labelings of the cycle-of-loops graph LOOP X C_8 having magic sum n, where LOOP is the 1-vertex, 1-loop-edge graph.

Original entry on oeis.org

1, 47, 650, 4726, 23219, 87677, 274132, 743724, 1806597, 4016683, 8306078, 16168802, 29904823, 52936313, 90209192, 148694104, 238002057, 371131047, 565361074, 843316046, 1234212155, 1775313397, 2513615996, 3507784580, 4830364045, 6570292131, 8835738822, 11757299770, 15491572031

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,-36,84,-126,126,-84,36,-9,1).

Cf. A019298, A006325, A244497, A244879, A244873, A289992.

A244880:=n->(630 + 3051*n + 6570*n^2 + 8211*n^3 + 6503*n^4 + 3339*n^5 + 1085*n^6 + 204*n^7 + 17*n^8) / 630: seq(A244880(n), n=0..50); # Wesley Ivan Hurt, Sep 16 2017

CoefficientList[Series[(1 + 38 (x + x^5) + 263 (x^2 + x^4) + 484 x^3 + x^6)/(1 - x)^9, {x, 0, 28}], x] (* Michael De Vlieger, Sep 15 2017 *)

Vec((1 + 6*x + x^2)*(1 + 32*x + 70*x^2 + 32*x^3 + x^4) / (1 - x)^9 + O(x^30)) \\ Colin Barker, Jan 12 2017

G.f.: (1+38*(x+x^5)+263*(x^2+x^4)+484*x^3+x^6) / (1-x)^9.
From Colin Barker, Jan 12 2017: (Start)
a(n) = (630 + 3051*n + 6570*n^2 + 8211*n^3 + 6503*n^4 + 3339*n^5 + 1085*n^6 + 204*n^7 + 17*n^8) / 630.
a(n) = 9*a(n-1) - 36*a(n-2) + 84*a(n-3) - 126*a(n-4) + 126*a(n-5) - 84*a(n-6) + 36*a(n-7) - 9*a(n-8) + a(n-9) for n>8.
(End)
(326*n^2-195*n+142)*a(n) +(-652*n^2-652*n-10725)*a(n-1) +(326*n^2+847*n+663)*a(n-2) +2*(-165*n^2-165*n-71)=0. - R. J. Mathar, Mar 10 2025

Name corrected by David J. Seal, Sep 13 2017

A289992 Number of magic labelings of the prism graph I X C_8 having magic sum n.

Original entry on oeis.org

1, 49, 746, 6122, 34067, 144963, 506772, 1524628, 4074949, 9898229, 22220990, 46695870, 92769495, 175610631, 318756136, 557659432, 944355593, 1553488697, 2489980818, 3898657938, 5976186139, 8985711691, 13274641084, 19296041660, 27634190285

Offset: 0

David J. Seal, Sep 13 2017

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).

Cf. A019298, A061927, A244497, A292281, A244873, A244880.

a(n) = A244880(n) + 2*Sum_{i=0..n-1} A244880(i).
From Colin Barker, Sep 13 2017: (Start)
G.f.: (1 + x)*(1 + 6*x + x^2)*(1 + 32*x + 70*x^2 + 32*x^3 + x^4) / (1 - x)^10.
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)
[Proof of the g.f. follows from the convolution formula and insertion of the g.f. A244880(x): Sum_{n>=0} a(n)x^n = Sum_{n>=0} A244880(n)*x^n +2*Sum_{n>=0} Sum_{i=0..n-1} A244880(i)*x^n = A244880(x) +2*Sum_{i>=0} Sum_{n>=i+1} A244880(i)*x^n = A244880(x) +2*Sum_{i>=0} A244880(i)*x^(i+1) Sum_{n>=0} x^n = A244880(x)+2*A244880(x)*x/(1+x) = A244880(x)*(1+2*x/(1-x)). R. J. Mathar, Mar 09 2025]

A292281 Number of magic labelings of the prism graph I X C_6 having magic sum n.

Original entry on oeis.org

1, 20, 167, 867, 3322, 10309, 27410, 64770, 139479, 278674, 523457, 933725, 1594008, 2620411, 4168756, 6444020, 9711165, 14307456, 20656363, 29283143, 40832198, 56086305, 75987814, 101661910, 134442035, 175897566, 227863845, 292474657, 372197252, 469870007, 588742824

Offset: 0

David J. Seal, Sep 13 2017

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).

Cf. A019298, A061927, A244497, A244879, A244873, A289992.

f[n_] := SeriesCoefficient[(1 + 11 x + 24 x^2 + 11 x^3 + x^4)/(1 - x)^7, {x, 0, n}]; Table[f[n] + 2 Sum[f[i], {i, 0, n - 1}], {n, 0, 24}] (* Michael De Vlieger, Sep 15 2017 *)

a(n) = A244879(n) + 2*Sum_{i=0..n-1} A244879(i).
From Colin Barker, Sep 13 2017: (Start)
G.f.: (1 + x)*(1 + 11*x + 24*x^2 + 11*x^3 + x^4) / (1 - x)^8.
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)
[Proof of the g.f. follows from the g.f. of A244879 with the resummation demonstrated in A289992: g.f. = A244879(x)*(1+2*x/(1-x)). - R. J. Mathar, Mar 09 2025]

A293311 Rectangular array read by antidiagonals: A(n,k) = number of magic labelings of the graph LOOP X C_n (see comments) having magic sum k, n >= 1, k >= 0.

Original entry on oeis.org

1, 2, 1, 3, 3, 1, 4, 6, 4, 1, 5, 10, 11, 7, 1, 6, 15, 23, 26, 11, 1, 7, 21, 42, 70, 57, 18, 1, 8, 28, 69, 155, 197, 129, 29, 1, 9, 36, 106, 301, 533, 571, 289, 47, 1, 10, 45, 154, 532, 1223, 1884, 1640, 650, 76, 1, 11, 55, 215, 876, 2494, 5103, 6604, 4726, 1460, 123, 1

Offset: 1

L. Edson Jeffery, Oct 06 2017

The graph LOOP X C_n is constructed by attaching a loop to each vertex of the cycle graph C_n.

The generating function for row n of this array was found via the "Omega" package for Mathematica authored by Axel Riese. The package can be downloaded from the link given in the article by G. E. Andrews, et al., and is needed for the Mathematica program below.

			Array begins:
.  1   2    3     4      5       6       7        8        9        10
.  1   3    6    10     15      21      28       36       45        55
.  1   4   11    23     42      69     106      154      215       290
.  1   7   26    70    155     301     532      876     1365      2035
.  1  11   57   197    533    1223    2494     4654     8105     13355
.  1  18  129   571   1884    5103   11998    25362    49347     89848
.  1  29  289  1640   6604   21122   57271   137155   298184    599954
.  1  47  650  4726  23219   87677  274132   743724  1806597   4016683
.  1  76 1460 13604  81555  363606 1310974  4029310 10936124  26868719
.  1 123 3281 39175 286555 1508401 6271378 21836366 66220705 179784715

G. E. Andrews, P. Paule and A. Riese, MacMahon's partition analysis III. The Omega package.
Eric Weisstein's World of Mathematics, Cycle Graph.
Eric Weisstein's World of Mathematics, Graph Loop.

Cf. A293311.

Cf. A000027, A000217, A019298, A006325, A244497, A244879, A244873, A244880, A293310, A293309 (rows 1..10).

(* Run this first: *)
<< Omega.m;
(* Then run the following in a different cell: *)
nmax = 11; Do[cond = {}; Do[If[n == 1, AppendTo[cond, Subscript[a, 1] + Subscript[a, 2] == Subscript[a, 3]]; Break[], AppendTo[cond, If[j == n, Subscript[a, 2*j - 1] + Subscript[a, 2*j] + Subscript[a, 1] == Subscript[a, 2*n + 1], Subscript[a, 2*j - 1] + Subscript[a, 2*j] + Subscript[a, 2*j + 1] == Subscript[a, 2*n + 1]]]], {j, n}]; f = OEqSum[Product[Subscript[x, i]^Subscript[a, i], {i, 2*n + 1}], cond, \[Lambda]][[1]] /. {Subscript[x, 2*n + 1] -> z} /. {Subscript[x, _] -> 1}; Do[f = OEqR[f, Subscript[\[Lambda], k]], {k, Length[cond]}]; Do[a293311[n, k] = Coefficient[Series[f, {z, 0, nmax - 1}], z, k], {k, 0, nmax - 1}], {n, nmax}];
(* Array: *)
Grid[Table[a293311[n, k], {n, nmax}, {k, 0, nmax - 1}]]
(* Array antidiagonals flattened (gives this sequence): *)
Flatten[Table[a293311[n, k - n], {k, 11}, {n, k}]]

A293309 Number of magic labelings of the graph LOOP X C_10 (see comments) having magic sum n, n >= 0.

Original entry on oeis.org

1, 123, 3281, 39175, 286555, 1508401, 6271378, 21836366, 66220705, 179784715, 445824731, 1025102013, 2211041131, 4514532465, 8789910980, 16416797116, 29556115153, 51502789451, 87162399205, 143684487475, 231291309931, 364347612673, 562724586326

Offset: 0

L. Edson Jeffery, Oct 05 2017

nonn

The graph LOOP X C_n is constructed by attaching a loop to each vertex of the cycle graph C_n.

The generating function for this sequence was found via the "Omega" package for Mathematica authored by Axel Riese. The package can be downloaded from the link given in the article by G. E. Andrews et al.

G. E. Andrews, P. Paule and A. Riese, MacMahon's partition analysis III. The Omega package.
Eric Weisstein's World of Mathematics, Cycle Graph.
Eric Weisstein's World of Mathematics, Graph Loop.
Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).

Cf. A293311, A293312.

Cf. A000027, A000217, A019298, A006325, A244497, A244879, A244873, A244880, A293310 (magic labelings of LOOP X C_k, for k=1..9).

CoefficientList[Series[(1 + 112*z + 1983*z^2 + 9684*z^3 + 16120*z^4 + 9684*z^5 + 1983*z^6 + 112*z^7 + z^8)/(1 - z)^11, {z, 0, 22}], z]
LinearRecurrence[{11, -55, 165, -330, 462, -462, 330, -165, 55, -11, 1}, {1, 123, 3281, 39175, 286555, 1508401, 6271378, 21836366, 66220705, 179784715, 445824731}, 25] (* Vincenzo Librandi, Oct 12 2017 *)

G.f.: (1 + 112*z + 1983*z^2 + 9684*z^3 + 16120*z^4 + 9684*z^5 + 1983*z^6 + 112*z^7 + z^8)/(1 - z)^11.

cp's OEIS Frontend

A006325 4-dimensional analog of centered polygonal numbers.

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A019298 Number of balls in pyramid with base either a regular hexagon or a hexagon with alternate sides differing by 1 (balls in hexagonal pyramid of height n taken from hexagonal close-packing).

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A244497 Number of magic labelings of the prism graph I X C_5 with magic sum n.

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A244879 Number of magic labelings of the cycle-of-loops graph LOOP X C_6 having magic sum n, where LOOP is the 1-vertex, 1-loop-edge graph.

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A061927 a(n) = n*(n+1)*(2*n+1)*(n^2+n+3)/30.

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A244880 Number of magic labelings of the cycle-of-loops graph LOOP X C_8 having magic sum n, where LOOP is the 1-vertex, 1-loop-edge graph.

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A289992 Number of magic labelings of the prism graph I X C_8 having magic sum n.

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A061927 a(n) = n(n+1)(2n+1)(n^2+n+3)/30.