A019298 -id:A019298 - cp's OEIS frontend

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.

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R. P. Stanley, Examples of Magic Labelings, Unpublished Notes, 1973 [Cached copy, with permission]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

A200763 T(n,k)=Number of 0..k arrays x(0..n-1) of n elements with nondecreasing average value.

Original entry on oeis.org

2, 3, 3, 4, 6, 4, 5, 10, 11, 5, 6, 15, 23, 19, 6, 7, 21, 42, 51, 32, 7, 8, 28, 69, 113, 110, 53, 8, 9, 36, 106, 219, 297, 233, 87, 9, 10, 45, 154, 388, 679, 767, 488, 142, 10, 11, 55, 215, 638, 1387, 2070, 1957, 1013, 231, 11, 12, 66, 290, 995, 2583, 4874, 6235, 4947, 2088

Offset: 1

R. H. Hardin Nov 22 2011

Table starts
..2...3....4.....5......6......7.......8.......9.......10.......11........12
..3...6...10....15.....21.....28......36......45.......55.......66........78
..4..11...23....42.....69....106.....154.....215......290......381.......489
..5..19...51...113....219....388.....638.....995.....1483.....2133......2975
..6..32..110...297....679...1387....2583....4500.....7410....11669.....17687
..7..53..233...767...2070...4874...10283...20012....36412....62780....103412
..8..87..488..1957...6235..16919...40437...87914...176767...333702....597390
..9.142.1013..4947..18608..58198..157577..382720...850389..1757813...3420112
.10.231.2088.12419..55148.198807..609826.1654657..4062796..9195619..19445435
.11.375.4278.31006.162532.675372.2347039.7114665.19304047.47842607.109955586

			Some solutions for n=8 k=8
..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0
..1....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0
..2....1....1....4....3....3....0....3....2....3....2....5....3....1....0....2
..2....7....6....2....1....2....3....2....1....4....7....3....4....5....3....6
..2....3....4....7....7....7....5....6....1....6....3....3....5....2....8....4
..4....7....6....6....4....4....4....5....1....6....4....3....3....7....3....4
..7....4....8....4....5....6....2....4....6....7....6....8....8....7....7....6
..5....5....5....6....6....5....4....7....6....4....4....4....7....4....6....8

R. H. Hardin, Table of n, a(n) for n = 1..9999

Column 2 is A001911(n+1)
Column 7 is A200707
Row 2 is A000217(n+1)
Row 3 is A019298(n+1)

A236364 Sum of all the middle parts in the partitions of 3n into 3 parts.

Original entry on oeis.org

1, 5, 18, 40, 80, 135, 217, 320, 459, 625, 836, 1080, 1378, 1715, 2115, 2560, 3077, 3645, 4294, 5000, 5796, 6655, 7613, 8640, 9775, 10985, 12312, 13720, 15254, 16875, 18631, 20480, 22473, 24565, 26810, 29160, 31672, 34295, 37089, 40000, 43091, 46305, 49708

Offset: 1

Wesley Ivan Hurt, Jan 23 2014

			Add second columns for a(n):
                                               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           5          18           40          80      ..   a(n)

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for sequences related to partitions
Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).

Cf. A000330, A019298, A235988.

A236364:=n->n*(n+1)*(2*n+1)/6 - floor((n-1)/2) * (4*floor((n-1)/2)^2 + (3*n+6)*floor((n-1)/2) - 6*n^2 + 3*n + 2)/6; seq(A236364(n), n=1..100);

Table[Sum[i^2, {i, n}] + Sum[(n + i) (n - 2 i), {i, Floor[(n - 1)/2]}], {n, 100}]
CoefficientList[Series[(x^4 + 3 x^3 + 7 x^2 + 3 x + 1)/ ((x - 1)^4 (x + 1)^2), {x, 0, 50}], x] (* Vincenzo Librandi, Feb 18 2014 *)

Vec(x*(x^4+3*x^3+7*x^2+3*x+1)/((x-1)^4*(x+1)^2) + O(x^100)) \\ Colin Barker, Jan 23 2014

a(n) = A000330(n) + Sum_{i=1..floor((n-1)/2)} (n + i)*(n - 2i).
a(n) = n*(n+1)*(2*n+1)/6 - floor((n-1)/2) * (4*floor((n-1)/2)^2 + 3*(n+2)*floor((n-1)/2) - 6*n^2 + 3*n + 2)/6.
G.f.: x*(x^4+3*x^3+7*x^2+3*x+1)/((x-1)^4*(x+1)^2). - Joerg Arndt, Jan 23 2014
a(n) = (n*(3-3*(-1)^n+10*n^2))/16. - Colin Barker, Jan 23 2014
a(n) = (n^3 + ceiling(n/2)^3 + floor(n/2)^3)/2. - Wesley Ivan Hurt, Apr 15 2016
a(n) = 2*a(n-1)+a(n-2)-4*a(n-3)+a(n-4)+2*a(n-5)-a(n-6). - Wesley Ivan Hurt, Nov 19 2021

Name clarified by Wesley Ivan Hurt, Apr 16 2016

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]

A380853 Number of ways to place six distinct positive integers on a triangle, three on the corners and three on the sides such that the sum of the three values on each side is n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 3, 5, 13, 14, 25, 37, 47, 58, 89, 98, 126, 159, 188, 219, 276, 303, 362, 423, 478, 536, 633, 688, 781, 881, 973, 1068, 1211, 1301, 1443, 1589, 1724, 1866, 2066, 2202, 2396, 2598, 2790, 2986, 3250, 3439, 3699, 3967, 4219, 4480, 4819, 5071

Offset: 1

Derek Holton and Alex Holton, Feb 06 2025

Solutions differing by only rotation or reflections are not counted separately.

If the numbers do not need to be distinct and rotations and reflections are counted separately we get A019298(n-2). If the numbers do not need to be distinct but rotations and reflections do not count separately we get A006918(n-2). If the six numbers must be distinct and reflections and rotations count separately we get 6*a(n). - R. J. Mathar, Feb 27 2025

			The a(9) = 1 solution is:
       1
     5   6
   3   4   2

R. J. Mathar, Illustrations - Examples (2025)
Index entries for linear recurrences with constant coefficients, signature (-1,0,2,3,2,0,-3,-4,-3,0,2,3,2,0,-1,-1).

Cf. A342467, A380105.

A380853 := proc(n)
    -3412+2353*n+30*n^3-480*n^2+(135*n-900)*(-1)^n ;
    %-144*(2*A010891(n)+A010891(n-1)+2*A010891(n-2)) ;
    %-160*(17*A049347(n)+8*A049347(n-1)) ;
    %-360*A057077(n) ;
    %+480*(-1)^n*(A099254(n)-A099254(n-1)) ;
    %/720 ;
end proc:
seq(A380853(n),n=1..40) ; # R. J. Mathar, Feb 27 2025

a(n) = my(c=0, t); for(x=3, n-5, t=n-x; for(y=2, min(x-1, t-1), for(z=1, y-1, if(#Set([x, y, z, t-y, t-z, n-y-z])==6, c++)))); c; \\ Jinyuan Wang, Feb 07 2025

G.f.: x^9*(1 + 4*x + 8*x^2 + 16*x^3 + 18*x^4 + 18*x^5 + 15*x^6 + 10*x^7)/((1 - x)^4*(1 + 2*x + 2*x^2 + x^3)^2*(1 + x + 2*x^2 + 2*x^3 + 2*x^4 + x^5 + x^6)). - Stefano Spezia, Feb 08 2025

A380105(n) = a(n)-a(n-3). - R. J. Mathar, Mar 13 2025

A053493 Number of symmetric 4 X 4 matrices of nonnegative integers with every row and column adding to n.

Original entry on oeis.org

1, 10, 56, 214, 641, 1620, 3616, 7340, 13825, 24510, 41336, 66850, 104321, 157864, 232576, 334680, 471681, 652530, 887800, 1189870, 1573121, 2054140, 2651936, 3388164, 4287361, 5377190, 6688696, 8256570, 10119425, 12320080, 14905856, 17928880, 21446401

Offset: 0

N. J. A. Sloane, Jan 15 2000; definition revised Jul 06 2014

R. P. Stanley, Enumerative Combinatorics, Wadsworth, Vol. 1, 1986; see Prop. 4.6.21, p. 235, G_4(lambda).

Colin Barker, Table of n, a(n) for n = 0..1000
L. Carlitz, Enumeration of symmetric arrays, Duke Math. J., Vol. 33 (1966), 771-782. MR0201332 (34 #1216).
R. P. Stanley, Magic labelings of graphs, symmetric magic squares,..., Duke Math. J. 43 (3) (1976) 511-531, F_4(x) in section 5.
Index entries for linear recurrences with constant coefficients, signature (6,-14,14,0,-14,14,-6,1).

Cf. A019298, A053494.

CoefficientList[Series[(1+4x+10x^2+4x^3+x^4)/((1-x)^7(1+x)), {x, 0, 30}], x] (* or *) LinearRecurrence[{6,-14,14,0,-14,14,-6,1},{1,10,56,214,641,1620,3616,7340},30] (* Harvey P. Dale, Oct 31 2011 *)

Vec((1+4*x+10*x^2+4*x^3+x^4) / ((1-x)^7*(1+x)) + O(x^40)) \\ Colin Barker, Jan 14 2017

G.f.: (1+4*x+10*x^2+4*x^3+x^4)/((1-x)^7*(1+x)).
a(0)=1, a(1)=10, a(2)=56, a(3)=214, a(4)=641, a(5)=1620, a(6)=3616, a(7)=7340, 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). - Harvey P. Dale, Oct 31 2011
a(n) = (9*(31+(-1)^n) + 768*n + 928*n^2 + 624*n^3 + 238*n^4 + 48*n^5 + 4*n^6) / 288. - Colin Barker, Jan 14 2017

A053494 Number of symmetric 5 X 5 matrices of nonnegative integers with every row and column adding to n.

Original entry on oeis.org

1, 26, 348, 2698, 14751, 62781, 222190, 681460, 1865715, 4655535, 10756921, 23290026, 47700173, 93104473, 174248451, 314246511, 548380980, 929209095, 1533389605, 2470568045, 3894914166, 6019752376, 9136114923, 13635769173, 20039850376, 29033765566

Offset: 0

N. J. A. Sloane, Jan 15 2000

R. P. Stanley, Enumerative Combinatorics, Wadsworth, Vol. 1, 1986; see Prop. 4.6.21, p. 235, G_5(lambda).

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]
R. P. Stanley, Magic labelings of graphs, symmetric magic squares,..., Duke Math. J. 43 (3) (1976) 511-531, F_5(x) in Section 5.
Index entries for linear recurrences with constant coefficients, signature (5,-4,-20,40,16,-100,44,110,-110,-44,100,-16,-40,20,4,-5,1).

Cf. A019298, A053493.

CoefficientList[Series[(1+21x+222x^2+1082x^3+3133x^4+5722x^5+7013x^6+5722x^7+3133x^8+1082x^9+222x^10+21x^11+x^12)/((1-x)^11(1+x)^6),{x,0,30}],x] (* or *) LinearRecurrence[ {5,-4,-20,40,16,-100,44,110,-110,-44,100,-16,-40,20,4,-5,1},{1,26,348,2698,14751,62781,222190,681460,1865715,4655535,10756921,23290026,47700173,93104473,174248451,314246511,548380980},30] (* Harvey P. Dale, Mar 05 2023 *)

Vec((1 + 21*x + 222*x^2 + 1082*x^3 + 3133*x^4 + 5722*x^5 + 7013*x^6 + 5722*x^7 + 3133*x^8 + 1082*x^9 + 222*x^10 + 21*x^11 + x^12) / ((1 - x)^11*(1 + x)^6) + O(x^30)) \\ Colin Barker, Jan 14 2017

G.f.: (1 + 21*x + 222*x^2 + 1082*x^3 + 3133*x^4 + 5722*x^5 + 7013*x^6 + 5722*x^7 + 3133*x^8 + 1082*x^9 + 222*x^10 + 21*x^11 + x^12) / ((1-x)^11*(1+x)^6).

a(n) = (189*(59981+5555*(-1)^n) + 18*(2345165+65331*(-1)^n)*n + (76615494+689850*(-1)^n)*n^2 + 40*(2138179+6237*(-1)^n)*n^3 + (63277966+47250*(-1)^n)*n^4 + 1260*(25421+3*(-1)^n)*n^5 + 11171664*n^6 + 2644080*n^7 + 405954*n^8 + 36500*n^9 + 1460*n^10) / 12386304. - Colin Barker, Jan 14 2017

Revised definition, Jul 06 2014

A212683 Number of (w,x,y,z) with all terms in {1,...,n} and |x-y| = w + |y-z|.

Original entry on oeis.org

0, 0, 2, 8, 22, 46, 84, 138, 212, 308, 430, 580, 762, 978, 1232, 1526, 1864, 2248, 2682, 3168, 3710, 4310, 4972, 5698, 6492, 7356, 8294, 9308, 10402, 11578, 12840, 14190, 15632, 17168, 18802, 20536, 22374, 24318, 26372, 28538, 30820

Offset: 0

Clark Kimberling, May 24 2012

For a guide to related sequences, see A211795.

Also the number of (w,x,y) with all terms in {0,...,n-1} and |w-x| < |x-y|, see A212959. - Clark Kimberling, Jun 02 2012

Index entries for linear recurrences with constant coefficients, signature (3,-2,-2,3,-1).

Cf. A019298, A211795, A212684.

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[Abs[x - y] == w + Abs[y - z], s = s + 1],
{w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 40]]   (* A212683 *)
%/2  (* A019298 *)
LinearRecurrence[{3, -2, -2, 3, -1}, {0, 0, 2, 8, 22}, 40]

a(n) = 2*A019298(n-1) for n>=1.
a(n) = 3*a(n-1) - 2*a(n-2) - 2*a(n-3) + 3*a(n-4) - a(n-5).
G.f.: (2*x^2 + 2*x^3 + 2*x^4)/(1 - 3*x + 2*x^2 + 2*x^3 - 3*x^4 + x^5).
a(n) + A212684(n) = n^3. - Clark Kimberling, Jun 02 2012 [corrected by Jason Yuen, Aug 19 2025]
a(n) = (2*n^3 - 3*n^2 + 2*n - (n mod 2))/4. - Ayoub Saber Rguez, Sep 02 2021

A236758 Number of partitions of 3*n into 3 parts with smallest part prime.

Original entry on oeis.org

0, 1, 3, 6, 10, 14, 20, 25, 32, 37, 45, 51, 61, 68, 79, 86, 98, 106, 120, 129, 144, 153, 169, 179, 196, 206, 223, 233, 251, 262, 282, 294, 315, 327, 348, 360, 382, 395, 418, 431, 455, 469, 495, 510, 537, 552, 580, 596, 625, 641, 670, 686, 716, 733, 764, 781

Offset: 1

Wesley Ivan Hurt, Jan 30 2014

			Count the primes in last column for a(n):
                                               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
---------------------------------------------------------------------
    0           1           3           6           10      ..   a(n)

Index entries for sequences related to partitions

Cf. A019298, A235988, A236364, A236762, A010051 (for function isprime).

with(numtheory); A236758:=n->sum((pi(n) - pi(n-1)) * (2*n - 2*i + 1 - floor((n - i + 1)/2)), i=1..n); seq(A236758(n), n=1..100);

Table[Sum[(PrimePi[i] - PrimePi[i - 1]) (2 n - 2 i + 1 - Floor[(n - i + 1)/2]), {i, n}], {n, 100}]

def a(n): return sum(1 for L in Partitions(3*n,length=3).list() if is_prime(L[2]))

a(n) = Sum_{i=1..n} A010051(i) * (2*n - 2*i + 1 - floor((n - i + 1)/2)).

cp's OEIS Frontend

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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A200763 T(n,k)=Number of 0..k arrays x(0..n-1) of n elements with nondecreasing average value.

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A236364 Sum of all the middle parts in the partitions of 3n into 3 parts.

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

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A380853 Number of ways to place six distinct positive integers on a triangle, three on the corners and three on the sides such that the sum of the three values on each side is n.

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A053493 Number of symmetric 4 X 4 matrices of nonnegative integers with every row and column adding to n.

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A053494 Number of symmetric 5 X 5 matrices of nonnegative integers with every row and column adding to n.

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