A027480 -id:A027480 - cp's OEIS frontend

A093445 The triangular triangle.

1, 3, 3, 6, 9, 6, 10, 18, 17, 10, 15, 30, 33, 27, 15, 21, 45, 54, 51, 39, 21, 28, 63, 80, 82, 72, 53, 28, 36, 84, 111, 120, 114, 96, 69, 36, 45, 108, 147, 165, 165, 150, 123, 87, 45, 55, 135, 188, 217, 225, 215, 190, 153, 107, 55, 66, 165, 234, 276, 294, 291, 270, 234

Offset: 1

Amarnath Murthy, Apr 02 2004

The n-th row of the triangular table begins by considering n triangular numbers (A000217) in order. Now segregate them into n groups beginning with n members in the first group, n-1 members in the second group, etc. Now sum each group. Thus the first term is the sum of first n numbers = n(n+1)/2, the second term is the sum of the next n-1 terms (from n+1 to 2n-1), the third term is the sum of the next n-2 terms (2n to 3n-3), etc. and the last term is simply n(n+1)/2. This triangle can be called a triangular triangle. The sequence contains the triangle by rows.

			Triangle begins:
   1
   3,  3
   6,  9,   6
  10, 18,  17,  10
  15, 30,  33,  27,  15
  21, 45,  54,  51,  39, 21
  28, 63,  80,  82,  72, 53, 28
  36, 84, 111, 120, 114, 96, 69, 36
The row for n = 4 is (1+2+3+4), (5+6+7), (8+9), 10 => 10 18 17 10.

Reinhard Zumkeller, Rows n = 1..100 of triangle, flattened

Cf. A000217, A093446. TT(n, 2) = A045943. TT(n, n-1) = A014209. TT(0, k) = A027480.

Cf. A005920 (central terms), A002817 (row sums).

a093445 n k = a093445_row n !! (k-1)
a093445_row n = f [n, n - 1 .. 1] [1 ..] where
   f [] _      = []
   f (x:xs) ys = sum us : f xs vs where (us,vs) = splitAt x ys
a093445_tabl = map a093445_row [1 ..]
-- Reinhard Zumkeller, Oct 03 2012

A093445 := proc(n,k)
    A000217(k*n-A000217(k-1))-A000217((k-1)*n-A000217(k-2)) ;
end proc:
seq(seq(A093445(n,k),k=1..n),n=1..10) ; # R. J. Mathar, Dec 09 2015

T[n_] := n(n + 1)/2; TT[n_, k_] := T[k*n - T[k - 1]] - T[(k - 1)*n - T[k - 2]]; Flatten[ Table[ TT[n, k], {n, 1, 11}, {k, 1, n}]] (* Robert G. Wilson v, Apr 24 2004 *)
Table[Total/@TakeList[Range[(n(n+1))/2],Range[n,1,-1]],{n,20}]//Flatten (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Feb 15 2019 *)

T(n) = A000217(n) is the n-th Triangular number. TT(n, k) is the k-th term of the n-th row, 0 < k <= n.
TT(n, k) = T(k*n - T(k - 1)) - T((k - 1)*n - T(k - 2)).
TT(n, 1) = TT(n, n) = T(n) = A000217(n).

Edited, corrected and extended by Robert G. Wilson v, Apr 24 2004

A129533 Array read by antidiagonals: T(n,k) = binomial(n+1,2)*binomial(n+k,n+1) for 0 <= k <= n.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 3, 3, 0, 0, 6, 12, 6, 0, 0, 10, 30, 30, 10, 0, 0, 15, 60, 90, 60, 15, 0, 0, 21, 105, 210, 210, 105, 21, 0, 0, 28, 168, 420, 560, 420, 168, 28, 0, 0, 36, 252, 756, 1260, 1260, 756, 252, 36, 0, 0, 45, 360, 1260, 2520, 3150, 2520, 1260, 360, 45, 0, 0, 55, 495

Offset: 0

Emeric Deutsch, Apr 22 2007

Previous name was: Triangle read by rows: T(n,k)=derivative of the q-binomial coefficient [n,k] evaluated at q=1 (0<=k<=n). - N. J. A. Sloane, Jan 06 2016
For example, T(5,2)=30 because [5,2] = q^6 + q^5 + 2*q^4 + 2*q^3 + 2*q^2 + q + 1 with derivative 6q^5 + 5q^4 + 8q^3 + 6q^2 + 4q + 1, having value 30 at q=1. - Emeric Deutsch, Apr 22 2007
Sum of entries in n-th antidiagonal = n(n-1)2^(n-3) = A001788(n-1).
T(n,k) = A094305(n-2, k-1) for n >= 2, k >= 1.
T(n,k) is total number of pips on a set of generalized linear dominoes with n cells (rather than two) and with the number of pips in each cell running from 0 to k (rather than 6). T(2,6) = 168 gives the total number of pips on a standard set of dominoes. We regard a generalized linear domino with n cells and up to k pips per cell as an ordered n-tuple [i_1, i_2, ..., i_n] with 0 <= i_1 <= i_2 <=  ... <= i_n <= k. - Alan Shore and N. J. A. Sloane, Jan 06 2016
T(n,k) can also be written more symmetrically as the trinomial coefficient (n+k; n-1, k-1, 2). - N. J. A. Sloane, Jan 06 2016
As a triangle read by rows, T(n,k) is the total number of inversions over all length n binary words having exactly k 1's. T(n,k) is also the total area above all North East lattice paths from the origin to the point (k,n-k). - Geoffrey Critzer, Mar 22 2018

			Array begins:
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ... (A000004)
0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, ... (A000217)
0, 3, 12, 30, 60, 105, 168, 252, 360, 495, 660, 858, ... (A027480)
0, 6, 30, 90, 210, 420, 756, 1260, 1980, 2970, 4290, ... (A033487)
0, 10, 60, 210, 560, 1260, 2520, 4620, 7920, 12870, ... (A266732)
0, 15, 105, 420, 1260, 3150, 6930, 13860, 25740, ... (A240440)
0, 21, 168, 756, 2520, 6930, 16632, 36036, ... (A266733)
...
If regarded as a triangle, this begins:
  0;
  0,  0;
  0,  1,  0;
  0,  3,  3,  0;
  0,  6, 12,  6,  0;
  0, 10, 30, 30, 10,  0;
  0, 15, 60, 90, 60, 15, 0;
  ...

G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976.

Cemil Karaçam and Alper Vural, Enumerating 2D and 3D lattice paths with arbitrary steps, Mathematica Bohemica, pp. 1-13 (2025). See p. 5.
Index entries for sequences related to dominoes.

Cf. A001788.
Rows give A000004, A000217, A027480, A033487, A266732, A240440, A266733.
A128503 and A094305 are very similar sequences.

dd:=proc(n,m) if m=0 or n=0 then 0 else (m+n)!/(2*(m-1)!*(n-1)!); fi; end;
f:=n->[seq(dd(n,m),m=0..30)];
for n from 0 to 10 do lprint(f(n)); od: # produces sequence as square array
T:=(n,k)->k*(k+1)*binomial(n,k+1)/2: for n from 0 to 12 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form

Table[Table[D[Expand[FunctionExpand[QBinomial[n, k, q]]], q] /. q -> 1, {k, 0, n}], {n, 0, 15}] // Grid (* Geoffrey Critzer, Mar 22 2018 *)

T(n,k) = (1/2)*k*(k+1)*binomial(n,k+1).

G.f.: G(q,z) = qz^2/(1-z-qz)^3.

Entry revised by N. J. A. Sloane, Jan 06 2016

A267370 Partial sums of A140091.

Original entry on oeis.org

0, 6, 21, 48, 90, 150, 231, 336, 468, 630, 825, 1056, 1326, 1638, 1995, 2400, 2856, 3366, 3933, 4560, 5250, 6006, 6831, 7728, 8700, 9750, 10881, 12096, 13398, 14790, 16275, 17856, 19536, 21318, 23205, 25200, 27306, 29526, 31863, 34320, 36900, 39606, 42441, 45408, 48510

Offset: 0

Bruno Berselli, Jan 13 2016

After 0, this sequence is the third column of the array in A185874.

Sequence is related to A051744 by A051744(n) = n*a(n)/3 - Sum_{i=0..n-1} a(i) for n>0.

			The sequence is also provided by the row sums of the following triangle (see the fourth formula above):
.  0;
.  1,  5;
.  4,  7, 10;
.  9, 11, 13, 15;
. 16, 17, 18, 19, 20;
. 25, 25, 25, 25, 25, 25;
. 36, 35, 34, 33, 32, 31, 30;
. 49, 47, 45, 43, 41, 39, 37, 35;
. 64, 61, 58, 55, 52, 49, 46, 43, 40;
. 81, 77, 73, 69, 65, 61, 57, 53, 49, 45, etc.
First column is A000290.
Second column is A027690.
Third column is included in A189834.
Main diagonal is A008587; other parallel diagonals: A016921, A017029, A017077, A017245, etc.
Diagonal 1, 11, 25, 43, 65, 91, 121, ... is A161532.

Bruno Berselli, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A005581, A051744, A105163, A140091, A185874.

Cf. similar sequences of the type n*(n+1)*(n+k)/2: A002411 (k=0), A006002 (k=1), A027480 (k=2), A077414 (k=3, with offset 1), A212343 (k=4, without the initial 0), this sequence (k=5).

Magma
```
[n*(n+1)*(n+5)/2: n in [0..50]];
```

Table[n (n + 1) (n + 5)/2, {n, 0, 50}]
LinearRecurrence[{4,-6,4,-1},{0,6,21,48},50] (* Harvey P. Dale, Jul 18 2019 *)

PARI
```
vector(50, n, n--; n*(n+1)*(n+5)/2)
```
Sage
```
[n*(n+1)*(n+5)/2 for n in (0..50)]
```

O.g.f.: 3*x*(2 - x)/(1 - x)^4.
E.g.f.: x*(12 + 9*x + x^2)*exp(x)/2.
a(n) = n*(n + 1)*(n + 5)/2.
a(n) = Sum_{i=0..n} n*(n - i) + 5*i, that is: a(n) = A002411(n) + A028895(n). More generally, Sum_{i=0..n} n*(n - i) + k*i = n*(n + 1)*(n + k)/2.
a(n) = 3*A005581(n+1).
a(n+1) - 3*a(n) + 3*a(n-1) = 3*A105163(n) for n>0.
From Amiram Eldar, Jan 06 2021: (Start)
Sum_{n>=1} 1/a(n) = 163/600.
Sum_{n>=1} (-1)^(n+1)/a(n) = 4*log(2)/5 - 253/600. (End)

A330298 a(n) is the number of subsets of {1..n} that contain exactly 1 odd and 2 even numbers.

Original entry on oeis.org

0, 0, 0, 0, 2, 3, 9, 12, 24, 30, 50, 60, 90, 105, 147, 168, 224, 252, 324, 360, 450, 495, 605, 660, 792, 858, 1014, 1092, 1274, 1365, 1575, 1680, 1920, 2040, 2312, 2448, 2754, 2907, 3249, 3420, 3800, 3990, 4410, 4620, 5082, 5313, 5819, 6072, 6624, 6900, 7500, 7800, 8450, 8775, 9477

Offset: 0

Enrique Navarrete, Feb 29 2020

The general formula for the number of subsets of {1..n} that contain exactly k odd and j even numbers is binomial(ceiling(n/2), k) * binomial(floor(n/2), j).

			For n=6, a(6) = 9 and the 9 subsets are: {1,2,4}, {1,2,6}, {1,4,6}, {2,3,4}, {2,3,6}, {2,4,5}, {2,5,6}, {3,4,6}, {4,5,6}.

Colin Barker, Table of n, a(n) for n = 0..1000
J.S. Seneschal, Oblong Prism Illustration
Index entries for linear recurrences with constant coefficients, signature (1,3,-3,-3,3,1,-1).

Cf. A330299, A330300.
Cf. A000217, A002378.
Interleaves the positive integers of A006002 and A027480.

a[n_] := Ceiling[n/2] * Binomial[Floor[n/2], 2]; Array[a, 55, 0] (* Amiram Eldar, Mar 01 2020 *)
Table[Length[Select[Subsets[Range[n],{3}],Total[Boole[OddQ[#]]]==1&]],{n,0,60}] (* Harvey P. Dale, Jul 26 2020 *)

a(n) = ceil(n/2) * binomial(floor(n/2), 2) \\ Andrew Howroyd, Mar 01 2020

concat([0,0,0,0], Vec(x^4*(2 + x) / ((1 - x)^4*(1 + x)^3) + O(x^40))) \\ Colin Barker, Mar 02 2020

a(n) = ceiling(n/2) * binomial(floor(n/2), 2).
From Colin Barker, Mar 01 2020: (Start)
G.f.: x^4*(2 + x) / ((1 - x)^4*(1 + x)^3).
a(n) = a(n-1) + 3*a(n-2) - 3*a(n-3) - 3*a(n-4) + 3*a(n-5) + a(n-6) - a(n-7) for n>6. (End)
E.g.f.: (x*(-3 + x + x^2)*cosh(x) + (3 - x + x^3)*sinh(x))/16. - Stefano Spezia, Mar 02 2020
a(n) = 3/32-5*n^2/32-n/32+n^3/16+(-)^n*(n-3+n^2)/32. - R. J. Mathar, Mar 31 2025

A161983 Irregular triangle read by rows: the group of 2n + 1 integers starting at A014105(n).

Original entry on oeis.org

0, 3, 4, 5, 10, 11, 12, 13, 14, 21, 22, 23, 24, 25, 26, 27, 36, 37, 38, 39, 40, 41, 42, 43, 44, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 136, 137

Offset: 0

Juri-Stepan Gerasimov, Jun 23 2009

The squares of numbers in each row can be gathered in an equation with the first n terms on one side, the next n+1 terms on the other. The third row, for example, could be rendered as 10^2 + 11^2 + 12^2 = 13^2 + 14^2.
This sequence contains all nonnegative integers that are within a distance of n from 2n^2 + 2n where n is any nonnegative integer. The nonnegative integers that are not in this sequence are of the form 2n^2 + k where n is any positive integer and -n <= k <= n-1. Also, when n is the product of two consecutive integers, a(n) = 2n; for example, a(20) = 40. See explicit formulas for the sequence in the formula section below. - Dennis P. Walsh, Aug 09 2013
Numbers k with the property that the largest Dyck path of the symmetric representation of sigma(k) has a central valley, n > 0. (Cf. A237593.) - Omar E. Pol, Aug 28 2018

			Triangle begins:
   0;
   3,  4,  5;
  10, 11, 12, 13, 14;
  21, 22, 23, 24, 25, 26, 27;
  36, 37, 38, 39, 40, 41, 42, 43, 44;
  55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65;
  78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90;
...

Michael Boardman, Proof Without Words: Pythagorean Runs, Math. Mag., 73 (2000), 59.

Union of A014105 and A317304.
The complement is A162917.
Column 1 gives A014105.
Right border gives A014106.
Row sums give the even-indexed terms of A027480.
Cf. A000290, A014105, A014106, A027480, A162917, A237593, A317304.

seq(seq(2*n^2+2*n+k,k=-n..n),n=0..10); # Dennis P. Walsh, Aug 09 2013
seq(n+floor(sqrt(n))*(floor(sqrt(n))+1),n=0..100); # Dennis P. Walsh, Aug 09 2013

As a triangle, T(n,k) = 2n^2 + 2n + k where -n <= k <= n and n = 0,1,... - Dennis P. Walsh, Aug 09 2013

As sequence, a(n) = n + floor(sqrt(n))*(floor(sqrt(n)) + 1); equivalently, a(n) = n + A000196(n)*(A000196(n)+1). - Dennis P. Walsh, Aug 09 2013

Definition clarified, 8th row terms corrected by R. J. Mathar, Jul 19 2009

A168641 Triangle read by rows: T(n,k) = [x^k] p(x,n), where p(x,n) = 3(x + 1)^n - 2(x^n + 1) - n*(x + x^(n - 1)) for n >= 2, p(x,0) = 1, and p(x,1) = x + 1.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 6, 6, 1, 1, 8, 18, 8, 1, 1, 10, 30, 30, 10, 1, 1, 12, 45, 60, 45, 12, 1, 1, 14, 63, 105, 105, 63, 14, 1, 1, 16, 84, 168, 210, 168, 84, 16, 1, 1, 18, 108, 252, 378, 378, 252, 108, 18, 1, 1, 20, 135, 360, 630, 756, 630, 360, 135, 20, 1

Offset: 0

Roger L. Bagula and Gary W. Adamson, Dec 01 2009

			Triangle begins:
  1;
  1,  1;
  1,  2,   1;
  1,  6,   6,   1;
  1,  8,  18,   8,   1;
  1, 10,  30,  30,  10,   1;
  1, 12,  45,  60,  45,  12,   1;
  1, 14,  63, 105, 105,  63,  14,   1;
  1, 16,  84, 168, 210, 168,  84,  16,  1;
  1, 18, 108, 252, 378, 378, 252, 108,  18,  1;
  1, 20, 135, 360, 630, 756, 630, 360, 135, 20, 1;
  ...

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A095151, A132046, A168643, A168644, A168646.

Columns (essentially): A005843 (k=1), A045943 (k=2), A027480 (k=3), A050534 (k=4), A253942 (k=5), A253943 (k=6), A253944 (k=7).

function f(n,k)
   if n le 2 then return 1;
   elif k eq 0 or k eq n then return 1;
   elif k eq 1 or k eq n-1 then return 2;
   else return 3;
   end if;
end function;
A168641:= func< n,k | Binomial(n,k)*f(n,k) >;
[A168641(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 24 2025

p[x_, n_]:= If[n==0, 1, If[n==1, 1+x, 3*(1+x)^n -(1+x^n) -(1+n*x +n*x^(n-1) + x^n)]];
Flatten[Table[CoefficientList[p[x, n], x], {n, 0, 10}]]
(* Second program *)
f[n_, k_]:= With[{b=Boole}, If[k<=n/2, b[k==0] +2*b[k==1] +3*b[2<=k<=n/2], f[n, n-k]]];
A168641[n_, k_]:= Binomial[n,k]*If[n<3,1,f[n,k]];
Table[A168641[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 24 2025 *)

T(n,k) := ratcoef(if n <= 2 then (1 + x)^n else 3*(x + 1)^n - (x^n + 1) - (x^n + n*x^(n - 1) + n*x + 1), x, k);
create_list(T(n, k), n, 0, 12, k, 0, n); /* Franck Maminirina Ramaharo, Jan 02 2019 */

def f(n,k):
    if (k<=n/2): return int(k==0) + 2*int(k==1) + 3*int(1A168641(n,k):
    if (n<3): return binomial(n,k)
    else: return binomial(n,k)*f(n,k)
print(flatten([[A168641(n,k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Mar 24 2025

From G. C. Greubel, Mar 24 2025: (Start)
T(n, k) = 3*binomial(n, k), for n >= 4 and 2 <= k <= n-2, otherwise T(n, 0) = T(n, n) = 1, T(n, 1) = T(n, n-1) = 2*A065475(n-1).
T(n, n-k) = T(n, k).
T(n, 1) = A005843(n) - [n=1] - 2*[n=2].
Columns: T(n, k) = 3*binomial(n,k) - 2*[n=k] - (k+1)*[n=k+1], k >= 2.
Sum_{k=0..n} T(n, k) = 2*A095151(n-1) - 2*[n=0] - 2*[n=1].
Sum_{k=0..n} (-1)^k*T(n, k) = (1+(-1)^n)*(n-2) + 5*[n=0]. (End)

Edited by Franck Maminirina Ramaharo, Jan 02 2019

A057587 Nonnegative numbers of form n*(n^2+-1)/2.

Original entry on oeis.org

0, 1, 3, 5, 12, 15, 30, 34, 60, 65, 105, 111, 168, 175, 252, 260, 360, 369, 495, 505, 660, 671, 858, 870, 1092, 1105, 1365, 1379, 1680, 1695, 2040, 2056, 2448, 2465, 2907, 2925, 3420, 3439, 3990, 4010, 4620, 4641, 5313, 5335, 6072, 6095, 6900, 6924, 7800

Offset: 0

N. J. A. Sloane, Oct 05 2000

Taking alternate terms gives A027480 and A006003. - Jeremy Gardiner, Apr 10 2005

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,3,-3,-3,3,1,-1).

Cf. A027480, A006003.

[(2*n^3+9*n^2+15*n+5+(3*n^2+n-5)*(-1)^n)/32 : n in [0..50]]; // Wesley Ivan Hurt, Mar 27 2015

A057587:=n->(2*n^3+9*n^2+15*n+5+(3*n^2+n-5)*(-1)^n)/32: seq(A057587(n), n=0..50); # Wesley Ivan Hurt, Mar 27 2015

CoefficientList[Series[x*(1 + 2 x - x^2 + x^3)/((x - 1)^4*(x + 1)^3), {x, 0, 50}], x] (* Wesley Ivan Hurt, Mar 27 2015 *)
Table[(2 n^3 + 9 n^2 + 15 n + 5 + (3 n^2 + n - 5) (-1)^n) / 32, {n, 0, 50}] (* Vincenzo Librandi, Mar 28 2015 *)
LinearRecurrence[{1,3,-3,-3,3,1,-1},{0,1,3,5,12,15,30},50] (* Harvey P. Dale, Nov 29 2022 *)

concat(0, Vec(x*(x^3-x^2+2*x+1)/((x-1)^4*(x+1)^3) + O(x^100))) \\ Colin Barker, Nov 18 2014

a(n) = (2*n^3+9*n^2+15*n+5+(3*n^2+n-5)*(-1)^n)/32. - Luce ETIENNE, Nov 18 2014
From Wesley Ivan Hurt, Mar 27 2015: (Start)
G.f.: x*(1 + 2 x - x^2 + x^3)/((x - 1)^4*(x + 1)^3).
a(n) = a(n-1)+3*a(n-2)-3*a(n-3)-3*a(n-4)+3*a(n-5)+a(n-6)-a(n-7). (End)
a(2*n+1) + a(2*n) = (a(2*n+1) - a(2*n))^3. - Greg Dresden, Feb 10 2022
a(2*n+1)^2 - a(2*n)^2 = (n+1)^4. - Adam Michael Bere, Feb 15 2024

A117560 a(n) = n*(n^2 - 1)/2 - 1.

Original entry on oeis.org

2, 11, 29, 59, 104, 167, 251, 359, 494, 659, 857, 1091, 1364, 1679, 2039, 2447, 2906, 3419, 3989, 4619, 5312, 6071, 6899, 7799, 8774, 9827, 10961, 12179, 13484, 14879, 16367, 17951, 19634, 21419, 23309, 25307, 27416, 29639, 31979, 34439, 37022, 39731, 42569

Offset: 2

Joseph Biberstine (jrbibers(AT)indiana.edu), Mar 29 2006

a(n-1) is an approximation for the lower bound of the "antimagic constant" of an antimagic square of order n. The antimagic constant here is defined as the least integer in the set of consecutive integers to which the rows, columns and diagonals of the square sum. By analogy with the magic constant. This approximation follows from the observation that (2*Sum_{k=1..n^2} k) + (m) + (m+1) <= Sum_{k=0..2*n+1} (m + k) where m is the antimagic constant for an antimagic square of order n. a(n) = A027480(n+1) - 1. Stricter bounds seem likely to exist. See A117561 for the upper bounds. Note there exist no antimagic squares of order two or three, but the values are indexed here for completeness.

			a(3) = 29 because the antimagic constant of an antimagic square of order 4 must be at least 29 (see comments).

Vincenzo Librandi, Table of n, a(n) for n = 2..5000
Eric Weisstein's World of Mathematics, Antimagic Square.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A027480, A049475, A050257, A117561.

[n*(n^2-1)/2 - 1: n in [2..50]]; // Vincenzo Librandi, Jun 20 2011

Mathematica
```
Table[n*(n^2-1)/2 - 1, {n, 2, 50}]
```

a(n) = n*(n^2 - 1)/2 - 1.
G.f.: x^2*(2 + 3*x - 3*x^2 + x^3)/(1-x)^4. - Colin Barker, Mar 29 2012
From Elmo R. Oliveira, Aug 19 2025: (Start)
E.g.f.: 1 + x + (1 + x)*(-2 + 2*x + x^2)*exp(x)/2.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n >= 6. (End)

A126275 Moment of inertia of all magic squares of order n.

Original entry on oeis.org

5, 60, 340, 1300, 3885, 9800, 21840, 44280, 83325, 147620, 248820, 402220, 627445, 949200, 1398080, 2011440, 2834325, 3920460, 5333300, 7147140, 9448285, 12336280, 15925200, 20345000, 25742925, 32284980, 40157460, 49568540, 60749925, 73958560, 89478400

Offset: 2

Jonathan Vos Post, Dec 23 2006

Michael De Vlieger, Table of n, a(n) for n = 2..10000
Peter Loly, The Invariance of the Moment of Inertia of Magic Squares, Mathematical Gazette, Vol. 88, No. 511 (March 2004), 151-153, JSTOR:3621372.
Ivars Peterson, Magic Square Physics, Science News online, Jul 01, 2006; Vol. 170, No. 1.
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).
Index entries for sequences related to moment of inertia.

Cf. A027480, A126276.

Array[(#^2*(#^4 - 1))/12 &, 31, 2] (* or *)
Drop[CoefficientList[Series[-5 x^2*(x + 1) (x^2 + 4 x + 1)/(x - 1)^7, {x, 0, 32}], x], 2] (* Michael De Vlieger, Apr 13 2021 *)
LinearRecurrence[{7,-21,35,-35,21,-7,1},{5,60,340,1300,3885,9800,21840},40] (* Harvey P. Dale, Apr 03 2023 *)

a(n) = (n^2 * (n^4 - 1))/12 \\ Felix Fröhlich, May 31 2021

Vec(-5*x^2*(x+1)*(x^2+4*x+1)/(x-1)^7 + O(x^30)) \\ Felix Fröhlich, May 31 2021

a(n) = (n^2 * (n^4 - 1))/12.
G.f.: -5*x^2*(x+1)*(x^2+4*x+1) / (x-1)^7. - Colin Barker, Dec 10 2012
a(n) = Sum_{i=0..n^2-1} (k+i)^2 - (k*n + A027480(n-1))^2. - Charlie Marion, May 08 2021
From Amiram Eldar, Jul 03 2025: (Start)
Sum_{n>=2} 1/a(n) = 3*Pi*coth(Pi) - 2*Pi^2 + 21/2.
Sum_{n>=2} (-1)^n/a(n) = 3*Pi*cosech(Pi) + Pi^2 - 21/2. (End)

More terms from Colin Barker, Dec 10 2012

A338645 Expansion of Product_{k>=1} (1 + x^k)^binomial(k+2,2).

Original entry on oeis.org

1, 3, 9, 29, 78, 207, 526, 1284, 3054, 7084, 16071, 35748, 78167, 168195, 356754, 746772, 1544145, 3157056, 6387114, 12795366, 25397760, 49977262, 97542936, 188912466, 363196750, 693424803, 1315161528, 2478648920, 4643337213, 8648452782, 16019345259, 29515269060, 54104712129

Offset: 0

Ilya Gutkovskiy, May 09 2021

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A000217, A027480, A028377, A217093, A219555, A343200, A344097, A344098.

nmax = 32; CoefficientList[Series[Product[(1 + x^k)^Binomial[k + 2, 2], {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, (1/n) Sum[Sum[(-1)^(k/d + 1) d Binomial[d + 2, 2], {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 32}]

a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} ( Sum_{d|k} (-1)^(k/d+1) * A027480(d) ) * a(n-k).

a(n) ~ (7/15)^(1/8) * 2^(-21/8) * n^(-5/8) * exp((2/3)*(7/15)^(1/4)*Pi * n^(3/4) + 9*sqrt(15/7)*zeta(3) * sqrt(n) / (2*Pi^2) + ((5/7)^(1/4)*Pi / (2*3^(3/4)) - 1215*(15/7)^(1/4)*zeta(3)^2 / (28*Pi^5)) * n^(1/4) + 54675*zeta(3)^3 / (98*Pi^8) - 45*zeta(3) / (28*Pi^2)). - Vaclav Kotesovec, May 12 2021

cp's OEIS Frontend

A093445 The triangular triangle.

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A129533 Array read by antidiagonals: T(n,k) = binomial(n+1,2)*binomial(n+k,n+1) for 0 <= k <= n.

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A267370 Partial sums of A140091.

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A330298 a(n) is the number of subsets of {1..n} that contain exactly 1 odd and 2 even numbers.

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A161983 Irregular triangle read by rows: the group of 2n + 1 integers starting at A014105(n).

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A168641 Triangle read by rows: T(n,k) = [x^k] p(x,n), where p(x,n) = 3*(x + 1)^n - 2*(x^n + 1) - n*(x + x^(n - 1)) for n >= 2, p(x,0) = 1, and p(x,1) = x + 1.

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A168641 Triangle read by rows: T(n,k) = [x^k] p(x,n), where p(x,n) = 3(x + 1)^n - 2(x^n + 1) - n*(x + x^(n - 1)) for n >= 2, p(x,0) = 1, and p(x,1) = x + 1.