A008794 -id:A008794 - cp's OEIS frontend

A032438 a(n) = n^2 - floor((n+1)/2)^2.

0, 0, 3, 5, 12, 16, 27, 33, 48, 56, 75, 85, 108, 120, 147, 161, 192, 208, 243, 261, 300, 320, 363, 385, 432, 456, 507, 533, 588, 616, 675, 705, 768, 800, 867, 901, 972, 1008, 1083, 1121, 1200, 1240, 1323, 1365, 1452, 1496, 1587, 1633, 1728, 1776, 1875, 1925

Offset: 0

N. J. A. Sloane

The answer to a question from Mike and Laurie Crain (2crains(AT)concentric.net): how many even numbers are there in an n X n multiplication table starting at 1 X 1?
a(n+1) is the number of pairs (x,y) with x and y in {0,...,n}, x and y of the same parity, and x+y >= n. - Clark Kimberling, Jul 02 2012
From J. M. Bergot, Aug 08 2013: (Start)
Define a triangle to have T(1,1)=0 and T(n,c) = n^2 - c^2. The difference of the sum of the terms in antidiagonal(n+1) and those in antidiagonal(n)=a(n).
Column 0 is vertical and T(n,n)=0. The first few rows are 0; 3,0; 8,5,0; 15,12,7,0; 24,21,16,9,0; 35,32,27,20,11,0; the first few antidiagonals are 0; 3; 8,0; 15,5; 24,12,0; 35,21,7; 48,32,16,0; the first few sum of terms in the antidiagonals are 0, 3, 8, 20, 36, 63, 96, 144, 200, 275, 360, 468, 588, 735, 896, 1088, 1296, 1539. (End)
Sum of the largest parts in the partitions of 2n into two distinct odd parts. For example, a(5) = 16; the partitions of 2(5) = 10 into two distinct odd parts are (9,1) and (7,3). The sum of the largest parts is then 9+7 = 16. - Wesley Ivan Hurt, Nov 27 2017

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

First differences are in A059029, partial sums in A143785.

Cf. A008794, A014255.

[n^2-Floor( (n+1)/2 )^2 : n in [0..60]]; // Vincenzo Librandi, Sep 27 2011

A032438:=n->n^2-floor((n+1)/2)^2; seq(A032438(n), n=0..100) # Wesley Ivan Hurt, Nov 25 2013

Table[n^2-Floor[((n+1)/2)]^2,{n,0,50}] (* or *) LinearRecurrence[ {1,2,-2,-1,1},{0,0,3,5,12},51]

a(n)=n^2 - ((n+1)\2)^2 \\ Charles R Greathouse IV, Feb 19 2017

a(n) = n^2 - A008794(n+1).
G.f.: x^2*(x^2 + 2*x + 3)/(1-x^2)^2/(1-x). - Ralf Stephan, Jun 10 2003
a(n) = (1/8)*(2*n*(3*n-1)+(2*n+1)*(-1)^n-1). a(-n-1) = A014255(n). - Bruno Berselli, Sep 27 2011
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5), n > 4. - Harvey P. Dale, Nov 24 2011
E.g.f.: (x*(1 + 3*x)*cosh(x) + (3*x^2 + 3*x - 1)*sinh(x))/4. - Stefano Spezia, Aug 01 2022

A236540 Triangle read by rows: T(n,k), n>=1, k>=1, in which column k lists k copies of the positive squares in nondecreasing order, except the first column which lists the triangular numbers, and the first element of column k is in row k(k+1)/2.

Original entry on oeis.org

0, 1, 3, 1, 6, 1, 10, 4, 15, 4, 1, 21, 9, 1, 28, 9, 1, 36, 16, 4, 45, 16, 4, 1, 55, 25, 4, 1, 66, 25, 9, 1, 78, 36, 9, 1, 91, 36, 9, 4, 105, 49, 16, 4, 1, 120, 49, 16, 4, 1, 136, 64, 16, 4, 1, 153, 64, 25, 9, 1, 171, 81, 25, 9, 1, 190, 81, 25, 9, 4, 210, 100, 36, 9, 4, 1

Offset: 1

Omar E. Pol, Jan 28 2014

Gives an identity for the sum of all aliquot divisors of all positive integers <= n.
Alternating sum of row n equals A153485(n), i.e., Sum_{k=1..A003056(n)} (-1)^(k-1)*T(n,k) = A153485(n).
Row n has length A003056(n) hence the first element of column k is in row A000217(k).
Column 1 is A000217. Columns 2-3 are A008794, A211547, but without the zeros.
Column k lists the partial sums of the k-th column of triangle A231347 which gives an identity for the sum of aliquot divisors of n. - Omar E. Pol, Nov 11 2014

			Triangle begins:
    0;
    1;
    3,   1;
    6,   1;
   10,   4;
   15,   4,   1;
   21,   9,   1;
   28,   9,   1;
   36,  16,   4;
   45,  16,   4,   1;
   55,  25,   4,   1;
   66,  25,   9,   1;
   78,  36,   9,   1;
   91,  36,   9,   4;
  105,  49,  16,   4,  1;
  120,  49,  16,   4,  1;
  136,  64,  16,   4,  1;
  153,  64,  25,   9,  1;
  171,  81,  25,   9,  1;
  190,  81,  25,   9,  4;
  210, 100,  36,   9,  4,  1;
  231, 100,  36,  16,  4,  1;
  253, 121,  36,  16,  4,  1;
  276, 121,  49,  16,  4,  1;
  ...
For n = 6 the divisors of all positive integers <= 6 are [1], [1, 2], [1, 3], [1, 2, 4], [1, 5], [1, 2, 3, 6] hence the sum of all aliquot divisors is [0] + [1] + [1] + [1+2] + [1] + [1+2+3] = 0 + 1 + 1 + 3 + 1 + 6 = 12. On the other hand the 6th row of triangle is 15, 4, 1, therefore the alternating row sum is 15 - 4 + 1 = 12, equaling the sum of all aliquot divisors of all positive integers <= 6.

Cf. A000203, A000217, A001065, A008794, A003056, A153485, A196020, A211547, A211343, A228813, A231345, A231347, A235791, A235794, A235799, A236104, A236106, A236112, A237591, A237593, A286001.

A354594 a(n) = n^2 + 2*floor(n/2)^2.

Original entry on oeis.org

0, 1, 6, 11, 24, 33, 54, 67, 96, 113, 150, 171, 216, 241, 294, 323, 384, 417, 486, 523, 600, 641, 726, 771, 864, 913, 1014, 1067, 1176, 1233, 1350, 1411, 1536, 1601, 1734, 1803, 1944, 2017, 2166, 2243, 2400, 2481, 2646, 2731, 2904

Offset: 0

David Lovler, Jun 01 2022

The first bisection is A033581, the second bisection is A080859. - Bernard Schott, Jun 07 2022

David Lovler, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A000290, A008794, A033581, A080859, A247375.

Cf. A213037, A322744 (diagonal), A354595, A354596.

a[n_] := n^2 + 2 Floor[n/2]^2
Table[a[n], {n, 0, 90}]    (* A354594 *)
LinearRecurrence[{1, 2, -2, -1, 1}, {0, 1, 6, 11, 24}, 60]

PARI
```
a(n) = n^2 + 2*(n\2)^2;
```

a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5), n >= 5.
a(n) = A000290(n) + 2*A008794(n).
G.f.: x*(1 + 5*x + 3*x^2 + 3*x^3)/((1 - x)^3*(1 + x)^2).
E.g.f.: (x*(1 + 3*x)*cosh(x) + (1 + 3*x + 3*x^2)*sinh(x))/2. - Stefano Spezia, Jun 07 2022

A354595 a(n) = n^2 + 4*floor(n/2)^2.

Original entry on oeis.org

0, 1, 8, 13, 32, 41, 72, 85, 128, 145, 200, 221, 288, 313, 392, 421, 512, 545, 648, 685, 800, 841, 968, 1013, 1152, 1201, 1352, 1405, 1568, 1625, 1800, 1861, 2048, 2113, 2312, 2381, 2592, 2665, 2888, 2965, 3200, 3281, 3528, 3613, 3872

Offset: 0

David Lovler, Jun 01 2022

The first bisection is A139098, the second bisection is A102083.

David Lovler, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A000290, A008794, A102083, A139098, A247375.

Cf. A213037, A327259 (diagonal), A354594, A354596.

a[n_] := n^2 + 4 Floor[n/2]^2
Table[a[n], {n, 0, 90}]    (* A354595 *)
LinearRecurrence[{1, 2, -2, -1, 1}, {0, 1, 8, 13, 32}, 60]

PARI
```
a(n) = n^2 + 4*(n\2)^2;
```

a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5), n >= 5.
a(n) = A000290(n) + 4*A008794(n).
G.f.: x*(1 + 7*x + 3*x^2 + 5*x^3)/((1 - x)^3*(1 + x)^2).
E.g.f.: 2*x^2*cosh(x) + (1 + 2*x + 2*x^2)*sinh(x). - Stefano Spezia, Jun 07 2022

A354596 Array T(n,k) = k^2 + (2n-4)*floor(k/2)^2, n >= 0, k >= 0, read by descending antidiagonals.

Original entry on oeis.org

0, 1, 0, 0, 1, 0, 5, 2, 1, 0, 0, 7, 4, 1, 0, 9, 8, 9, 6, 1, 0, 0, 17, 16, 11, 8, 1, 0, 13, 18, 25, 24, 13, 10, 1, 0, 0, 31, 36, 33, 32, 15, 12, 1, 0, 17, 32, 49, 54, 41, 40, 17, 14, 1, 0, 0, 49, 64, 67, 72, 49, 48, 19, 16, 1, 0, 21, 50, 81, 96, 85, 90, 57, 56, 21, 18, 1, 0

Offset: 0

David Lovler, Jun 01 2022

Column k is an arithmetic progression with difference 2*A008794(k).
Odd rows of A133728 triangle are contained in row 0.
For i = 0 through 4, row i is 0 and the diagonal of A319929, A322630 = A213037, A003991, A322744, and A327259, respectively. In general, row i is 0 and the diagonal of array U(i;n,k) described in A327263.

			T(n,k) begins:
  0,   1,   0,   5,   0,   9,   0,  13, ...
  0,   1,   2,   7,   8,  17,  18,  31, ...
  0,   1,   4,   9,  16,  25,  36,  49, ...
  0,   1,   6,  11,  24,  33,  54,  67, ...
  0,   1,   8,  13,  32,  41,  72,  85, ...
  0,   1,  10,  15,  40,  49,  90, 103, ...
  0,   1,  12,  17,  48,  57, 108, 121, ...
  ...

David Lovler, Table of n, a(n) for n = 0..5150

Cf. A000290, A008794, A133728, A213037, A247375.
Cf. A266222, A266439.
Cf. A319929, A322630, A322744, A327259, A327263, A354594, A354595.

T[n_, k_] := k^2 + (2*n - 4)*Floor[k/2]^2; Table[T[n - k, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* Amiram Eldar, Jun 20 2022 *)

PARI
```
T(n,k) = k^2 + (2*n-4)*(k\2)^2;
```

T(n,k) = U(n;k,k) (see A327263).
For each row, T(n,k) = T(n,k-1) + 2*T(n,k-2) - 2*T(n,k-3) - T(n,k-4) + T(n,k-5), k >= 5.
G.f. for row n: x*(1 + (2*n-1)*x + 3*x^2 + (2*n-3)*x^3)/((1 - x)^3*(1 + x)^2). When n = 2, this reduces to x*(1 + x)/(1 - x)^3.
E.g.f. for row n: (((4-n)*x + n*x^2)*cosh(x) + (n-2 + n*x + n*x^2)*sinh(x))/2. When n = 2, this reduces to (x + x^2)*cosh(x) + (x + x^2)*sinh(x) = (x + x^2)*exp(x).

A128180 A002260 * A097807.

Original entry on oeis.org

1, -1, 2, 2, -1, 3, -2, 3, -1, 4, 3, -2, 4, -1, 5, -3, 4, -2, 5, -1, 6, 4, -3, 5, -2, 6, -1, 7, -4, 5, -3, 6, -2, 7, -1, 8, 5, -4, 6, -3, 7, -2, 8, -1, 9, -5, 6, -4, 7, -3, 8, -2, 9, -1, 10

Offset: 1

Gary W. Adamson, Feb 17 2007

Row sums = A008794: (1, 1, 4, 4, 9, 9, 16, 16, ...).
Unsigned row sums = the triangular sequence, A000217: (1, 3, 6, 10, ...) by virtue of the fact that each row is a permutation of the natural numbers.
A128179 = A097807 * A002260.

			Triangle begins:
   1;
  -1,  2;
   2, -1,  3;
  -2,  3, -1,  4;
   3, -2,  4, -1,  5;
  -3,  4, -2,  5, -1,  6;
   4, -3,  5, -2,  6, -1,  7;
  ...

Cf. A002260, A097807, A008794, A128179.

T(n,k)=(2*k-1+(-1)^(n-k)*(2*n+1))/4 \\ Franklin T. Adams-Watters, Apr 12 2011

A002260 * A097807 as infinite lower triangular matrices.
From Franklin T. Adams-Watters, Apr 12 2011: (Start)
T(n,k) = (2k - 1 + (-1)^(n-k)*(2n+1))/4.
|T(n,k)| = (2n+1 + (-1)^(n-k)*(2k-1))/4. (End)

A194801 Square array read by antidiagonals: T(n,k) = k((n+1)k-n+1)/2, k = 0, +- 1, +- 2,..., n >= 0.

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 0, 1, 1, 3, 0, 1, 2, 4, 1, 0, 1, 3, 5, 4, 6, 0, 1, 4, 6, 7, 9, 3, 0, 1, 5, 7, 10, 12, 9, 10, 0, 1, 6, 8, 13, 15, 15, 16, 6, 0, 1, 7, 9, 16, 18, 21, 22, 16, 15, 0, 1, 8, 10, 19, 21, 27, 28, 26, 25, 10, 0, 1, 9, 11, 22, 24, 33, 34, 36, 35

Offset: 0

Omar E. Pol, Feb 05 2012

Note that a single formula gives several types of numbers. Row 0 lists 0 together the Molien series for 3-dimensional group [2,k]+ = 22k. Row 1 lists, except first zero, the squares repeated. If n >= 2, row n lists the generalized (n+3)-gonal numbers, for example: row 2 lists the generalized pentagonal numbers A001318. See some other examples in the cross-references section.

			Array begins:
(A008795): 0, 1,  0,  3,  1,  6,  3, 10,   6,  15,  10...
(A008794): 0, 1,  1,  4,  4,  9,  9, 16,  16,  25,  25...
A001318:   0, 1,  2,  5,  7, 12, 15, 22,  26,  35,  40...
A000217:   0, 1,  3,  6, 10, 15, 21, 28,  36,  45,  55...
A085787:   0, 1,  4,  7, 13, 18, 27, 34,  46,  55,  70...
A001082:   0, 1,  5,  8, 16, 21, 33, 40,  56,  65,  85...
A118277:   0, 1,  6,  9, 19, 24, 39, 46,  66,  75, 100...
A074377:   0, 1,  7, 10, 22, 27, 45, 52,  76,  85, 115...
A195160:   0, 1,  8, 11, 25, 30, 51, 58,  86,  95, 130...
A195162:   0, 1,  9, 12, 28, 33, 57, 64,  96, 105, 145...
A195313:   0, 1, 10, 13, 31, 36, 63, 70, 106, 115, 160...
A195818:   0, 1, 11, 14, 34, 39, 69, 76, 116, 125, 175...

Rows (0-11): 0 together with A008795, (truncated A008794), A001318, A000217, A085787, A001082, A118277, A074377, A195160, A195162, A195313, A195818
Columns (0-9): A000004, A000012, A001477, (truncated A000027), A016777, (truncated A008585), A016945, (truncated A016957), A017341, (truncated A017329).
Cf. A139600.

A236631 Triangle read by rows: T(j,k), j>=1, k>=1, in which column k lists the positive squares repeated k-1 times, except the column 1 which is A123327. The elements of the even-indexed columns are multiplied by -1. The first element of column k is in row k(k+1)/2.

Original entry on oeis.org

1, 3, 5, -1, 8, -1, 10, -4, 15, -4, 1, 16, -9, 1, 23, -9, 1, 25, -16, 4, 31, -16, 4, -1, 34, -25, 4, -1, 45, -25, 9, -1, 42, -36, 9, -1, 55, -36, 9, -4, 60, -49, 16, -4, 1, 67, -49, 16, -4, 1, 69, -64, 16, -4, 1, 86, -64, 25, -9, 1, 84, -81, 25, -9, 1, 103

Offset: 1

Omar E. Pol, Jan 29 2014

T(j,k) which row j has length A003056(j) hence the first element of column k is in row A000217(j).
Row sums give A000203.
Interpreted as a sequence with index n this is also the first differences of A236630. If a(n) is positive then a(n) is the number of cells turned ON at n-th stage in the structure of A236630. If a(n) is negative then a(n) is the number of cells turned OFF at n-th stage in the structure of A236630.

			Written as an irregular triangle the sequence begins:
1;
3;
5,     -1;
8,     -1;
10,    -4;
15,    -4,    1;
16,    -9,    1;
23,    -9,    1;
25,   -16,    4;
31,   -16,    4,   -1;
34,   -25,    4,   -1;
45,   -25,    9,   -1;
42,   -36,    9,   -1;
55,   -36,    9,   -4;
60,   -49,   16,   -4,   1;
67,   -49,   16,   -4,   1;
69,   -64,   16,   -4,   1;
86,   -64,   25,   -9,   1;
84,   -81,   25,   -9,   1;
103,  -81,   25,   -9,   4;
102, -100,   36,   -9,   4,  -1;
113, -100,   36,  -16,   4,  -1;
122, -121,   36,  -16,   4,  -1;
145, -121,   49,  -16,   4,  -1;
...
For j = 15 the divisors of 15 are 1, 3, 5, 15, therefore the sum of divisors of 15 is 1 + 3 + 5 + 15 = 24. On the other hand the 15th row of triangle is 60, -49, 16, -4, 1, therefore the row sum is 60 - 49 + 16 - 4 + 1 = 24, equalling the sum of divisors of 15.

Cf. A000203, A000217, A000290, A003056, A004125, A024916, A008794, A123327, A196020, A211547, A236104, A236630, A237593.

T(n,1) = A000203(n) + A004125(n).

A293783 Triangle of numbers of squares {i^2}, i = 0,1..ceiling(n/2), in permutations of {1..n} in A293857.

Original entry on oeis.org

0, 1, 0, 1, 2, 0, 2, 8, 0, 4, 0, 24, 0, 12, 0, 108, 0, 36, 576, 0, 720, 0, 144, 4608, 0, 4032, 0, 576, 0, 31680, 0, 31680, 0, 2880, 0, 288000, 0, 201600, 0, 14400, 2505600, 0, 2764800, 0, 1987200, 0, 86400, 30067200, 0, 28512000, 0, 14515200, 0, 518400

Offset: 1

Peter J. C. Moses and Vladimir Shevelev, Oct 18 2017

From Shevelev's comment in A008794 it follows that the last entries of rows, corresponding to the maximal possible i^2, form sequence A010551, n >= 1. Also note that the last entry of each row is the gcd of all its entries (for a proof see comment in A293857). - Vladimir Shevelev, Oct 26 2017

			Triangle begins
         0,      1;
         0,      1;
         2,      0,        2;
         8,      0,        4;
         0,     24,        0,     12;
         0,    108,        0,     36;
       576,      0,      720,      0,      144;
      4608,      0,     4032,      0,      576;
         0,  31680,        0,  31680,        0,  2880;
         0, 288000,        0, 201600,        0, 14400;
   2505600,      0,  2764800,      0,  1987200,     0,  86400;
  30067200,      0, 28512000,      0, 14515200,     0, 518400;
The compressed triangle resulting from the division of each entry by the last entry of its row begins as follows. If i is the index of the row, starting with i = 1 then this last entry is floor(i/2)! * (i - floor(i/2))!.
    0,   1;
    0,   1;
    1,   0,   1;
    2,   0,   1;
    0,   2,   0,   1;
    0,   3,   0,   1;
    4,   0,   5,   0,   1;
    8,   0,   7,   0,   1;
    0,  11,   0,  11,   0,   1;
    0,  20,   0,  14,   0,   1;
   29,   0,  32,   0,  23,   0,   1;
   58,   0,  55,   0,  28,   0,   1;
    0,  88,   0,  94,   0,  46,   0,   1;
    0, 169,   0, 146,   0,  53,   0,   1;
  263,   0, 282,   0, 283,   0,  86,   0,   1;
  526,   0, 515,   0, 383,   0,  97,   0,   1;
For the sense of the entries of this triangle see the [Shevelev] link (with a continuation there). Let B(n,i) be the set of permutations C of 1..n for which c_1 - c_2 + ... + (-1)^(n-1)*c_n = i^2, i >= 0. Then |B(n,i)| is the entry in the n-th row and i-th column of the first triangle. Let us call two permutations C_1 and C_2 equivalent if one of them is obtained from another by a permutation of its elements with odd indices and/or separately with even indices. Let b(n,i) be the entry in the n-th row and i-th column of the second triangle. Then b(n,i) is the maximal possible number of pairwise non-equivalent permutations which could be chosen in B(n,i). On the other hand, it is the smallest number of non-equivalent permutations in B(n,i) such that every other permutation in B(n,i) is equivalent to one of them. So in some sense b(n,i) is the dimension of B(n,i). In particular, b(n,i) = 0 corresponds to empty B(n,i). - _Vladimir Shevelev_, Nov 13 2017

Peter J. C. Moses, Table of the first 100 rows
Vladimir Shevelev, Basis in subsets of permutations described in A293783, SeqFan post Oct 27 2017.

Cf. A008794, A010551, A293857, A293984.

a293783=Flatten[Table[PadLeft[Riffle[#,Table[0,{Floor[(n-1)/4]}]/.{}->0],1+Floor[(1+n)/2]](Floor[n/2]!*(n-Floor[n/2])!)&[Reverse[Map[SeriesCoefficient[QBinomial[n,Floor[(n+1)/2],q],{q,0,#}]&,Map[2#(Floor[(n+1)/2] - #)&,Range[0,Floor[(n+1)/4]]]]]],{n,20}]] (* Peter J. C. Moses, Nov 01 2017 *)

Row sums of triangle give A293857.

If C = {c_1..c_n} is a permutation of {1..n}, then c_1 - c_2 + ... has the same parity as 1 + 2 + ... + n = n*(n+1)/2. So adjacent rows in the triangle for odd and even n have the same positions of 0's. These positions follow through one, beginning from the first position for n == 1,2 (mod 4) and from the second position for n == 3,0 (mod 4). - David A. Corneth and Vladimir Shevelev, Oct 19 2017

A317300 Sequence obtained by taking the general formula for generalized k-gonal numbers: m((k - 2)m - k + 4)/2, where m = 0, +1, -1, +2, -2, +3, -3, ... and k >= 5. Here k = 0.

Original entry on oeis.org

0, 1, -3, 0, -8, -3, -15, -8, -24, -15, -35, -24, -48, -35, -63, -48, -80, -63, -99, -80, -120, -99, -143, -120, -168, -143, -195, -168, -224, -195, -255, -224, -288, -255, -323, -288, -360, -323, -399, -360, -440, -399, -483, -440, -528, -483, -575, -528, -624, -575, -675, -624, -728, -675, -783

Offset: 0

Omar E. Pol, Jul 29 2018

Taking the same formula with k = 1 we have A317301.
Taking the same formula with k = 2 we have A001057 (canonical enumeration of integers).
Taking the same formula with k = 3 we have 0 together with A008795 (Molien series for 3-dimensional representation of dihedral group D_6 of order 6).
Taking the same formula with k = 4 we have A008794 (squares repeated) except the initial zero.
Taking the same formula with k >= 5 we have the generalized k-gonal numbers (see Crossrefs section).

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Row 0 of A303301.
Cf. A001057, A008795, A008794, A174474, A317301.
Sequences of generalized k-gonal numbers: A001318 (k=5), A000217 (k=6), A085787 (k=7), A001082 (k=8), A118277 (k=9), A074377 (k=10), A195160 (k=11), A195162 (k=12), A195313 (k=13), A195818 (k=14), A277082 (k=15), A274978 (k=16), A303305 (k=17), A274979 (k=18), A303813 (k=19), A218864 (k=20), A303298 (k=21), A303299 (k=22), A303303 (k=23), A303814 (k=24), A303304 (k=25), A316724 (k=26), A316725 (k=27), A303812 (k=28), A303815 (k=29), A316729 (k=30).

concat(0, Vec(x*(1 - 4*x + x^2) / ((1 - x)^3*(1 + x)^2) + O(x^50))) \\ Colin Barker, Aug 01 2018

a(n) = -A174474(n+1).
From Colin Barker, Aug 01 2018: (Start)
G.f.: x*(1 - 4*x + x^2) / ((1 - x)^3*(1 + x)^2).
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n>4.
a(n) = -n*(n + 4) / 4 for n even.
a(n) = -(n - 3)*(n + 1) / 4 for n odd.
(End)

cp's OEIS Frontend

A032438 a(n) = n^2 - floor((n+1)/2)^2.

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A236540 Triangle read by rows: T(n,k), n>=1, k>=1, in which column k lists k copies of the positive squares in nondecreasing order, except the first column which lists the triangular numbers, and the first element of column k is in row k(k+1)/2.

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A354594 a(n) = n^2 + 2*floor(n/2)^2.

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A354595 a(n) = n^2 + 4*floor(n/2)^2.

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A354596 Array T(n,k) = k^2 + (2n-4)*floor(k/2)^2, n >= 0, k >= 0, read by descending antidiagonals.

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A128180 A002260 * A097807.

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A194801 Square array read by antidiagonals: T(n,k) = k*((n+1)*k-n+1)/2, k = 0, +- 1, +- 2,..., n >= 0.

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A236631 Triangle read by rows: T(j,k), j>=1, k>=1, in which column k lists the positive squares repeated k-1 times, except the column 1 which is A123327. The elements of the even-indexed columns are multiplied by -1. The first element of column k is in row k(k+1)/2.

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A194801 Square array read by antidiagonals: T(n,k) = k((n+1)k-n+1)/2, k = 0, +- 1, +- 2,..., n >= 0.

A317300 Sequence obtained by taking the general formula for generalized k-gonal numbers: m((k - 2)m - k + 4)/2, where m = 0, +1, -1, +2, -2, +3, -3, ... and k >= 5. Here k = 0.