A008794 -id:A008794 - cp's OEIS frontend

A317301 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 = 1.

0, 1, -2, 1, -5, 0, -9, -2, -14, -5, -20, -9, -27, -14, -35, -20, -44, -27, -54, -35, -65, -44, -77, -54, -90, -65, -104, -77, -119, -90, -135, -104, -152, -119, -170, -135, -189, -152, -209, -170, -230, -189, -252, -209, -275, -230, -299, -252, -324, -275, -350, -299, -377, -324, -405, -350, -434

Offset: 0

Omar E. Pol, Jul 29 2018

Taking the same formula with k = 0 we have A317300.
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 1 of A303301.
Cf. A000096, A001057, A008794, A008795, A317300.
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).

/* By definition: */ k:=1; [0] cat [m*i*((k-2)*m*i-k+4)/2: i in [1,-1], m in [1..30]]; // Bruno Berselli, Jul 30 2018

Table[(-2 n (n + 1) - 5 (2 n + 1) (-1)^n + 5)/16, {n, 0, 60}] (* Bruno Berselli, Jul 30 2018 *)

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

From Bruno Berselli, Jul 30 2018: (Start)
O.g.f.: x*(1 - 3*x + x^2)/((1 + x)^2*(1 - x)^3).
E.g.f.: (-5*(1 + 2*x) + (5 - 2*x^2)*exp(2*x))*exp(-x)/16.
a(n) = a(-n+1) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5).
a(n) = (-2*n*(n + 1) - 5*(2*n + 1)*(-1)^n + 5)/16. Therefore:
a(n) = -n*(n + 6)/8 for even n;
a(n) = -(n - 5)*(n + 1)/8 for odd n. Also:
a(n) = a(n-5) for odd n > 3.
2*(2*n - 1)*a(n) + 2*(2*n + 1)*a(n-1) + n*(n^2 - 3) = 0. (End)

A329708 Triangle, read by rows, where the n-th row lists the (2n+1) coefficients of (1+2x+...+(n+1)x^n)^2.

Original entry on oeis.org

1, 1, 4, 4, 1, 4, 10, 12, 9, 1, 4, 10, 20, 25, 24, 16, 1, 4, 10, 20, 35, 44, 46, 40, 25, 1, 4, 10, 20, 35, 56, 70, 76, 73, 60, 36, 1, 4, 10, 20, 35, 56, 84, 104, 115, 116, 106, 84, 49, 1, 4, 10, 20, 35, 56, 84, 120, 147, 164, 170, 164, 145, 112, 64

Offset: 0

Seiichi Manyama, Feb 29 2020

			Triangle begins:
  1;
  1, 4,  4;
  1, 4, 10, 12,  9;
  1, 4, 10, 20, 25, 24, 16;
  1, 4, 10, 20, 35, 44, 46, 40, 25;
  ...

Seiichi Manyama, Rows n = 0..99, flattened

Row sums give A000537(n+1).
T(n,2n) gives A000290(n+1).
Cf. A000292, A002260, A084608, A008794, A329709.

row[n_]:=CoefficientList[Series[(Sum[(i+1)x^i,{i,0,n}])^2,{x,0,2n}],x]; Array[row,8,0]//Flatten (* Stefano Spezia, Feb 15 2025 *)

for(n=0, 10, print(Vecrev(sum(k=0, n, (k+1)*x^k)^2), ", "))

T(n,k) = A000292(k+1) for k=0..n.

Sum_{k=0..2n} (-1)^k * T(n,k) = A008794(n+2). - Alois P. Heinz, Feb 14 2025

A115012 Sum_{i=1..n, gcd(5,i)=1} i.

Original entry on oeis.org

1, 3, 6, 10, 10, 16, 23, 31, 40, 40, 51, 63, 76, 90, 90, 106, 123, 141, 160, 160, 181, 203, 226, 250, 250, 276, 303, 331, 360, 360, 391, 423, 456, 490, 490, 526, 563, 601, 640, 640, 681, 723, 766, 810, 810, 856, 903, 951, 1000, 1000, 1051, 1103, 1156, 1210, 1210, 1266

Offset: 1

N. J. A. Sloane, Feb 24 2006

nonn

Replacing 5 in the definition by 2, 3, 4, 5, 6, 7, 8, 9 gives respectively A008794, A068626, A008794, this sequence, A115014, A115015, A008794, A068626.

A122656 a(n) = n*floor(n/2)^2.

Original entry on oeis.org

0, 0, 2, 3, 16, 20, 54, 63, 128, 144, 250, 275, 432, 468, 686, 735, 1024, 1088, 1458, 1539, 2000, 2100, 2662, 2783, 3456, 3600, 4394, 4563, 5488, 5684, 6750, 6975, 8192, 8448, 9826, 10115, 11664, 11988, 13718, 14079, 16000, 16400, 18522, 18963, 21296, 21780

Offset: 0

N. J. A. Sloane, Sep 22 2006

Szeged index of cycle of length n.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Janez Žerovnik, Szeged index of symmetric graphs, J. Chem. Inf. Comput. Sci., 39 (1999), 77-80; alternative link.
Index entries for linear recurrences with constant coefficients, signature (1,3,-3,-3,3,1,-1).

Cf. A002117, A013661, A008794.

[n*Floor(n/2)^2: n in [0..50]]; // Vincenzo Librandi, May 31 2014

Table[n Floor[n/2]^2,{n,0,50}] (* or *) LinearRecurrence[ {1,3,-3,-3,3,1,-1},{0,0,2,3,16,20,54},50] (* Harvey P. Dale, May 31 2014 *)

a(n) = (n*(1-(-1)^n+2*(-1+(-1)^n)*n+2*n^2))/8. G.f.: x^2*(x^4+x^3+7*x^2+x+2) / ((x-1)^4*(x+1)^3). - Colin Barker, Sep 20 2013
a(n) = n*A008794(n). - R. J. Mathar, Mar 04 2018
Sum_{n>=2} 1/a(n) = zeta(3)/2 + zeta(2) + 4*(log(2)-1). - Amiram Eldar, May 15 2024

A262742 Irregular table read by rows: T(n,k) is the number of binary symmetric n X n matrices with exactly k 1's; n>=0, 0<=k<=n^2. Where the symmetry axes are in horizontal and vertical.

Original entry on oeis.org

1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 2, 2, 2, 2, 2, 2, 1, 1, 1, 0, 0, 0, 4, 0, 0, 0, 6, 0, 0, 0, 4, 0, 0, 0, 1, 1, 1, 4, 4, 10, 10, 20, 20, 31, 31, 40, 40, 44, 44, 40, 40, 31, 31, 20, 20, 10, 10, 4, 4, 1, 1, 1, 0, 0, 0, 9, 0, 0, 0, 36, 0, 0, 0, 84, 0, 0, 0, 126, 0, 0

Offset: 0

Kival Ngaokrajang, Sep 29 2015

The row length of this irregular triangle is n^2+1 = A002522(n).
Inspired by A262666, but rotating the diagonal and antidiagonal symmetry axis to horizontal and vertical axes.
From Wolfdieter Lang, Oct 12 2015 (Start):
Double symmetry of n X n matrix M: M(i, j) = M(n-i+1, j) = M(i, n-j+1) (= M(n-i+1, n-j+1)), here with entries from {0, 1}.
Due to 0 <-> 1 flip the rows are symmetric.
The number of independent entries in such an n X n doubly symmetric matrix is A008794(n+1) (squares repeated). Therefore, the row sums give repeated  A002416 (omitting the first 1): 1, 2, 2, 16, 16, 512, 512, ... (End) - Wolfdieter Lang, Oct 12 2015

			Irregular table begins:
n\k 0   1   2   3   4   5   6   7   8   9   ...
0:  1
1:  1   1
2:  1   0   0   0   1
3:  1   1   2   2   2   2   2   2   1   1
...
Row 4: 1, 0, 0, 0, 4, 0, 0, 0, 6, 0, 0, 0, 4, 0, 0, 0, 1;
Row 5: 1, 1, 4, 4, 10, 10, 20, 20, 31, 31, 40, 40, 44, 44, 40, 40, 31, 31, 20, 20, 10, 10, 4, 4, 1, 1.
...

Kival Ngaokrajang, Illustration of initial terms

Cf. A262666,A002522, A008794, A002416.

More terms from Alois P. Heinz, Sep 29 2015

A295643 Squares repeated 4 times; a(n) = floor(n/4)^2.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 1, 1, 4, 4, 4, 4, 9, 9, 9, 9, 16, 16, 16, 16, 25, 25, 25, 25, 36, 36, 36, 36, 49, 49, 49, 49, 64, 64, 64, 64, 81, 81, 81, 81, 100, 100, 100, 100, 121, 121, 121, 121, 144, 144, 144, 144, 169, 169, 169, 169, 196, 196, 196, 196, 225, 225, 225

Offset: 0

Wesley Ivan Hurt, Nov 25 2017

a(n+1) is the sum of the smallest odd parts of the partitions of n into two distinct parts. For example, a(11) = 4; the partitions of 10 into two distinct parts are (9,1), (8,2), (7,3) and (6,4). The sum of the smallest odd parts in these partitions is then 1+3 = 4.

a(n+2) is the sum of the smallest odd parts of the partitions of n into two parts. For example, a(8) = 4; the partitions of 6 into two parts are (5,1), (4,2) and (3,3). The sum of the smallest odd parts is then 1+3 = 4.

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

Cf. A000290, A002265, A004526, A008794.

A136523 Triangle T(n,k) = A053120(n,k) + A053120(n-1,k), read by rows.

Original entry on oeis.org

1, 1, 1, -1, 1, 2, -1, -3, 2, 4, 1, -3, -8, 4, 8, 1, 5, -8, -20, 8, 16, -1, 5, 18, -20, -48, 16, 32, -1, -7, 18, 56, -48, -112, 32, 64, 1, -7, -32, 56, 160, -112, -256, 64, 128, 1, 9, -32, -120, 160, 432, -256, -576, 128, 256, -1, 9, 50, -120, -400, 432, 1120, -576, -1280, 256, 512

Offset: 0

Roger L. Bagula, Mar 23 2008

			Triangle begins as:
   1;
   1,  1;
  -1,  1,   2;
  -1, -3,   2,    4;
   1, -3,  -8,    4,    8;
   1,  5,  -8,  -20,    8,   16;
  -1,  5,  18,  -20,  -48,   16,   32;
  -1, -7,  18,   56,  -48, -112,   32,   64;
   1, -7, -32,   56,  160, -112, -256,   64,   128;
   1,  9, -32, -120,  160,  432, -256, -576,   128, 256;
  -1,  9,  50, -120, -400,  432, 1120, -576, -1280, 256, 512;

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

Cf. A000007, A001792, A001793, A001794, A000330, A008794, A011782.
Cf. A025192, A040000 (row sums), A053120, A057077, A081277, A109613.
Cf. A124182.

function A053120(n,k)
  if ((n+k) mod 2) eq 1 then return 0;
  elif n eq 0 then return 1;
  else return (-1)^Floor((n-k)/2)*(n/(n+k))*Binomial(Floor((n+k)/2), k)*2^k;
  end if;
end function;
A136523:= func< n,k | A053120(n,k) + A053120(n-1,k) >;
[A136523(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 26 2023

A053120[n_, k_]:= Coefficient[ChebyshevT[n,x], x, k];
T[n_, k_]:= T[n, k]= A053120[n,k] + A053120[n-1,k];
Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten

def A053120(n,k):
    if (n+k)%2==1: return 0
    elif n==0: return 1
    else: return floor((-1)^((n-k)//2)*(n/(n+k))*binomial((n+k)//2, k)*2^k)
def A136523(n,k): return A053120(n,k) + A053120(n-1,k)
flatten([[A136523(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jul 26 2023

T(n, k) = A053120(n,k) + A053120(n-1,k).
Sum_{k=0..n} T(n, k) = A040000(n).
From G. C. Greubel, Jul 26 2023: (Start)
T(n, 0) = A057077(n).
T(n, 1) = (-1)^floor((n-1)/2) * A109613(n-1).
T(n, 2) = (-1)^floor((n-2)/2) * A008794(n-1).
T(n, 3) = (-1)^floor((n+1)/2) * A000330(n-1).
T(n, n) = A011782(n).
T(n, n-1) = A011782(n-1).
T(n, n-2) = -A001792(n-2).
T(n, n-4) = A001793(n-3).
T(n, n-6) = -A001794(n-6).
Sum_{k=0..n} (-1)^k*T(n,k) = A000007(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = A000007(n) + [n=1].
Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = (-1)^floor(n/2)*A025192(floor(n/2)). (End)

Edited by G. C. Greubel, Jul 26 2023

A141543 Triangle T(n,k) read by brows: T(n,2k)=k, T(n,2k+1) = k-n, for 0<=k<=n.

Original entry on oeis.org

0, 0, -1, 0, -2, 1, 0, -3, 1, -2, 0, -4, 1, -3, 2, 0, -5, 1, -4, 2, -3, 0, -6, 1, -5, 2, -4, 3, 0, -7, 1, -6, 2, -5, 3, -4, 0, -8, 1, -7, 2, -6, 3, -5, 4, 0, -9, 1, -8, 2, -7, 3, -6, 4, -5, 0, -10, 1, -9, 2, -8, 3, -7, 4, -6, 5

Offset: 0

Paul Curtz, Aug 16 2008

In each row, two bisections count up.

			Triangle begins as:
  0;
  0, -1;
  0, -2, 1;
  0, -3, 1, -2;
  0, -4, 1, -3, 2;
  0, -5, 1, -4, 2, -3;
  0, -6, 1, -5, 2, -4, 3;

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

Sums include: A000217 (signed row), A008794 (row), A159915 (diagonal).

Cf. A014682, A130472.

A141543:= func< n,k | ((k+1) mod 2)*Floor(k/2) + (k mod 2)*(-n + Floor((k-1)/2)) >;
[A141543(n,k): k in [0..n], n in [0..15]]; // G. C. Greubel, Sep 17 2024

A141543 := proc(n,k) if type(k,'even') then k/2; else (k-1)/2-n ; end if; end proc:
seq(seq(A141543(n,k),k=0..n),n=0..15) ; # R. J. Mathar, Jul 07 2011

Flatten[Table[If[EvenQ[k],k/2,(k-1)/2-n],{n,0,10},{k,0,n}]] (* Harvey P. Dale, Sep 24 2013 *)

def A141543(n,k): return ((k+1)%2)*(k//2) + (k%2)*(-n + ((k-1)//2))
flatten([[A141543(n,k) for k in range(n+1)] for n in range(16)]) # G. C. Greubel, Sep 17 2024

From G. C. Greubel, Sep 17 2024: (Start)
T(n, k) = (1/2)*((1+(-1)^k)*floor(k/2) + (1-(-1)^k)*(-n + floor((k - 1)/2)) ).
T(n, n) = A130472(n).
T(2*n, n) = (-1)^n*A014682(n).
Sum_{k=0..n} T(n, k) = (-1)*A008794(n+1).
Sum_{k=0..n} (-1)^k*T(n, k) = A000217(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = A159915(n+1).
Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = (1/32)*(6*n^2 + 6*n - 5 + (-1)^n*(2*n + 1) + 2*(1 - i)*(-i)^n + 2*(1 + i)*i^n). (End)

A355011 Array read by ascending antidiagonals: T(n, k) is the number of self-conjugate n-core partitions with k corners.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 4, 1, 1, 3, 4, 5, 1, 1, 3, 9, 5, 7, 1, 1, 4, 9, 15, 7, 8, 1, 1, 4, 16, 15, 27, 8, 10, 1, 1, 5, 16, 34, 27, 37, 10, 11, 1, 1, 5, 25, 34, 76, 37, 55, 11, 13, 1, 1, 6, 25, 65, 76, 124, 55, 69, 13, 14, 1, 1, 6, 36, 65, 175, 124, 216, 69, 93, 14, 16, 1, 1

Offset: 2

Stefano Spezia, Jun 15 2022

T(n, k) is also equal to the number of cornerless symmetric Motzkin paths of length 2*k + n - 1 with n - 1 flat steps (see Theorem 3.7 and Proposition 3.8 at pp. 16 - 17 in Cho et al.).

			The array begins:
  1,  1,  1,   1,   1,   1,    1,    1, ...
  1,  1,  1,   1,   1,   1,    1,    1, ...
  2,  4,  5,   7,   8,  10,   11,   13, ...
  2,  4,  5,   7,   8,  10,   11,   13, ...
  3,  9, 15,  27,  37,  55,   69,   93, ...
  3,  9, 15,  27,  37,  55,   69,   93, ...
  4, 16, 34,  76, 124, 216,  309,  471, ...
  4, 16, 34,  76, 124, 216,  309,  471, ...
  5, 25, 65, 175, 335, 675, 1095, 1875, ...
  5, 25, 65, 175, 335, 675, 1095, 1875, ...
  ...

Hyunsoo Cho, JiSun Huh, Hayan Nam, and Jaebum Sohn, Combinatorics on bounded free Motzkin paths and its applications, arXiv:2205.15554 [math.CO], 2022.

Cf. A000012 (n = 2,3), A001651, A004526 (k = 1), A008794 (k = 2), A247643 (n = 6,7), A355010.

T[n_,k_]:=Sum[Binomial[Floor[(k-1)/2],Floor[(i-1)/2]]Binomial[Floor[k/2],Floor[i/2]]Binomial[Floor[n/2]+k-i,k],{i,Min[k,Floor[n/2]]}]; Flatten[Table[T[n-k+1,k],{n,2,13},{k,1,n-1}]]

T(n, k) = Sum_{i=1..min(k,floor(n/2))} binomial(floor((k-1)/2), floor((i-1)/2))*binomial(floor(k/2), floor(i/2))*binomial(floor(n/2)+k-i, k). (See proposition 3.8 in Cho et al.).

T(4, n) = T(5, n) = A001651(n+1).

A360610 Triangle read by rows: T(n,k) is the number of squares of side length k that can be placed inside a square of side length n without overlap, 1 <= k <= n.

Original entry on oeis.org

1, 4, 1, 9, 1, 1, 16, 4, 1, 1, 25, 4, 1, 1, 1, 36, 9, 4, 1, 1, 1, 49, 9, 4, 1, 1, 1, 1, 64, 16, 4, 4, 1, 1, 1, 1, 81, 16, 9, 4, 1, 1, 1, 1, 1, 100, 25, 9, 4, 4, 1, 1, 1, 1, 1, 121, 25, 9, 4, 4, 1, 1, 1, 1, 1, 1, 144, 36, 16, 9, 4, 4, 1, 1, 1, 1, 1, 1, 169, 36, 16, 9, 4, 4, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Torlach Rush, Feb 13 2023

T(n,k) is square 1 <= k <= n.
Alternative triangle construction: Write each column k as each square repreated k times.
T(*,1) is A000290.
T(*,2) is A008794.
T(*,3) is A211547.
T(*,4) is A295643(n+4).
T(*,5) is A287392(n+1).
Row sums of triangle are A222548.
This assumes the sides of the small squares are parallel to those of the large square.  If the small squares are allowed to be rotated, better packings may exist (see e.g. the Friedman link).

			Sum_{T(1,*)} = A222548(1) = 1;
Sum_{T(2,*)} = A222548(2) = 5;
Sum_{T(3,*)} = A222548(3) = 11.
Triangle begins:
    1;
    4,  1;
    9,  1, 1;
   16,  4, 1, 1;
   25,  4, 1, 1, 1;
   36,  9, 4, 1, 1, 1;
   49,  9, 4, 1, 1, 1, 1;
   64, 16, 4, 4, 1, 1, 1, 1;
   81, 16, 9, 4, 1, 1, 1, 1, 1;
  100, 25, 9, 4, 4, 1, 1, 1, 1, 1;
  ...

E. Friedman, Packing Unit Squares in Squares: A Survey and New Results, The Electronic Journal of Combinatorics 5 (2009), DS#7.

Cf. A000290, A008794, A211547, A222548, A287392, A295643.

Python
```
def T(n, k): return (n//k)**2
```

T(n,k) = floor(n/k)^2.

cp's OEIS Frontend

A317301 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 = 1.

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A329708 Triangle, read by rows, where the n-th row lists the (2n+1) coefficients of (1+2*x+...+(n+1)*x^n)^2.

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A115012 Sum_{i=1..n, gcd(5,i)=1} i.

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A122656 a(n) = n*floor(n/2)^2.

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A262742 Irregular table read by rows: T(n,k) is the number of binary symmetric n X n matrices with exactly k 1's; n>=0, 0<=k<=n^2. Where the symmetry axes are in horizontal and vertical.

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A295643 Squares repeated 4 times; a(n) = floor(n/4)^2.

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A136523 Triangle T(n,k) = A053120(n,k) + A053120(n-1,k), read by rows.

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A141543 Triangle T(n,k) read by brows: T(n,2k)=k, T(n,2k+1) = k-n, for 0<=k<=n.

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A317301 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 = 1.

A329708 Triangle, read by rows, where the n-th row lists the (2n+1) coefficients of (1+2x+...+(n+1)x^n)^2.