A101688 -id:A101688 - cp's OEIS frontend

A111967 Inverse of number triangle A101688.

1, 0, 1, 0, -1, 1, 0, 1, -1, 1, 0, 0, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, 0, 0, 0, 0, -1, 1, 0, 1, -1, 1, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1

Offset: 0

Paul Barry, Aug 23 2005

Row sums are the Fredholm-Rueppel sequence A036987 [conjecture].

			Triangle begins
1,
0, 1,
0, -1, 1,
0, 1, -1, 1,
0, 0, 0, -1, 1,
0, -1, 1, 0, -1, 1,
0, 0, 0, 0, 0, -1, 1,
0, 1, -1, 1, 0, 0, -1, 1,
0, 0, 0, 0, 0, 0, 0, -1, 1,
0, 0, 0, -1, 1, 0, 0, 0, -1, 1,
0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1,
0, -1, 1, 0, -1, 1, 0, 0, 0, 0, -1, 1,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1,
0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, -1, 1,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1,
0, 1, -1, 1, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, -1, 1,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1

R. J. Mathar, OEIS A111967

Cf. A036987, A101688

T(n, k) = if(binomial(k, n-k)>0, 1, 0); \\ A101688
mrepeat(nn) = matrix(nn, nn, n, k, T(n-1, k-1)); \\ A101688
lista(nn) = my(m=mrepeat(nn+1), im = 1/m, list = List()); for (n = 1, nn, listput(list, vector(n, k, im[n,k]));); Vec(list); \\ Michel Marcus, Nov 12 2022

G.f. of k-th column is x^k*if(k=0,1,x*Sum_{j>=0} (-1)^j*x^(-2^(j/2)*(((k+2)/(2*sqrt(2))-(k+1))(-1)^j-(k+2)/(2*sqrt(2))-(k+1))-(k+2))+1-x). - Paul Barry, Jan 30 2007

A144734 Triangle read by rows, A054533 * transpose(A101688) (matrix product) provided A101688 is read as a square array by antidiagonals upwards.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 0, 2, 2, 0, 1, 2, 3, 4, 0, -1, 0, 2, 3, 2, 0, 1, 2, 3, 4, 5, 6, 0, 0, 0, 0, 4, 4, 4, 4, 0, 0, 0, 3, 3, 3, 6, 6, 6, 0, -1, 0, -1, 0, 4, 5, 4, 5, 4, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 0, 0, -2, -2, 0, 0, 4, 4, 6, 6, 4, 4, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0, -1, 0

Offset: 1

Gary W. Adamson, Sep 20 2008

Right border = A000010, phi(n).

Row sums = A023896: (1, 1, 3, 4, 10, 6, 21, ...).

			First few rows of the triangle are as follows:
  1;
  0,  1;
  0,  1,  2;
  0,  0,  2,  2;
  0,  1,  2,  3,  4;
  0, -1,  0,  2,  3,  2;
  0,  1,  2,  3,  4,  5,  6;
  0,  0,  0,  0,  4,  4,  4,  4;
  0,  0,  0,  3,  3,  3,  6,  6,  6;
  0, -1,  0, -1,  0,  4,  5,  4,  5,  4;
  0,  1,  2,  3,  4,  5,  6,  7,  8,  9, 10;
  ...
row 4 = (0, 0, 2, 2) = partial sums from the right of row 4 of triangle A054533: (0, -2, 0, 2).

Jinyuan Wang, Table of n, a(n) for n = 1..10000

Cf. A000010, A023896, A054533, A101688, A157658 (column 2).

Triangle read by rows, A054533 * transpose(A101688) (matrix product); i.e., partial sums from of the right of triangle A054533 (because A101688 can be viewed as an upper triangular matrix of 1's).
From Petros Hadjicostas, Jul 28 2019: (Start)
T(n,k) = Sum_{m = k..n} A054533(n,m) = Sum_{d|n} d * mu(n/d) * ((n/d) - ceiling(k/d) + 1) for n >= 1 and 1 <= k <= n.
T(n,k) = phi(n) - Sum_{d|n} d * mu(n/d) * ceiling(k/d) for n >= 2 and 1 <= k <= n.
(End)

Name edited by and more terms from Petros Hadjicostas, Jul 28 2019

A168258 Triangle read by rows, A101688 * A000012 as infinite lower triangular matrices.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 2, 2, 2, 1, 3, 3, 3, 2, 1, 3, 3, 3, 3, 2, 1, 4, 4, 4, 4, 3, 2, 1, 4, 4, 4, 4, 4, 3, 2, 1, 5, 5, 5, 5, 5, 4, 3, 2, 1, 5, 5, 5, 5, 5, 5, 4, 3, 2, 1, 6, 6, 6, 6, 6, 6, 5, 4, 3, 2, 1, 6, 6, 6, 6, 6, 6, 6, 5, 4, 3, 2, 1

Offset: 1

Gary W. Adamson, Nov 21 2009

Row sums = A001318, general pentagonal numbers: (1, 2, 5, 12, 15, 22, ...).
Eigensequence of the triangle = A168259: (1, 2, 6, 14, 38, 96, 254, 656, ...).
The operation A101688 * A000012 transforms rows of A101688 into sequence terms by taking partial sums from the right of A101688 rows. For example, row 3 of A101688 (0, 0, 1, 1) becomes (2, 2, 2, 1). - Gary W. Adamson, Nov 15 2022

			First few rows of the triangle:
  1;
  1, 1;
  2, 2, 1;
  2, 2, 2, 1;
  3, 3, 3, 2, 1;
  3, 3, 3, 3, 2, 1;
  4, 4, 4, 4, 3, 2, 1;
  4, 4, 4, 4, 4, 3, 2, 1;
  5, 5, 5, 5, 5, 4, 3, 2, 1;
  5, 5, 5, 5, 5, 5, 4, 3, 2, 1;
  6, 6, 6, 6, 6, 6, 5, 4, 3, 2, 1;
  6, 6, 6, 6, 6, 6, 6, 5, 4, 3, 2, 1;
  7, 7, 7, 7, 7, 7, 7, 6, 5, 4, 3, 2, 1;
  7, 7, 7, 7, 7, 7, 7, 7, 6, 5, 4, 3, 2, 1;
  8, 8, 8, 8, 8, 8, 8, 8, 7, 6, 5, 4, 3, 2, 1;
  ...

Cf. A001318, A101688, A000012, A168259, A004736, A204164.

T(n, k) = if(binomial(k, n-k)>0, 1, 0); \\ A101688
lista(nn) = my(ma=matrix(nn+1, nn, n, k, T(n-1, k-1)), mb=matrix(nn, nn, n, k, n>=k)); my(m=ma*mb, list=List()); for (n=1, nn, listput(list, vector(n, k, m[n,k]))); Vec(list); \\ Michel Marcus, Nov 16 2022

Triangle read by rows, A101688 * A000012 as infinite lower triangular matrices.

a(n) = min(A004736, A204164); a(n) = min(j, floor((t+2)/2)), where j=(t*t+3*t+4)/2-n, t=floor((-1+sqrt(8*n-7))/2). - Boris Putievskiy, Apr 18 2013

Name corrected by Gary W. Adamson, Nov 15 2022

A168316 Triangle read by rows, square of triangle A101688.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 1, 2, 1, 0, 1, 3, 2, 1, 0, 0, 2, 3, 2, 1, 0, 0, 2, 4, 3, 2, 1, 0, 0, 1, 3, 4, 3, 2, 1, 0, 0, 1, 3, 5, 4, 3, 2, 1, 0, 0, 0, 2, 4, 5, 4, 3, 2, 1, 0, 0, 0, 2, 4, 6, 5, 4, 3, 2, 1, 0, 0, 0, 1, 3, 5, 6, 5, 4, 3, 2, 1, 0, 0, 0, 1, 3, 5, 7, 6, 5, 4, 3, 2, 1

Offset: 1

Gary W. Adamson, Nov 22 2009

Row sums = A129819 starting (1, 1, 3, 4, 7, 8, 12,...).

Eigensequence of the triangle = A168317: (1, 1, 3, 6, 16, 39, 103, 263, 690,...).

			First few rows of the triangle =
1;
0, 1;
0, 2, 1;
0, 1, 2, 1;
0, 1, 3, 2, 1;
0, 0, 2, 3, 2, 1;
0, 0, 2, 4, 3, 2, 1;
0, 0, 1, 3, 4, 3, 2, 1;
0, 0, 1, 3, 5, 4, 3, 2, 1;
0, 0, 0, 2, 4, 5, 4, 3, 2, 1;
0, 0, 0, 2, 4, 6, 5, 4, 3, 2, 1;
0, 0, 0, 1, 3, 5, 6, 5, 4, 3, 2, 1;
0, 0, 0, 1, 3, 5, 7, 6, 5, 4, 3, 2, 1;
0, 0, 0, 0, 2, 4, 6, 7, 6, 5, 4, 3, 2, 1;
0, 0, 0, 0, 2, 4, 6, 8, 7, 6, 5, 4, 3, 2, 1;
0, 0, 0, 0, 1, 3, 5, 7, 8, 7, 6, 5, 4, 3, 2, 1;
0, 0, 0, 0, 1, 3, 5, 7, 9, 8, 7, 6, 5, 4, 3, 2, 1;
0, 0, 0, 0, 0, 2, 4, 6, 8, 9, 8, 7, 6, 5, 4, 3, 2, 1;
0, 0, 0, 0, 0, 2, 4, 6, 8, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1;
0, 0, 0, 0, 0, 1, 3, 5, 7, 9, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1;
...

Cf. A101688, A129819, A168317, A168318.

rows = 15; A = Array[If[#1 <= #2, 1, 0]&, {rows, rows}];
M = Table[PadRight[Table[A[[i-j+1, j]], {j, 1, i}], rows], {i, 1, rows}];
M2 = MatrixPower[M, 2];
Table[M2[[i, j]], {i, 1, rows}, {j, 1, i}] // Flatten (* Jean-François Alcover, May 04 2017 *)

Triangle read by rows, (A101688)^2, as an infinite lower triangular matrix.

Missing row inserted by Jean-François Alcover, May 04 2017

A168508 Triangle read by rows: A101688 * A051731.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 4, 2, 1, 1, 1, 1, 1, 4, 2, 1, 1, 1, 1, 1, 1, 5, 2, 2, 1, 1, 1, 1, 1, 1, 5, 3, 2, 1, 1, 1, 1, 1, 1, 1, 6, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1, 6, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 7, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 4, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Gary W. Adamson, Nov 27 2009

More precisely, form the product of the lower triangular matrix T defined in A101688 and the lower triangular matrix T defined in A051731. - N. J. A. Sloane, Dec 05 2020
Row sums = A060831: (1, 2, 4, 5, 7, 9, 11, 12, 15,...).
A168509 = A051731 * A101688

			First few rows of the triangle =
1;
1, 1;
2, 1, 1;
2, 1, 1, 1;
3, 1, 1, 1, 1;
3, 2, 1, 1, 1, 1;
4, 2, 1, 1, 1, 1, 1;
4, 2, 1, 1, 1, 1, 1, 1;
5, 2, 2, 1, 1, 1, 1, 1, 1;
5, 3, 2, 1, 1, 1, 1, 1, 1, 1;
6, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1;
6, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1;
7, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1;
7, 4, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
8, 4, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
8, 4, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
9, 4, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
...

Cf. A051731, A060831, A101688, A168509.

a(29) = 1 inserted and more terms from Georg Fischer, May 29 2023

A168509 Triangle read by rows, A051731 * A101688.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Gary W. Adamson, Nov 27 2009

Row sums = A079247: (1, 2, 3, 4, 4, 7, 5, 8, 8, 10,...).

A168508 = A101688 * A051731

			First few rows of the triangle =
1;
1, 1;
1, 1, 1;
1, 1, 1, 1;
1, 0, 1, 1, 1;
1, 2, 1, 1, 1, 1,
1, 0, 0, 1, 1, 1, 1;
1, 1, 1, 1, 1, 1, 1, 1;
1, 1, 1, 0, 1, 1, 1, 1, 1;
1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1;
1, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1;
1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1;
1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
1, 1, 2, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1;
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1;
...

Cf. A051731, A101688, A079247, A168508

Triangle read by rows, inverse Mobius transform of A101688; where A051731 = the

inverse Mobius transform operator.

A168313 Triangle read by rows, retain 1's as rightmost diagonal of A101688 and replace all other 1's with 2's.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 0, 2, 1, 0, 0, 2, 2, 1, 0, 0, 0, 2, 2, 1, 0, 0, 0, 2, 2, 2, 1, 0, 0, 0, 0, 2, 2, 2, 1, 0, 0, 0, 0, 2, 2, 2, 2, 1, 0, 0, 0, 0, 0, 2, 2, 2, 2, 1, 0, 0, 0, 0, 0, 2, 2, 2, 2, 2, 1

Offset: 1

Gary W. Adamson, Nov 22 2009

Row sums = odd integers repeated: (1, 1, 3, 3, 5, 5,...).

Eigensequence of the triangle = A168314: (1, 1, 3, 5, 13, 29, 71, 165, 401,...).

			First few rows of the triangle =
1;
0, 1;
0, 2, 1;
0, 0, 2, 1;
0, 0, 2, 2, 1;
0, 0, 0, 2, 2, 1;
0, 0, 0, 2, 2, 2, 1;
0, 0, 0, 0, 2, 2, 2, 1;
0, 0, 0, 0, 2, 2, 2, 2, 1;
0, 0, 0, 0, 0, 2, 2, 2, 2, 1;
0, 0, 0, 0, 0, 2, 2, 2, 2, 2, 1;
0, 0, 0, 0, 0, 0, 2, 2, 2, 2, 2, 1;
...

Boris Putievskiy, Transformations (of) Integer Sequences And Pairing Functions, arXiv:1212.2732 [math.CO], 2012.

Cf. A101688, A168314, A168315

rows = 11;
A = Array[Which[#1 == 1, 1, #1 <= #2, 2, True, 0]&, {rows, rows}];
Table[A[[i-j+1, j]], {i, 1, rows}, {j, 1, i}] // Flatten (* Jean-François Alcover, Aug 08 2018 *)

Triangle read by rows, retain 1's as rightmost diagonal of A101688 and replace all other 1's with 2's.
From Boris Putievskiy, Jan 09 2013: (Start)
a(n) = 2*A101688(n)-A023531(n).
a(n) = 2*floor((2*A002260(n)+1)/(A003056(n)+3))*A002260(n)-A023531(n).
a(n) = 2*floor((2*n-t*(t+1)+1)/(t+3))*(n-t*(t+1)/2) - floor((sqrt(8*n+1)-1)/2) + t, where t = floor((-1+sqrt(8*n-7))/2). (End)

A159936 Triangle read by rows, A051731 * A054533 * transpose(A101688), provided A101688 is read as a square array.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 3, 4, 1, 1, 2, 2, 3, 2, 1, 1, 2, 3, 4, 5, 6, 1, 1, 2, 2, 4, 4, 4, 4, 1, 1, 2, 3, 3, 3, 6, 6, 6, 1, 1, 2, 2, 4, 4, 5, 4, 5, 4, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 1, 2, 2, 3, 2, 4, 4, 6, 6, 4, 4, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 1, 1, 2, 2, 4, 4, 6, 6, 7, 6, 7, 6, 7, 6

Offset: 1

Gary W. Adamson, Apr 26 2009

Row sums = A057661: (1, 2, 4, 6, 11, 11, 22,...). Right border = A000010, phi(n).

			First few rows of the triangle are as follows:
  1;
  1, 1;
  1, 1, 2;
  1, 1, 2, 2;
  1, 1, 2, 3, 4;
  1, 1, 2, 2, 3, 2;
  1, 1, 2, 3, 4, 5, 6;
  1, 1, 2, 2, 4, 4, 4, 4;
  1, 1, 2, 3, 3, 3, 6, 6, 6;
  1, 1, 2, 2, 4, 4, 5, 4, 5, 4;
  1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10;
  1, 1, 2, 2, 3, 2, 4, 4, 6, 6,  4,  4;
  1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12;
  1, 1, 2, 2, 4, 4, 6, 6, 7, 6,  7,  6,  7, 6;
  1, 1, 2, 3, 3, 3, 5, 4, 3, 5,  9,  8, 10, 9, 8;
  1, 1, 2, 2, 4, 4, 4, 4, 8, 8,  8,  8,  8, 8, 8, 8;
  ...

Cf. A000010, A051731, A054532, A054533, A054534, A054535, A057661, A101688.

Triangle read by rows, A051731 * A054533 * A000012. A051731 = the inverse Mobius transform. A054533 = the lower left half of the Ramanujan sum table. The operation (* transpose(A101688)) takes partial sums of (A051731 * A054533) starting from the right. [Edited by Petros Hadjicostas, Jul 30 2019]

Name edited by Petros Hadjicostas, Jul 30 2019

A167948 Triangle read by rows, A101688 * (an infinite lower triangular matrix with A002083 as the main diagonal and the rest zeros).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 0, 1, 2, 0, 0, 1, 2, 3, 0, 0, 0, 2, 3, 6, 0, 0, 0, 2, 3, 6, 11, 0, 0, 0, 0, 3, 6, 11, 22, 0, 0, 0, 0, 3, 6, 11, 22, 42, 0, 0, 0, 0, 0, 6, 11, 22, 42, 84, 0, 0, 0, 0, 0, 6, 11, 22, 42, 84, 165, 0, 0, 0, 0, 0, 0, 11, 22, 42, 84, 165, 330

Offset: 1

Gary W. Adamson, Nov 15 2009

Row sums = A002083(n+1): (1, 1, 2, 3, 6, 11, 22, 42, 84, 165,...).
Sum of n-th row terms = rightmost term of next row.
Eigensequence of triangle A101688 = A002083 starting with offset 1: (1, 1, 2, 3, 6, 11, 22, 42,...).

			First few rows of the triangle:
  1;
  0, 1;
  0, 1, 1;
  0, 0, 1, 2;
  0, 0, 1, 2, 3;
  0, 0, 0, 2, 3, 6;
  0, 0, 0, 2, 3, 6, 11;
  0, 0, 0, 0, 3, 6, 11, 22;
  0, 0, 0, 0, 3, 6, 11, 22, 42;
  0, 0, 0, 0, 0, 6, 11, 22, 42, 84;
  0, 0, 0, 0, 0, 6, 11, 22, 42, 84, 165;
  0, 0, 0, 0, 0, 0, 11, 22, 42, 84, 165, 330;
  0, 0, 0, 0, 0, 0, 11, 22, 42, 84, 165, 330, 654;
  0, 0, 0, 0, 0, 0, .0, 22, 42, 84, 165, 330, 654, 1308;
  0, 0, 0, 0, 0, 0, .0, 22, 42, 84, 165, 330, 654, 1308, 2605;
  ...

Cf. A101688, A002083.

Equals M * Q, where M = triangle A101688, Q = an infinite lower triangular matrix with (1, 1, 1, 2, 3, 6, 11, 22, 42,...) as the main diagonal and the rest zeros.

A001318 Generalized pentagonal numbers: m(3m - 1)/2, m = 0, +-1, +-2, +-3, ....

Original entry on oeis.org

0, 1, 2, 5, 7, 12, 15, 22, 26, 35, 40, 51, 57, 70, 77, 92, 100, 117, 126, 145, 155, 176, 187, 210, 222, 247, 260, 287, 301, 330, 345, 376, 392, 425, 442, 477, 495, 532, 551, 590, 610, 651, 672, 715, 737, 782, 805, 852, 876, 925, 950, 1001, 1027, 1080, 1107, 1162, 1190, 1247, 1276, 1335

Offset: 0

N. J. A. Sloane

Partial sums of A026741. - Jud McCranie; corrected by Omar E. Pol, Jul 05 2012
From R. K. Guy, Dec 28 2005: (Start)
"Conway's relation twixt the triangular and pentagonal numbers: Divide the triangular numbers by 3 (when you can exactly):
0 1 3 6 10 15 21 28 36 45 55 66 78 91 105 120 136 153 ...
0 - 1 2 .- .5 .7 .- 12 15 .- 22 26 .- .35 .40 .- ..51 ...
.....-.-.....+..+.....-..-.....+..+......-...-.......+....
"and you get the pentagonal numbers in pairs, one of positive rank and the other negative.
"Append signs according as the pair have the same (+) or opposite (-) parity.
"Then Euler's pentagonal number theorem is easy to remember:
"p(n-0) - p(n-1) - p(n-2) + p(n-5) + p(n-7) - p(n-12) - p(n-15) ++-- = 0^n
where p(n) is the partition function, the left side terminates before the argument becomes negative and 0^n = 1 if n = 0 and = 0 if n > 0.
"E.g. p(0) = 1, p(7) = p(7-1) + p(7-2) - p(7-5) - p(7-7) + 0^7 = 11 + 7 - 2 - 1 + 0 = 15."
(End)
The sequence may be used in order to compute sigma(n), as described in Euler's article. - Thomas Baruchel, Nov 19 2003
Number of levels in the partitions of n + 1 with parts in {1,2}.
a(n) is the number of 3 X 3 matrices (symmetrical about each diagonal) M = {{a, b, c}, {b, d, b}, {c, b, a}} such that a + b + c = b + d + b = n + 2, a,b,c,d natural numbers; example: a(3) = 5 because (a,b,c,d) = (2,2,1,1), (1,2,2,1), (1,1,3,3), (3,1,1,3), (2,1,2,3). - Philippe Deléham, Apr 11 2007
Also numbers a(n) such that 24*a(n) + 1 = (6*m - 1)^2 are odd squares: 1, 25, 49, 121, 169, 289, 361, ..., m = 0, +-1, +-2, ... . - Zak Seidov, Mar 08 2008
From Matthew Vandermast, Oct 28 2008: (Start)
Numbers n for which A000326(n) is a member of A000332. Cf. A145920.
This sequence contains all members of A000332 and all nonnegative members of A145919. For values of n such that n*(3*n - 1)/2 belongs to A000332, see A145919. (End)
Starting with offset 1 = row sums of triangle A168258. - Gary W. Adamson, Nov 21 2009
Starting with offset 1 = Triangle A101688 * [1, 2, 3, ...]. - Gary W. Adamson, Nov 27 2009
Starting with offset 1 can be considered the first in an infinite set generated from A026741. Refer to the array in A175005. - Gary W. Adamson, Apr 03 2010
Vertex number of a square spiral whose edges have length A026741. The two axes of the spiral forming an "X" are A000326 and A005449. The four semi-axes forming an "X" are A049452, A049453, A033570 and the numbers >= 2 of A033568. - Omar E. Pol, Sep 08 2011
A general formula for the generalized k-gonal numbers is given by n*((k - 2)*n - k + 4)/2, n=0, +-1, +-2, ..., k >= 5. - Omar E. Pol, Sep 15 2011
a(n) is the number of 3-tuples (w,x,y) having all terms in {0,...,n} and 2*w = 2*x + y. - Clark Kimberling, Jun 04 2012
Generalized k-gonal numbers are second k-gonal numbers and positive terms of k-gonal numbers interleaved, k >= 5. - Omar E. Pol, Aug 04 2012
a(n) is the sum of the largest parts of the partitions of n+1 into exactly 2 parts. - Wesley Ivan Hurt, Jan 26 2013
Conway's relation mentioned by R. K. Guy is a relation between triangular numbers and generalized pentagonal numbers, two sequences from different families, but as triangular numbers are also generalized hexagonal numbers in this case we have a relation between two sequences from the same family. - Omar E. Pol, Feb 01 2013
Start with the sequence of all 0's. Add n to each value of a(n) and the next n - 1 terms. The result is the generalized pentagonal numbers. - Wesley Ivan Hurt, Nov 03 2014
(6k + 1) | a(4k). (3k + 1) | a(4k+1). (3k + 2) | a(4k+2). (6k + 5) | a(4k+3). - Jon Perry, Nov 04 2014
Enge, Hart and Johansson proved: "Every generalised pentagonal number c >= 5 is the sum of a smaller one and twice a smaller one, that is, there are generalised pentagonal numbers a, b < c such that c = 2a + b." (see link theorem 5). - Peter Luschny, Aug 26 2016
The Enge, et al. result for c >= 5 also holds for c >= 2 if 0 is included as a generalized pentagonal number. That is, 2 = 2*1 + 0. - Michael Somos, Jun 02 2018
Suggestion for title, where n actually matches the list and b-file: "Generalized pentagonal numbers: k(n)*(3*k(n) - 1)/2, where k(n) = A001057(n) = [0, 1, -1, 2, -2, 3, -3, ...], n >= 0" - Daniel Forgues, Jun 09 2018 & Jun 12 2018
Generalized k-gonal numbers are the partial sums of the sequence formed by the multiples of (k - 4) and the odd numbers (A005408) interleaved, with k >= 5. - Omar E. Pol, Jul 25 2018
The last digits form a symmetric cycle of length 40 [0, 1, 2, 5, ..., 5, 2, 1, 0], i.e., a(n) == a(n + 40) (mod 10) and a(n) == a(40*k - n - 1) (mod 10), 40*k > n. - Alejandro J. Becerra Jr., Aug 14 2018
Only 2, 5, and 7 are prime. All terms are of the form k*(k+1)/6, where 3 | k or 3 | k+1. For k > 6, the value divisible by 3 must have another factor d > 2, which will remain after the division by 6. - Eric Snyder, Jun 03 2022
8*a(n) is the product of two even numbers one of which is n + n mod 2. - Peter Luschny, Jul 15 2022
a(n) is the dot product of [1, 2, 3, ..., n] and repeat[1, 1/2]. a(5) = 12 = [1, 2, 3, 4, 5] dot [1, 1/2, 1, 1/2, 1] = [1 + 1 + 3 + 2 + 5]. - Gary W. Adamson, Dec 10 2022
Every nonnegative number is the sum of four terms of this sequence [S. Realis]. - N. J. A. Sloane, May 07 2023
From Peter Bala, Jan 06 2025: (Start)
The sequence terms are the exponents in the expansions of the following infinite products:
1) Product_{n >= 1} (1 - s(n)*q^n) = 1 + q + q^2 + q^5 + q^7 + q^12 + q^15 + ..., where s(n) = (-1)^(1 + mod(n+1,3)).
2) Product_{n >= 1} (1 - q^(2*n))*(1 - q^(3*n))^2/((1 - q^n)*(1 - q^(6*n))) = 1 + q + q^2 + q^5 + q^7 + q^12 + q^15 + ....
3) Product_{n >= 1} (1 - q^n)*(1 - q^(4*n))*(1 - q^(6*n))^5/((1 - q^(2*n))*(1 - q^(3*n))*(1 - q^(12*n)))^2 = 1 - q + q^2 - q^5 - q^7 + q^12 - q^15 + q^22 + q^26 - q^35 + ....
4) Product_{n >= 1} (1 - q^(2*n))^13/((1 - (-1)^n*q^n)*(1 - q^(4*n)))^5 = 1 - 5*q + 7*q^2 - 11*q^5 + 13*q^7 - 17*q^12 + 19*q^15 - + .... See Oliver, Theorem 1.1. (End)

			G.f. = x + 2*x^2 + 5*x^3 + 7*x^4 + 12*x^5 + 15*x^6 + 22*x^7 + 26*x^8 + 35*x^9 + ...

Enoch Haga, A strange sequence and a brilliant discovery, chapter 5 of Exploring prime numbers on your PC and the Internet, first revised ed., 2007 (and earlier ed.), pp. 53-70.
Ross Honsberger, Ingenuity in Mathematics, Random House, 1970, p. 117.
Donald E. Knuth, The Art of Computer Programming, vol. 4A, Combinatorial Algorithms, (to appear), section 7.2.1.4, equation (18).
Ivan Niven and Herbert S. Zuckerman, An Introduction to the Theory of Numbers, 2nd ed., Wiley, NY, 1966, p. 231.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..1000
G. E. Andrews and J. A. Sellers, Congruences for the Fishburn Numbers, arXiv preprint arXiv:1401.5345 [math.NT], 2014.
Paul Barry, On sequences with {-1, 0, 1} Hankel transforms, arXiv preprint arXiv:1205.2565 [math.CO], 2012. - From _N. J. A. Sloane_, Oct 18 2012
Burkard Polster (Mathologer), The hardest "What comes next?" (Euler's pentagonal formula), Youtube video, Oct 17 2020.
S. Cooper and M. D. Hirschhorn, Results of Hurwitz type for three squares, Discrete Math., Vol. 274, No. 1-3 (2004), pp. 9-24. See P(q).
Stephen Eberhart, Letter to N. J. A. Sloane, Jan 19 1978.
John Elias, Illustration of Initial Terms: Generalized Penthexagrams.
John Elias, Illustration: Star Number Fractal.
John Elias, Illustration: Generalized Pentagonals In Generalized-Octagonal-Hexagrams.
Andreas Enge, William Hart, and Fredrik Johansson, Short addition sequences for theta functions, arXiv:1608.06810 [math.NT], 2016.
Leonhard Euler, Découverte d'une loi tout extraordinaire des nombres par rapport à la somme de leurs diviseurs, Opera Omnia, Series I, Vol. 2 (1751), pp. 241-253.
Leonhard Euler, On the remarkable properties of the pentagonal numbers, arXiv:math/0505373 [math.HO], 2005.
Leonhard Euler, De mirabilibus proprietatibus numerorum pentagonalium, par. 2
Leonhard Euler, Observatio de summis divisorum p. 8.
Leonhard Euler, An observation on the sums of divisors, p. 8, arXiv:math/0411587 [math.HO], 2004.
Alex Fink, Richard K. Guy, and Mark Krusemeyer, Partitions with parts occurring at most thrice, Contrib. Discr. Math., Vol. 3, No. 2 (2008), pp. 76-114.
Silvia Heubach and Toufik Mansour, Counting rises, levels and drops in compositions, arXiv:math/0310197 [math.CO], 2003.
Alfred Hoehn, Illustration of initial terms.
Barbara H. Margolius, Permutations with inversions, J. Integ. Seq., Vol. 4 (2001), Article 01.2.4.
Johannes W. Meijer, Euler's Ship on the Pentagonal Sea, pdf and jpg.
Johannes W. Meijer and Manuel Nepveu, Euler's ship on the Pentagonal Sea, Acta Nova, Vol. 4, No. 1 (December 2008), pp. 176-187.
Mircea Merca, The Lambert series factorization theorem, The Ramanujan Journal, January 2017; DOI: 10.1007/s11139-016-9856-3.
Mircea Merca and Maxie D. Schmidt, New Factor Pairs for Factorizations of Lambert Series Generating Functions, arXiv:1706.02359 [math.CO], 2017. See Remark 2.2.
Mircea Merca, Euler's partition function in terms of 2-adic valuation, Bol. Soc. Mat. Mex. 30, 76 (2024). See p. 3.
Ivan Niven, Formal power series, Amer. Math. Monthly, Vol. 76, No. 8 (1969), pp. 871-889.
Robert J. Lemke Oliver, Eta quotients and theta functions, Advances in Mathematics, Vol. 241, Jul. 2013, pp. 1-17.
Vladimir Pletser, Congruence conditions on the number of terms in sums of consecutive squared integers equal to squared integers, arXiv:1409.7969 [math.NT], 2014.
Vladimir Pletser, Finding all squared integers expressible as the sum of consecutive squared integers using generalized Pell equation solutions with Chebyshev polynomials, arXiv preprint arXiv:1409.7972 [math.NT], 2014.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992, arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
S. Realis, Question 271, Nouv. Corresp. Math., 4 (1878) 27-29.
Steven J. Schlicker, Numbers Simultaneously Polygonal and Centered Polygonal, Mathematics Magazine, Vol. 84, No. 5 (December 2011), pp. 339-350.
André Weil, Two lectures on number theory, past and present, L'Enseign. Math., Vol. XX (1974), pp. 87-110; Oeuvres III, pp. 279-302.
Eric Weisstein's World of Mathematics, Pentagonal numbers, Partition Function P.
Eric Weisstein's World of Mathematics, Pentagonal Number Theorem.
Wikipedia, Pentagonal number theorem.
M. Wohlgemuth, Pentagon, Kartenhaus und Summenzerlegung.
Keke Zhang, Generalized Catalan numbers, arXiv:2011.09593 [math.CO], 2020.
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A080995 (characteristic function), A026741 (first differences), A034828 (partial sums), A165211 (mod 2).
Cf. A000326 (pentagonal numbers), A005449 (second pentagonal numbers), A000217 (triangular numbers).
Indices of nonzero terms of A010815, i.e., the (zero-based) indices of 1-bits of the infinite binary word to which the terms of A068052 converge.
Union of A036498 and A036499.
Cf. A153384, A168258, A101688, A174739, A175005.
Cf. A074378, A057569, A057570, A007310.
Sequences of generalized k-gonal numbers: this sequence (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).
Column 1 of A195152.
Squares in APs: A221671, A221672.
Cf. A054440, A260664, A260672.
Quadrisection: A049453(k), A033570(k), A033568(k+1), A049452(k+1), k >= 0.
Cf. A002620.

a:=[0,1,2,5];; for n in [5..60] do a[n]:=2*a[n-2]-a[n-4]+3; od; a; # Muniru A Asiru, Aug 16 2018

a001318 n = a001318_list !! n
a001318_list = scanl1 (+) a026741_list -- Reinhard Zumkeller, Nov 15 2015

[(6*n^2 + 6*n + 1 - (2*n + 1)*(-1)^n)/16 : n in [0..50]]; // Wesley Ivan Hurt, Nov 03 2014

[(3*n^2 + 2*n + (n mod 2) * (2*n + 1)) div 8: n in [0..70]]; // Vincenzo Librandi, Nov 04 2014

A001318 := -(1+z+z**2)/(z+1)**2/(z-1)**3; # Simon Plouffe in his 1992 dissertation; gives sequence without initial zero
A001318 := proc(n) (6*n^2+6*n+1)/16-(2*n+1)*(-1)^n/16 ; end proc: # R. J. Mathar, Mar 27 2011

Table[n*(n+1)/6, {n, Select[Range[0, 100], Mod[#, 3] != 1 &]}]
Select[Accumulate[Range[0,200]]/3,IntegerQ] (* Harvey P. Dale, Oct 12 2014 *)
CoefficientList[Series[x (1 + x + x^2) / ((1 + x)^2 (1 - x)^3), {x, 0, 70}], x] (* Vincenzo Librandi, Nov 04 2014 *)
LinearRecurrence[{1,2,-2,-1,1},{0,1,2,5,7},70] (* Harvey P. Dale, Jun 05 2017 *)
a[ n_] := With[{m = Quotient[n + 1, 2]}, m (3 m + (-1)^n) / 2]; (* Michael Somos, Jun 02 2018 *)

{a(n) = (3*n^2 + 2*n + (n%2) * (2*n + 1)) / 8}; /* Michael Somos, Mar 24 2011 */

{a(n) = if( n<0, n = -1-n); polcoeff( x * (1 - x^3) / ((1 - x) * (1-x^2))^2 + x * O(x^n), n)}; /* Michael Somos, Mar 24 2011 */

{a(n) = my(m = (n+1) \ 2); m * (3*m + (-1)^n) / 2}; /* Michael Somos, Jun 02 2018 */

def a(n):
    p = n % 2
    return (n + p)*(3*n + 2 - p) >> 3
print([a(n) for n in range(60)])  # Peter Luschny, Jul 15 2022

def A001318(n): return n*(n+1)-(m:=n>>1)*(m+1)>>1 # Chai Wah Wu, Nov 23 2024

@CachedFunction
def A001318(n):
    if n == 0 : return 0
    inc = n//2 if is_even(n) else n
    return inc + A001318(n-1)
[A001318(n) for n in (0..59)] # Peter Luschny, Oct 13 2012

Euler: Product_{n>=1} (1 - x^n) = Sum_{n=-oo..oo} (-1)^n*x^(n*(3*n - 1)/2).
A080995(a(n)) = 1: complement of A090864; A000009(a(n)) = A051044(n). - Reinhard Zumkeller, Apr 22 2006
Euler transform of length-3 sequence [2, 2, -1]. - Michael Somos, Mar 24 2011
a(-1 - n) = a(n) for all n in Z. a(2*n) = A005449(n). a(2*n - 1) = A000326(n). - Michael Somos, Mar 24 2011. [The extension of the recurrence to negative indices satisfies the signature (1,2,-2,-1,1), but not the definition of the sequence m*(3*m -1)/2, because there is no m such that a(-1) = 0. - Klaus Purath, Jul 07 2021]
a(n) = 3 + 2*a(n-2) - a(n-4). - Ant King, Aug 23 2011
Product_{k>0} (1 - x^k) = Sum_{k>=0} (-1)^k * x^a(k). - Michael Somos, Mar 24 2011
G.f.: x*(1 + x + x^2)/((1 + x)^2*(1 - x)^3).
a(n) = n*(n + 1)/6 when n runs through numbers == 0 or 2 mod 3. - Barry E. Williams
a(n) = A008805(n-1) + A008805(n-2) + A008805(n-3), n > 2. - Ralf Stephan, Apr 26 2003
Sequence consists of the pentagonal numbers (A000326), followed by A000326(n) + n and then the next pentagonal number. - Jon Perry, Sep 11 2003
a(n) = (6*n^2 + 6*n + 1)/16 - (2*n + 1)*(-1)^n/16; a(n) = A034828(n+1) - A034828(n). - Paul Barry, May 13 2005
a(n) = Sum_{k=1..floor((n+1)/2)} (n - k + 1). - Paul Barry, Sep 07 2005
a(n) = A000217(n) - A000217(floor(n/2)). - Pierre CAMI, Dec 09 2007
If n even a(n) = a(n-1) + n/2 and if n odd a(n) = a(n-1) + n, n >= 2. - Pierre CAMI, Dec 09 2007
a(n)-a(n-1) = A026741(n) and it follows that the difference between consecutive terms is equal to n if n is odd and to n/2 if n is even. Hence this is a self-generating sequence that can be simply constructed from knowledge of the first term alone. - Ant King, Sep 26 2011
a(n) = (1/2)*ceiling(n/2)*ceiling((3*n + 1)/2). - Mircea Merca, Jul 13 2012
a(n) = (A008794(n+1) + A000217(n))/2 = A002378(n) - A085787(n). - Omar E. Pol, Jan 12 2013
a(n) = floor((n + 1)/2)*((n + 1) - (1/2)*floor((n + 1)/2) - 1/2). - Wesley Ivan Hurt, Jan 26 2013
From Oskar Wieland, Apr 10 2013: (Start)
a(n) = a(n+1) - A026741(n),
a(n) = a(n+2) - A001651(n),
a(n) = a(n+3) - A184418(n),
a(n) = a(n+4) - A007310(n),
a(n) = a(n+6) - A001651(n)*3 = a(n+6) - A016051(n),
a(n) = a(n+8) - A007310(n)*2 = a(n+8) - A091999(n),
a(n) = a(n+10)- A001651(n)*5 = a(n+10)- A072703(n),
a(n) = a(n+12)- A007310(n)*3,
a(n) = a(n+14)- A001651(n)*7. (End)
a(n) = (A007310(n+1)^2 - 1)/24. - Richard R. Forberg, May 27 2013; corrected by Zak Seidov, Mar 14 2015; further corrected by Jianing Song, Oct 24 2018
a(n) = Sum_{i = ceiling((n+1)/2)..n} i. - Wesley Ivan Hurt, Jun 08 2013
G.f.: x*G(0), where G(k) = 1 + x*(3*k + 4)/(3*k + 2 - x*(3*k + 2)*(3*k^2 + 11*k + 10)/(x*(3*k^2 + 11*k + 10) + (k + 1)*(3*k + 4)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 16 2013
Sum_{n>=1} 1/a(n) = 6 - 2*Pi/sqrt(3). - Vaclav Kotesovec, Oct 05 2016
a(n) = Sum_{i=1..n} numerator(i/2) = Sum_{i=1..n} denominator(2/i). - Wesley Ivan Hurt, Feb 26 2017
a(n) = A000292(A001651(n))/A001651(n), for n>0. - Ivan N. Ianakiev, May 08 2018
a(n) = ((-5 + (-1)^n - 6n)*(-1 + (-1)^n - 6n))/96. - José de Jesús Camacho Medina, Jun 12 2018
a(n) = Sum_{k=1..n} k/gcd(k,2). - Pedro Caceres, Apr 23 2019
Quadrisection. For r = 0,1,2,3: a(r + 4*k) = 6*k^2 + sqrt(24*a(r) + 1)*k + a(r), for k >= 1, with inputs (k = 0) {0,1,2,5}. These are the sequences A049453(k), A033570(k), A033568(k+1), A049452(k+1), for k >= 0, respectively. - Wolfdieter Lang, Feb 12 2021
a(n) = a(n-4) + sqrt(24*a(n-2) + 1), n >= 4. - Klaus Purath, Jul 07 2021
Sum_{n>=1} (-1)^(n+1)/a(n) = 6*(log(3)-1). - Amiram Eldar, Feb 28 2022
a(n) = A002620(n) + A008805(n-1). Gary W. Adamson, Dec 10 2022
E.g.f.: (x*(7 + 3*x)*cosh(x) + (1 + 5*x + 3*x^2)*sinh(x))/8. - Stefano Spezia, Aug 01 2024

cp's OEIS Frontend

A111967 Inverse of number triangle A101688.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A144734 Triangle read by rows, A054533 * transpose(A101688) (matrix product) provided A101688 is read as a square array by antidiagonals upwards.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

Extensions

A168258 Triangle read by rows, A101688 * A000012 as infinite lower triangular matrices.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Formula

Extensions

A168316 Triangle read by rows, square of triangle A101688.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

Extensions

A168508 Triangle read by rows: A101688 * A051731.

Views

Author

Keywords

Comments

Examples

Crossrefs

Extensions

A168509 Triangle read by rows, A051731 * A101688.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A168313 Triangle read by rows, retain 1's as rightmost diagonal of A101688 and replace all other 1's with 2's.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A159936 Triangle read by rows, A051731 * A054533 * transpose(A101688), provided A101688 is read as a square array.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

Extensions

A001318 Generalized pentagonal numbers: m(3m - 1)/2, m = 0, +-1, +-2, +-3, ....