A000012 -id:A000012 - cp's OEIS frontend

A134867 A010766 * A000012.

1, 3, 1, 5, 2, 1, 8, 4, 2, 1, 10, 5, 3, 2, 1, 14, 8, 5, 3, 2, 1, 16, 9, 6, 4, 3, 2, 1, 20, 12, 8, 6, 4, 3, 2, 1, 23, 14, 10, 7, 5, 4, 3, 2, 1, 27, 17, 12, 9, 7, 5, 4, 3, 2, 1, 29, 18, 13, 10, 8, 6, 5, 4, 3, 2, 1, 35, 23, 17, 13, 10, 8, 6, 5, 4, 3, 2, 1, 37, 24, 18, 14, 11, 9, 7, 6, 5, 4, 3, 2, 1

Offset: 1

Gary W. Adamson, Nov 14 2007

			First few rows of the triangle:
   1;
   3,  1;
   5,  2,  1;
   8,  4,  2, 1;
  10,  5,  3, 2, 1;
  14,  8,  5, 3, 2, 1;
  16,  9,  6, 4, 3, 2, 1;
  20, 12,  8, 6, 4, 3, 2, 1;
  23, 14, 10, 7, 5, 4, 3, 2, 1;
  27, 17, 12, 9, 7, 5, 4, 3, 2, 1;
  ...

Column k=1..4 give: A006218, A002541, A366968, A366972.
Row sums give A024916.
Cf. A010766, A135539.

t = Table[Sum[Floor[n/h], {h, k, n}], {n, 0, 10}, {k, 1, n}];
u = Flatten[t]  (* A134867 array *)
TableForm[t]    (* A134867 sequence *)
(* Clark Kimberling, Oct 11 2014 *)

T(n, k) = sum(j=k, n, n\j); \\ Seiichi Manyama, Oct 30 2023

A010766 * A000012 as infinite lower triangular matrices.
Triangle read by rows, partial row sums of A010766 starting fromt the right.
G.f. of column k: 1/(1-x) * Sum_{j>=1} x^(k*j)/(1-x^j) = 1/(1-x) * Sum_{j>=k} x^j/(1-x^j). - Seiichi Manyama, Oct 30 2023

More terms from Seiichi Manyama, Oct 30 2023

A143037 Triangle read by rows, A000012 * A127773 * A000012. A000012 is an infinite lower triangular matrix with all 1's, A127773 = (1; 0,3; 0,0,6; 0,0,0,10; ...).

Original entry on oeis.org

1, 3, 4, 6, 9, 10, 10, 16, 19, 20, 15, 25, 31, 34, 35, 21, 36, 46, 52, 55, 56, 28, 49, 64, 74, 80, 83, 84, 36, 64, 85, 100, 110, 116, 119, 120, 45, 81, 109, 130, 145, 155, 161, 164, 165, 55, 100, 136, 164, 185, 200, 210, 216, 219, 220

Offset: 1

Gary W. Adamson & Roger L. Bagula, Jul 18 2008

Right border = tetrahedral numbers, left border = triangular numbers.
Alternatively this is the square array A(n,k)
   1,   3,   6,  10,  15,  21,  28,  36,  45,  55, ...
   4,   9,  16,  25,  36,  49,  64,  81, 100, 121, ...
  10,  19,  31,  46,  64,  85, 109, 136, 166, 199, ...
  20,  34,  52,  74, 100, 130, 164, 202, 244, 290, ...
  35,  55,  80, 110, 145, 185, 230, 280, 335, 395, ...
  56,  83, 116, 155, 200, 251, 308, 371, 440, 515, ...
  ...
read by antidiagonals where A(n,k) = n*(n^2 + 3*k*n + 3*k^2 - 1)/6 is the sum of n triangular numbers starting at A000217(k). - R. J. Mathar, May 06 2015

			First few rows of the triangle:
   1;
   3,  4;
   6,  9, 10;
  10, 16, 19,  20;
  15, 25, 31,  34,  35;
  21, 36, 46,  52,  55,  56;
  28, 49, 64,  74,  80,  83,  84;
  36, 64, 85, 100, 110, 116, 119, 120;
  ...

Cf. A001296 (row sums).

A143037 := proc(n,k)
    k*(k^2-3*k*n-3*k+3*n^2+6*n+2) / 6 ;
end proc:
seq(seq(A143037(n,k),k=1..n),n=1..12) ; # R. J. Mathar, Aug 31 2022

T[n_,k_] = k*(k^2-3*k*n-3*k+3*n^2+6*n+2) / 6;Table[T[n,k],{n,10},{k,n}]//Flatten (* James C. McMahon, Aug 13 2025 *)

T(n,k) = k*(k^2-3*k*n-3*k+3*n^2+6*n+2) / 6. - R. J. Mathar, Aug 31 2022

A143061 Triangle read by rows, A000012 * A127647 * A000012.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 3, 5, 6, 7, 5, 8, 10, 11, 12, 8, 13, 16, 18, 19, 20, 13, 21, 26, 29, 31, 32, 33, 21, 34, 42, 47, 50, 52, 53, 54, 34, 55, 68, 76, 81, 84, 86, 87, 88, 55, 89, 110, 123, 131, 136, 139, 141, 142, 143, 89, 144, 178, 199, 212, 220, 225, 228, 230, 231

Offset: 1

Gary W. Adamson, Jul 20 2008

Row sums = A014286 (1, 3, 9, 21, 46, 94, ...); left border = Fibonacci numbers.

			First few rows of the triangle are:
  1;
  1,  2;
  2,  3,  4;
  3,  5,  6,  7;
  5,  8, 10, 11, 12;
  8, 13, 16, 18, 19, 20;
  ...

Robert Israel, Table of n, a(n) for n = 1..10011 (rows 1 to 141, flattened)

Cf. A000012, A000045, A014286, A127647.

seq(seq(combinat:-fibonacci(i+2)-combinat:-fibonacci(i+2-j),j=1..i),i=1..20); # Robert Israel, Nov 06 2016

From Robert Israel, Nov 06 2016: (Start)
T(n,k) = A000045(n+2) - A000045(n+2-k) for 1 <= k <= n.
G.f. as triangle: x*y*(1+x^2*y)/((1-x*y)*(1-x-x^2)*(1-x*y-x^2*y^2)). (End)

Corrected by Dintle N Kagiso, Nov 06 2016

A152205 Triangle read by rows, A000012 * A152204.

Original entry on oeis.org

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

Offset: 1

Gary W. Adamson, Nov 29 2008

Row sums = A000292, the tetrahedral numbers.
From Gary W. Adamson, Feb 14 2010: (Start)
Let the triangle = M. Then lim_{n->inf} M^n = A173277 as a left-shifted vector: (1, 4, 13, 32, 74, 152, 298, ...) = A(x), where A(x) satisfies A000290 = A(x)/A(x^2), A000290 = integer squares.
M * [1, 2, 3, ...] = A001752: (1, 4, 11, 24, 46, 80, 130, ...).
M * [1, 3, 6, 10, ...] = A028346: (1, 4, 12, 28, 58, 108, ...). (End)

			First few rows of the triangle:
    1;
    4;
    9,   1;
   16,   4;
   25,   9,   1;
   36,  16,   4;
   49,  25,   9,   1;
   64,  36,  16,   4;
   81,  49,  25,   9,   1;
  100,  64,  36,  16,   4;
  121,  81,  49,  25,   9,   1;
  144, 100,  64,  36,  16,   4;
  169, 121,  81,  49,  25,   9,   1;
  ...

Cf. A152204, A000292.

Cf. A173277, A001752, A028346. - Gary W. Adamson, Feb 14 2010

A000012 * A152204 = partial sums of A152204 by columns.

A230850 A054541 and A000012 interleaved.

Original entry on oeis.org

2, 1, 1, 1, 2, 1, 2, 1, 4, 1, 2, 1, 4, 1, 2, 1, 4, 1, 6, 1, 2, 1, 6, 1, 4, 1, 2, 1, 4, 1, 6, 1, 6, 1, 2, 1, 6, 1, 4, 1, 2, 1, 6, 1, 4, 1, 6, 1, 8, 1, 4, 1, 2, 1, 4, 1, 2, 1, 4, 1, 14, 1, 4, 1, 6, 1, 2, 1, 10, 1, 2, 1, 6, 1, 6, 1, 4, 1, 6, 1, 6, 1, 2, 1, 10, 1, 2, 1

Offset: 1

Omar E. Pol, Oct 31 2013

nonn

a(n) is also the length of the n-th edge of a staircase which represents the function pi(x) on the first quadrant of the square grid, see A000720.
a(2n-1) is the length of the n-th horizontal edge in the staircase.
a(2n) is the length of the n-th vertical edge in the staircase.
For another version see A230849.

			Illustration of initial terms, n = 1..22:
.
1                                                              _ _|
1                                                  _ _ _ _ _ _|
1                                          _ _ _ _|
1                                      _ _|
1                              _ _ _ _|
1                          _ _|
1                  _ _ _ _|
1              _ _|
1          _ _|
1        _|
1    _ _|
.
.      2 1   2   2       4   2       4   2       4           6   2
.
Drawing vertical line segments below the staircase (as shown below) we have that the number of cells in the vertical bars gives 0 together A000720.
Drawing horizontal line segments above the staircase we have that the number of cells in the k-th horizontal bar is A000040(k).
.    _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
31  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
29  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | |
23  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | | | | | | | |
19  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | | | | | | | | | | | |
17  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | | | | | | | | | | | | | |
13  |_ _ _ _ _ _ _ _ _ _ _ _ _| | | | | | | | | | | | | | | | | | |
11  |_ _ _ _ _ _ _ _ _ _ _| | | | | | | | | | | | | | | | | | | | |
7   |_ _ _ _ _ _ _| | | | | | | | | | | | | | | | | | | | | | | | |
5   |_ _ _ _ _| | | | | | | | | | | | | | | | | | | | | | | | | | |
3   |_ _ _| | | | | | | | | | | | | | | | | | | | | | | | | | | | |
2   |_ _|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|
.
.    0 0 1 2 2 3 3 4 4 4 4 5 5 6 6 6 6 7 7 8 8 8 8 9 9 9 9 9 9 10 10
.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000012, A000040, A000720, A001223, A007504, A046992, A054541, A141042, A152535, A182986, A230849.

Riffle[Join[{2},Differences[Prime[Range[100]]]],1] (* Paolo Xausa, Oct 31 2023 *)

A230850(n) = if(1==n,2,if((n%2),prime((n+1)/2)-prime(((n+1)/2)-1),1)); \\ Antti Karttunen, Dec 23 2018

a(1) = 2; for n > 1, a(n) = A230849(n). - Antti Karttunen, Dec 23 2018

A302764 Pascal-like triangle with A000012 as the left border and A080956 as the right border.

Original entry on oeis.org

1, 1, 1, 1, 2, 0, 1, 3, 2, -2, 1, 4, 5, 0, -5, 1, 5, 9, 5, -5, -9, 1, 6, 14, 14, 0, -14, -14, 1, 7, 20, 28, 14, -14, -28, -20, 1, 8, 27, 48, 42, 0, -42, -48, -27, 1, 9, 35, 75, 90, 42, -42, -90, -75, -35, 1, 10, 44, 110, 165, 132, 0, -132, -165, -110, -44

Offset: 1

Gregory Gerard Wojnar, Apr 12 2018

Number the rows of the triangle beginning with n=0. For each row construct a degree n polynomial with regularly decreasing powers, denoting the polynomial as f_n(x); e.g., for row 2 we have f_2(x)=1x^2+2x+0. Then construct g_n(x)=x^2*f_{n-1}(x)-(n+1)x+1. It obtains that g_n(x)=(1-x)(2-(1+x)^n). These g_n(x) are the denominators of the generating functions for the following sequences: A024537 (n=2); A195350 (n=3); A301417 (n=4); A301420 (n=5); A301421 (n=6); A301424 (n=7). For these sequences the asymptotic term-to-term ratios are 1/(2^(1/n)-1). The numerators of the generating functions are 1-x(x+1)^(n-1).

			Triangle begins:
  1;
  1, 1;
  1, 2,  0;
  1, 3,  2, -2;
  1, 4,  5,  0, -5;
  1, 5,  9,  5, -5,  -9;
  1, 6, 14, 14,  0, -14, -14;
  1, 7, 20, 28, 14, -14, -28, -20;
  ...

Cf. A000012, A007318, A024537, A080956, A195350, A301417, A301420, A301421, A301424.

T(n,k) = if (k==0, 1, if (k==n, (n+1)*(2-n)/2, if (k>n, 0, T(n-1,k) + T(n-1,k-1))));
tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n, k), ", ")); print); \\ Michel Marcus, Apr 21 2018

T(n,k) = T(n-1,k) + T(n-1,k-1) with T(n, 0) = 1 and T(n, n) = (n+1)*(2-n)/2.

A118923 Triangle T(n,k) built by placing T(n,0)=A000012(n) in the left edge, T(n,n)=A079978(n) on the right edge and filling the body with the Pascal recurrence T(n,k) = T(n-1,k) + T(n-1,k-1).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 1, 1, 3, 3, 2, 0, 1, 4, 6, 5, 2, 0, 1, 5, 10, 11, 7, 2, 1, 1, 6, 15, 21, 18, 9, 3, 0, 1, 7, 21, 36, 39, 27, 12, 3, 0, 1, 8, 28, 57, 75, 66, 39, 15, 3, 1, 1, 9, 36, 85, 132, 141, 105, 54, 18, 4, 0, 1, 10, 45, 121, 217, 273, 246, 159, 72, 22, 4, 0, 1, 11, 55, 166

Offset: 0

Alford Arnold, May 05 2006

The fourth diagonal is 1, 2, 5, 11, 21, ..., which is 1 + A000292. The fifth diagonal is 0, 2, 7, 18, 39, 75, 132, 217, 338, 504, 725, 1012, ..., which is A051743.
The array A007318 is generated by placing A000012 on both edges with the same Pascal-like recurrence, and the array A059259 uses edges defined by A000012 and A059841. - R. J. Mathar, Jan 21 2008
From Michael A. Allen, Nov 30 2021: (Start)
T(n,n-k) is the (n,k)-th entry of the (1/(1-x^3), x/(1-x)) Riordan array.
Sums of rows give A077947.
Sums of antidiagonals give A079962. (End)

			The table begins
  1
  1  0
  1  1  0
  1  2  1  1
  1  3  3  2  0
  1  4  6  5  2  0
  1  5 10 11  7  2  1
  1  6 15 21 18  9  3  0

Cf. A000292, A079978, A008620, A079998, A051743, A077947.

Generalization of A007318, A059259.

A000012 := proc(n) 1 ; end: A079978 := proc(n) if n mod 3 = 0 then 1; else 0 ; fi ; end: A118923 := proc(n,k) if k = 0 then A000012(n); elif k = n then A079978(n) ; else A118923(n-1,k)+A118923(n-1,k-1) ; fi ; end: for n from 0 to 15 do for k from 0 to n do printf("%d, ",A118923(n,k)) ; od: od: # R. J. Mathar, Jan 21 2008

Flatten@Table[CoefficientList[Series[1/((1 + x*y + x^2*y^2)(1 - x - x*y)), {x, 0, 23}, {y, 0, 11}], {x, y}][[n + 1, k + 1]], {n, 0, 11}, {k, 0, n}] (* Michael A. Allen, Nov 30 2021 *)

From Michael A. Allen, Nov 30 2021: (Start)
For 0 <= k < n, T(n,k) = (n-k)*Sum_{j=0..floor(k/3)} binomial(n-3*j,n-k)/(n-3*j).
G.f.: 1/((1+x*y+(x*y)^2)*(1-x-x*y)). (End)

Edited and extended by R. J. Mathar, Jan 21 2008

Offset changed by Michael A. Allen, Nov 30 2021

A127949 A000012 as an infinite lower triangular matrix with all 1's; A127899 = a simple transform; then A000012 * A127899. Given A051340, change all 1's to -1. Triangle read by rows, (n-1) -1's followed by "n".

Original entry on oeis.org

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

Offset: 1

Gary W. Adamson, Feb 09 2007

For the inverse of A127949 see A126615, a harmonic triangle.

This is one way to define an inverse to A000217. - R. J. Mathar, Apr 30 2010

			First few rows of the triangle are:
1;
-1, 2;
-1, -1, 3;
-1, -1, -1, 4;
...

Cf. A000012, A127899, A051340, A126615.

A127949 := proc(n) if issqr(1+8*n) then (sqrt(1+8*n)-1)/2 ; else -1 ; end if; end proc: seq(A127949(n),n=1..120) ; # R. J. Mathar, Apr 30 2010

More terms from R. J. Mathar, Apr 30 2010

A128623 Triangle read by rows: A128621 * A000012 as infinite lower triangular matrices.

Original entry on oeis.org

1, 2, 2, 6, 3, 3, 8, 8, 4, 4, 15, 10, 10, 5, 5, 18, 18, 12, 12, 6, 6, 28, 21, 21, 14, 14, 7, 7, 32, 32, 24, 24, 16, 16, 8, 8, 45, 36, 36, 27, 27, 18, 18, 9, 9, 50, 50, 40, 40, 30, 30, 20, 20, 10, 10, 66, 55, 55, 44, 44, 33, 33, 22, 22, 11, 11, 72, 72, 60, 60, 48, 48, 36, 36, 24, 24, 12, 12, 91, 78, 78, 65, 65, 52, 52, 39, 39, 26, 26, 13, 13

Offset: 1

Gary W. Adamson, Mar 14 2007

			First few rows of the triangle are:
   1;
   2,  2;
   6,  3,  3;
   8,  8,  4,  4;
  15, 10, 10,  5,  5;
  18, 18, 12, 12,  6, 6;
  28, 21, 21, 14, 14, 7, 7;
  ...

G. C. Greubel, Rows n = 1..100 of the triangle, flattened

Cf. A000027, A093005, A093353, A115514, A128621, A245524.

Cf. A128624 (row sums).

[n*Floor((n-k+2)/2): k in [1..n], n in [1..15]]; // G. C. Greubel, Mar 13 2024

Table[n*Floor[(n-k+2)/2], {n,15}, {k,n}]//Flatten (* G. C. Greubel, Mar 13 2024 *)

flatten([[n*((n-k+2)//2) for k in range(1,n+1)] for n in range(1,16)]) # G. C. Greubel, Mar 13 2024

Sum_{k=1..n} T(n, k) = A128624(n) (row sums).
T(n,k) = n*(1+floor((n-k)/2)), 1 <= k <= n. - R. J. Mathar, Jun 27 2012
From G. C. Greubel, Mar 13 2024: (Start)
T(n, k) = n*A115514(n, k).
T(n, k) = Sum_{j=k..n} A128621(n, j).
T(n, 1) = A093005(n).
T(n, 2) = A093353(n-1), n >= 2.
T(n, n) = A000027(n).
T(2*n-1, n) = A245524(n).
Sum_{k=1..n} (-1)^k*T(n, k) = (1/2)*(1-(-1)^n)*A000384(floor((n+1)/2)). (End)

a(41) = 27 inserted and more terms from Georg Fischer, Jun 05 2023

A131054 Triangle read by rows: A000012 * A119467 as infinite lower triangular matrices.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 4, 1, 1, 3, 4, 7, 1, 1, 3, 9, 7, 11, 1, 1, 4, 9, 22, 11, 16, 1, 1, 4, 16, 22, 46, 16, 22, 1, 1, 5, 16, 50, 46, 86, 22, 29, 1, 1, 5, 25, 50, 130, 86, 148, 29, 37, 1, 1, 6, 25, 95, 130, 296, 148, 239, 37, 46, 1, 1

Offset: 0

Gary W. Adamson, Jun 12 2007

Row sums = 2^n.

			First few rows of the triangle are:
1;
1, 1;
2, 1, 1;
2, 4, 1, 1;
3, 4, 7, 1, 1;
3, 9, 7, 11, 1, 1;
4, 9, 22, 11, 16, 1, 1;
4, 16, 22, 46, 16, 22, 1, 1;
...

Cf. A000012, A119467.

a(46) = 5 inserted and more terms from Georg Fischer, May 29 2023

cp's OEIS Frontend

A134867 A010766 * A000012.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A143037 Triangle read by rows, A000012 * A127773 * A000012. A000012 is an infinite lower triangular matrix with all 1's, A127773 = (1; 0,3; 0,0,6; 0,0,0,10; ...).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

Formula

A143061 Triangle read by rows, A000012 * A127647 * A000012.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Formula

Extensions

A152205 Triangle read by rows, A000012 * A152204.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A230850 A054541 and A000012 interleaved.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A302764 Pascal-like triangle with A000012 as the left border and A080956 as the right border.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Formula

A118923 Triangle T(n,k) built by placing T(n,0)=A000012(n) in the left edge, T(n,n)=A079978(n) on the right edge and filling the body with the Pascal recurrence T(n,k) = T(n-1,k) + T(n-1,k-1).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A127949 A000012 as an infinite lower triangular matrix with all 1's; A127899 = a simple transform; then A000012 * A127899. Given A051340, change all 1's to -1. Triangle read by rows, (n-1) -1's followed by "n".

Views

Author