A158405 -id:A158405 - cp's OEIS frontend

A158894 Sawtooth pattern of one, then two, then three, then four etc. consecutive odd numbers, starting each time at 3.

3, 3, 5, 3, 5, 7, 3, 5, 7, 9, 3, 5, 7, 9, 11, 3, 5, 7, 9, 11, 13, 3, 5, 7, 9, 11, 13, 15, 3, 5, 7, 9, 11, 13, 15, 17, 3, 5, 7, 9, 11, 13, 15, 17, 19, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 3, 5, 7, 9, 11, 13, 15, 17, 19

Offset: 1

Paul Curtz, Mar 29 2009

Obtained from A158405 by deleting the 1's, or from A141620 by deleting the positive numbers and flipping all signs.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A158405, A141620, A000124.

sp[n_]:=Module[{cl=Range[3,51,2],l={}},Table[l=Join[l,Take[cl,i]], {i,n}];l ]; sp[15] (* Harvey P. Dale, Oct 30 2011 *)
Table[Range[3,2n+1,2],{n,15}]//Flatten (* Harvey P. Dale, Jun 27 2025 *)

forstep(m=3,9,2,forstep(n=3,m,2,print1(n", "))) \\ Charles R Greathouse IV, Oct 30 2011

a(A000124(i))=3, i>=0. - R. J. Mathar, Apr 04 2009

Edited by R. J. Mathar, Apr 04 2009

Corrected by Harvey P. Dale, Oct 30 2011

A261819 Encoded symmetrical antidiagonal square binary matrices with either 1 or 2 ones.

Original entry on oeis.org

1, 6, 16, 40, 384, 576, 4096, 10240, 17408, 393216, 589824, 1081344, 16777216, 41943040, 71303168, 136314880, 6442450944, 9663676416, 17716740096, 34628173824, 1099511627776

Offset: 0

Eric Werley, Sep 24 2015

nonn

We encode square matrices that have zeros everywhere except the antidiagonal where the antidiagonal is symmetric with either 1 or 2 ones in it. We do this by reading off digits antidiagonally to get a binary number and then convert the number to a base 10 number.

			The 3 X 3 matrix
0 0 0
0 1 0
0 0 0
gives 000010000. Writing this as a base 10 number gives a(2)=16.
The 4 X 4 matrix
0 0 0 0
0 0 1 0
0 1 0 0
0 0 0 0
gives 0000000110000000. Writing this as a base 10 number gives a(4)=384.
The 5 X 5 matrix
0 0 0 0 0
0 0 0 1 0
0 0 0 0 0
0 1 0 0 0
0 0 0 0 0
gives 0000000000010100000000000. Writing this as a base 10 number gives a(7)=10240.

a(n) = A261195(2^n).

a(n) = 2^(A000217(floor(sqrt(4*n + 1)) - 1)) * (((A262769(floor(n/2)) * 2^((floor(sqrt(4*n + 1)) - 2*A002260(+1))/2)) * (1+(-1)^(floor(sqrt(4*n + 1))))/2) + ((A262777(floor(n/2)) * 2^((floor(sqrt(4*n + 1)) - A158405(+1))/2)) * (1-(-1)^(floor(sqrt(4*n + 1))))/2)).

A095873 Triangle T(n,k) = (2k-1)(n+k-1)*(n-k+1) read by rows, 1<=k<=n.

Original entry on oeis.org

1, 4, 9, 9, 24, 25, 16, 45, 60, 49, 25, 72, 105, 112, 81, 36, 105, 160, 189, 180, 121, 49, 144, 225, 280, 297, 264, 169, 64, 189, 300, 385, 432, 429, 364, 225, 81, 240, 385, 504, 585, 616, 585, 480, 289, 100, 297, 480, 637, 756, 825

Offset: 1

Gary W. Adamson, Jun 10 2004

Matrix square of A158405.

			[1 0 0 / 1 3 0 / 1 3 5]^2 = [1 0 0 / 4 9 0 / 9 24 25]. Delete the zeros and
read by rows:
1;
4, 9;
9, 24, 25;
16,45, 60, 49;
25,72,105,112, 81;

Albert H. Beiler, "Recreations in the Theory of Numbers", Dover, 1966.

Cf. A094728, A095871, A095872.

A095873 := proc(n,k)
        (2*k-1)*(n+k-1)*(n-k+1) ;
end proc:
seq(seq(A095873(n,k),k=1..n),n=1..13) ; # R. J. Mathar, Oct 30 2011

Table[(2k-1)(n+k-1)(n-k+1),{n,10},{k,n}]//Flatten (* Harvey P. Dale, May 03 2018 *)

T(n,k) = (2*k-1)*A094728(n,k).

Sum_{k=1..n} T(n,k)= n*(n+1)*(3*n^2+n-1)/6 = A103220(n). - R. J. Mathar, Oct 30 2011

Definition in closed form by R. J. Mathar, Oct 30 2011

A211197 Table T(n,k) = 2n + ((-1)^n)(1/2 - (k-1) mod 2) - 1/2; n, k > 0, read by antidiagonals.

Original entry on oeis.org

1, 2, 4, 1, 3, 5, 2, 4, 6, 8, 1, 3, 5, 7, 9, 2, 4, 6, 8, 10, 12, 1, 3, 5, 7, 9, 11, 13, 2, 4, 6, 8, 10, 12, 14, 16, 1, 3, 5, 7, 9, 11, 13, 15, 17, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22

Offset: 1

Boris Putievskiy, Feb 03 2013

In general, let B and C be sequences. By b(n) and c(n) denote elements B and C respectively. Table T(n,k) = (1-(-1)^k)*b(n)/2+(1+(-1)^k)*c(n)/2 read by antidiagonals.
For this sequence b(n)=2*n-1, b(n)=A005408(n), c(n)=2*n, c(n)=A005843(n).
If n is odd   row T(n,k) is alternation b(n) and c(n) starts from b(n).
If n is even  row T(n,k) is alternation c(n) and b(n) starts from c(n).
For this sequence if n is odd   alternation numbers  2*n-1 and 2*n   starts from  2*n-1.
For this sequence if n is even  alternation numbers  2*n   and 2*n-1 starts from  2*n.
T(n,k) is replication of the first and the second columns that are “a braid” from sequences B and C.

			The start of the sequence as table for general case:
  b(1)..c(1)..b(1)..c(1)..b(1)..c(1)..b(1)..c(1)..
  c(2)..b(2)..c(2)..b(2)..c(2)..b(2)..c(2)..b(2)..
  b(3)..c(3)..b(3)..c(3)..b(3)..c(3)..b(3)..c(3)..
  c(4)..b(4)..c(4)..b(4)..c(4)..b(4)..c(4)..b(4)..
  b(5)..c(5)..b(5)..c(5)..b(5)..c(5)..b(5)..c(5)..
  c(6)..b(6)..c(6)..b(6)..c(6)..b(6)..c(6)..b(6)..
  b(7)..c(7)..b(7)..c(7)..b(7)..c(7)..b(7)..c(7)..
  c(8)..b(8)..c(8)..b(8)..c(8)..b(8)..c(8)..b(8)..
  . . .
The start of the sequence as triangle array read by rows for general case:
  b(1);
  c(1),c(2);
  b(1),b(2),b(3);
  c(1),c(2),c(3),c(4);
  b(1),b(2),b(3),b(4),b(5);
  c(1),c(2),c(3),c(4),c(5),c(6);
  b(1),b(2),b(3),b(4),b(5),b(6),b(7);
  c(1),c(2),c(3),c(4),c(5),c(6),c(7),c(8);
. . .
Row number r contains r numbers.
If r is odd  b(1),b(2),...,b(r).
If r is even c(1),c(2),...,c(r).
The start of the sequence as table for b(n)=2*n-1 and c(n)=2*n:
  1....2...1...2...1...2...1...2...
  4....3...4...3...4...3...4...3...
  5....6...5...6...5...6...5...6...
  8....7...8...7...8...7...8...7...
  9...10...9..10...9..10...9..10...
  12..11..12..11..12..11..12..11...
  13..14..13..14..13..14..13..14...
  16..15..16..15..16..15..16..15...
  . . .
The start of the sequence as triangle array read by rows for  b(n)=2*n-1 and c(n)=2*n:
  1;
  2,4;
  1,3,5;
  2,4,6,8;
  1,3,5,7,9;
  2,4,6,8,10,12;
  1,3,5,7,9,11,13;
  2,4,6,8,10,12,14,16;
  . . .
Row number r contains r numbers.
If r is odd  1,3,...2*r-1 - coincides with the elements row number r triangle array read by rows for sequence 2*A002260-1.
If r is even 2,4,...,2*r  - coincides with the elements row number r triangle array read by rows for sequence 2*A002260.

Boris Putievskiy, Rows n = 1..140 of triangle, flattened
Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions, arXiv:1212.2732 [math.CO], 2012.

Cf. A000027, A003056, A002260, A004736, A210530, A158405, A005408, A005843, A057211.

t=int((math.sqrt(8*n-7) - 1)/ 2)
i=n-t*(t+1)/2
j=(t*t+3*t+4)/2-n
result =2*i+((-1)**i)*(0.5 - (j-1) % 2) - 0.5

a211197_list = [2*n - k%2 for k in range(1, 13) for n in range(1, k+1)] # David Radcliffe, Jun 01 2025

For the general case:
As a table: T(n,k) = (1-(-1)^k)*b(n)/2+(1+(-1)^k)*c(n)/2.
As a linear sequence: a(n) = (1-(-1)^j)*b(i)/2+(1+(-1)^j)*c(i)/2, where i = n-t*(t+1)/2, j = (t*t+3*t+4)/2-n, t = floor((-1+sqrt(8*n-7))/2).
where b(n) = 2*n-1 and c(n) = 2*n.
As a table: T(n,k) = 2*n+((-1)^n)*(1/2- (k-1) mod 2) - 1/2.
As a linear sequence:
a(n) = 2*A002260(n) + ((-1)^A002260(n))*(1/2- (A004736(n)-1) mod 2) -1/2.
a(n) = -(1+(-1)^A003056(n))*A002260(n) +(1+(-1)^A003056(n))*(2*A002260(n)-1)/2.
a(n) =  2*i+((-1)^i)*(1/2- (j-1) mod 2) - 1/2
a(n) = -(1+(-1)^t)*i +(1+(-1)^t)*(2*i-1)/2,
where i = n-t*(t+1)/2, j = (t*t+3*t+4)/2-n, t = floor((-1+sqrt(8*n-7))/2).
T(n, k) = 2*n - 1 + (n+k mod 2); a(n) = 2*A002260(n) - A057211(n). - David Radcliffe, Jun 01 2025

A271668 Triangle read by rows. The first column is A000217(n+1). From the second row we apply - A002262(n) for the following terms of the row.

Original entry on oeis.org

1, 3, 3, 6, 6, 5, 10, 10, 9, 7, 15, 15, 14, 12, 9, 21, 21, 20, 18, 15, 11, 28, 28, 27, 25, 22, 18, 13, 36, 36, 35, 33, 30, 26, 21, 15, 45, 45, 44, 42, 39, 35, 30, 24, 17, 55, 55, 54, 52, 49, 45, 40, 34, 27, 19, 66, 66, 65, 63, 60, 56, 51, 45, 38, 30, 21

Offset: 0

Paul Curtz, Apr 12 2016

Row sums: A084990(n+1).
A158405(n) = A002262(n) + A002260(n). See the formula.
(Without its first column, A094728 is A120070, which could be built from positive A005563 and -A158894.)

			a(0) = 1, a(1) = 3, a(2) =3-0 = 3,  a(3) = 6, a(4) =6-0= 6, a(5) =6-1= 5, ... .
Triangle:
1,
3,   3,
6,   6,  5,
10, 10,  9,  7,
15, 15, 14, 12,  9,
21, 21, 20, 18, 15, 11,
28, 28, 27, 25, 22, 18, 13,
36, 36, 35, 33, 30, 26, 21, 15,
etc.

Cf. A000096, A000217, A002260, A002262, A005408, A005563, A049780, A084990, A094728, A120070, A158405, A158894.

Table[(n^2 - n)/2 - Prepend[Accumulate@ Range[0, n - 3], 0], {n, 12}] // Flatten (* Michael De Vlieger, Apr 12 2016 *)

a(n) = A094728(n+1) - A049780(n).

A158897 The elements of A059100 at indices of triangular numbers, padded with zeros.

Original entry on oeis.org

6, 0, 11, 0, 0, 18, 0, 0, 0, 27, 0, 0, 0, 0, 38, 0, 0, 0, 0, 0, 51, 0, 0, 0, 0, 0, 0, 66, 0, 0, 0, 0, 0, 0, 0, 83, 0, 0, 0, 0, 0, 0, 0, 0, 102, 0, 0, 0, 0, 0, 0, 0, 0, 0, 123, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 146, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 171, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 198, 0, 0, 0, 0, 0, 0

Offset: 1

Paul Curtz, Mar 29 2009

nonn

Cf. A144396.

a(t)=A059100(i+1) if t=A000217(i), else a(t)=0.

a(n)=A141620(n-1)+A158405(n+1).

Edited and extended by R. J. Mathar, Apr 04 2009

A172049 Irregular triangle T(n,k) = 2k-1 with A008794(n+2) values in row n.

Original entry on oeis.org

1, 1, 1, 3, 5, 7, 1, 3, 5, 7, 1, 3, 5, 7, 9, 11, 13, 15, 17, 1, 3, 5, 7, 9, 11, 13, 15, 17, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43

Offset: 1

Paul Curtz, Jan 24 2010

These are the values A172002(2*n)-A172002(2*n-1) arranged in a table by adding line breaks.

The last (and largest) numbers in the lines are in A056220(floor(n+1)/2).

			1;
1;
1, 3, 5, 7;
1, 3, 5, 7;
1, 3, 5, 7, 9, 11, 13, 15, 17;
1, 3, 5, 7, 9, 11, 13, 15, 17;
1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31;
1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31;
1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43,...

Cf. A158405.

A174028 Triangle T(n,k) = 2+4k read by rows.

Original entry on oeis.org

2, 2, 6, 2, 6, 10, 2, 6, 10, 14, 2, 6, 10, 14, 18, 2, 6, 10, 14, 18, 22, 2, 6, 10, 14, 18, 22, 26, 2, 6, 10, 14, 18, 22, 26, 30, 2, 6, 10, 14, 18, 22, 26, 30, 34, 2, 6, 10, 14, 18, 22, 26, 30, 34, 38

Offset: 0

Paul Curtz, Mar 06 2010

			2;
2, 6;
2, 6, 10;
2, 6, 10, 14;
2, 6, 10, 14, 18;
2, 6, 10, 14, 18, 22;
2, 6, 10, 14, 18, 22, 26;

Wikipedia, Electron shell

Cf. A001105 (row sums), A167268.

T(n,k) = A016825(k).

T(n,k) = 2*A158405(n+1,k).

cp's OEIS Frontend

A158894 Sawtooth pattern of one, then two, then three, then four etc. consecutive odd numbers, starting each time at 3.

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A261819 Encoded symmetrical antidiagonal square binary matrices with either 1 or 2 ones.

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A095873 Triangle T(n,k) = (2*k-1)*(n+k-1)*(n-k+1) read by rows, 1<=k<=n.

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A211197 Table T(n,k) = 2*n + ((-1)^n)*(1/2 - (k-1) mod 2) - 1/2; n, k > 0, read by antidiagonals.

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A271668 Triangle read by rows. The first column is A000217(n+1). From the second row we apply - A002262(n) for the following terms of the row.

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A158897 The elements of A059100 at indices of triangular numbers, padded with zeros.

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A172049 Irregular triangle T(n,k) = 2k-1 with A008794(n+2) values in row n.

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A174028 Triangle T(n,k) = 2+4k read by rows.

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A095873 Triangle T(n,k) = (2k-1)(n+k-1)*(n-k+1) read by rows, 1<=k<=n.

A211197 Table T(n,k) = 2n + ((-1)^n)(1/2 - (k-1) mod 2) - 1/2; n, k > 0, read by antidiagonals.