A000012 -id:A000012 - cp's OEIS frontend

A131333 A131332 * A000012.

1, 1, 1, 2, 1, 1, 3, 2, 2, 1, 5, 3, 4, 2, 1, 8, 5, 7, 4, 3, 1, 13, 8, 12, 7, 7, 3, 1, 21, 13, 20, 12, 14, 7, 4, 1, 34, 21, 33, 20, 26, 14, 11, 4, 1, 55, 34, 54, 33, 46, 26, 25, 11, 5, 1

Offset: 1

Gary W. Adamson, Jun 29 2007

Leftmost two columns = Fibonacci numbers. Row sums = A029907: (1, 2, 4, 8, 15, 28, 51,...).

			First few rows of the triangle are:
1;
1, 1;
2, 1, 1;
3, 2, 2, 1;
5, 3, 4, 2, 1;
8, 5, 7, 4, 3, 1;
13, 8, 12, 7, 7, 3, 1;
21, 13, 20, 12, 14, 7, 4, 1;
...

Cf. A131332, A131334, A131335, A131336.

A131332 * A000012, where A000012 = (1; 1,1; 1,1,1;...). Partial sums of A131332 rows starting from the right.

A131334 A000012(signed) * A065941.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 2, 1, 1, 0, 1, 2, 2, 2, 1, 1, 0, 3, 2, 4, 2, 1, 0, 1, 3, 3, 6, 4, 3, 1, 1, 0, 4, 3, 9, 6, 7, 3, 1, 0, 1, 4, 4, 12, 9, 13, 7, 4, 1

Offset: 1

Gary W. Adamson, Jun 29 2007

Row sums = Fibonacci numbers.

			First few rows of the triangle are:
1;
0, 1;
1, 0, 1;
0, 1, 1, 1;
1, 0, 2, 1, 1;
0, 1, 2, 2, 2, 1;
1, 0, 3, 2, 4, 2, 1;
0, 1, 3, 3, 6, 4, 3, 1;
1, 0, 4, 3, 9, 6, 7, 3, 1;
...

Cf. A131332, A131333, A131335, A131336.

A000012(signed by columns, + - + -...) * A065941.

A131335 A000012 * A131334.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 2, 2, 1, 3, 2, 4, 2, 1, 3, 3, 6, 4, 3, 1, 4, 3, 9, 6, 7, 3, 1, 4, 4, 12, 9, 13, 7, 4, 1, 5, 4, 16, 12, 22, 13, 11, 4, 1, 5, 5, 20, 16, 34, 22, 24, 11, 5, 1

Offset: 1

Gary W. Adamson, Jun 29 2007

Row sums = A000071, Fibonacci numbers - 1, starting (1, 2, 4, 7, 12, 20, 33,...).

			First few rows of the triangle are:
1;
1, 1;
2, 1, 1;
2, 2, 2, 1;
3, 2, 4, 2, 1;
3, 3, 6, 4, 3, 1;
4, 3, 9, 6, 7, 3, 1;
...

Cf. A000071, A131334, A000012, A131332, A131333.

A000012 * A131334 as infinite lower triangular matrices, where A000012 = (1; 1,1; 1,1,1;...).

A132823 A007318 + 2A103451 - 2A000012.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 2, 4, 2, 1, 1, 3, 8, 8, 3, 1, 1, 4, 13, 18, 13, 4, 1, 1, 5, 19, 33, 33, 19, 5, 1, 1, 6, 26, 54, 68, 54, 26, 6, 1, 1, 7, 34, 82, 124, 124, 82, 34, 7, 1, 1, 8, 43, 118, 208, 250, 208, 118, 43, 8, 1, 1, 9, 53, 163, 328, 460, 460, 328, 163, 53, 9, 1

Offset: 0

Gary W. Adamson, Sep 02 2007

Row sums = A132824: (1, 2, 2, 4, 10, 24, 54, 116, 242, ...).

			First few rows of the triangle:
  1;
  1, 1;
  1, 0,  1;
  1, 1,  1,  1;
  1, 2,  4,  2,   1;
  1, 3,  8,  8,   3,   1;
  1, 4, 13, 18,  13,   4,  1;
  1, 5, 19, 33,  33,  19,  5,  1;
  1, 6, 26, 54,  68,  54, 26,  6, 1;
  1, 7, 34, 82, 124, 124, 82, 34, 7, 1;
  ...

Cf. A007318, A103451, A000012, A132824.

A(2n,n) gives A115112 for n>0.

A007318 + 2*A103451 - 2*A000012 as infinite lower triangular matrices.

One missing 1 inserted and more terms added by Alois P. Heinz, Feb 10 2019

A058393 A square array based on 1^n (A000012) with each term being the sum of 2 consecutive terms in the previous row.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 0, 1, 2, 1, 1, 1, 2, 3, 1, 0, 1, 2, 4, 4, 1, 1, 1, 2, 4, 7, 5, 1, 0, 1, 2, 4, 8, 11, 6, 1, 1, 1, 2, 4, 8, 15, 16, 7, 1, 0, 1, 2, 4, 8, 16, 26, 22, 8, 1, 1, 1, 2, 4, 8, 16, 31, 42, 29, 9, 1, 0, 1, 2, 4, 8, 16, 32, 57, 64, 37, 10, 1, 1, 1, 2, 4, 8, 16, 32, 63, 99, 93, 46, 11, 1, 0

Offset: 0

Henry Bottomley, Nov 24 2000

Changing the formula by replacing T(0,2n)=T(1,n) by T(0,2n)=T(m,n) for some other value of m, would make the generating function change to coefficient of x^n in expansion of (1+x)^k/(1-x^2)^m. This would produce A058394, A058395, A057884, (and effectively A007318).

			Rows are (1,0,1,0,1,0,1,...), (1,1,1,1,1,1,...), (1,2,2,2,2,2,...), (1,3,4,4,4,...) etc.

Rows are A000035 (A000012 with zeros), A000012, A040000 etc. Columns are A000012, A001477, A000124, A000125, A000127, A006261, A008859, A008860, A008861, A008862, A008863 etc. Diagonals include A000079, A000225, A000295, A002662, A002663, A002664, A035038, A035039, A035040, A035041, etc. The triangles A008949, A054143 and A055248 also appear in the half of the array which is not powers of 2.

T(n, k)=T(n-1, k-1)+T(n, k-1) with T(0, k)=1, T(1, 1)=1, T(0, 2n)=T(1, n) and T(0, 2n+1)=0. Coefficient of x^n in expansion of (1+x)^k/(1-x^2).

A130303 A130296 * A000012.

Original entry on oeis.org

1, 3, 1, 5, 2, 1, 7, 3, 2, 1, 9, 4, 3, 2, 1, 11, 5, 4, 3, 2, 1, 13, 6, 5, 4, 3, 2, 1, 15, 7, 6, 5, 4, 3, 2, 1, 17, 8, 7, 6, 5, 4, 3, 2, 1, 19, 9, 8, 7, 6, 5, 4, 3, 2, 1

Offset: 1

Gary W. Adamson, May 20 2007

			1;
3, 1;
5, 2, 1;
7, 3, 2, 1;
9, 4, 3, 2, 1;
11, 5, 4, 3, 2, 1;
13, 6, 5, 4, 3, 2, 1;
15, 7, 6, 5, 4, 3, 2, 1;
17, 8, 7, 6, 5, 4, 3, 2, 1;
19, 9, 8, 7, 6, 5, 4, 3, 2, 1;

H. S. M. Coxeter, Regular Polytopes, 3rd ed., Dover, NY, 1973, pp 159-162

Cf. A130296, A000012, A034856 (row sums), A130302 (commuted matrix product)

Clear[e, n, k];
e[n_, 0] := 2*n - 1;
e[n_, k_] := 0 /; k >= n;
e[n_, k_] := (e[n - 1, k]*e[n, k - 1] + 1)/e[n - 1, k - 1];
Table[Table[e[n, k], {k, 0, n - 1}], {n, 1, 10}];
Flatten[%]

A130296 * A000012 as infinite lower triangular matrices. (1,3,5,...) as the left border; (1,2,3,...) in all other columns.

e(n,k)= (e(n - 1, k)*e(n, k - 1) + 1)/e(n - 1, k - 1)

Additional comments from Roger L. Bagula and Gary W. Adamson, Mar 28 2009

A131336 A131334 * A000012.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 3, 3, 2, 1, 5, 4, 4, 2, 1, 8, 8, 7, 5, 3, 1, 13, 12, 12, 9, 7, 3, 1, 21, 21, 20, 17, 14, 8, 4, 1, 34, 33, 33, 29, 26, 17, 11, 4, 1, 55, 55, 54, 50, 46, 34, 25, 12, 5, 1

Offset: 1

Gary W. Adamson, Jun 29 2007

Left border = Fibonacci numbers. Row sums = A131337.

			First few rows of the triangle are:
1;
1, 1;
2, 1, 1;
3, 3, 2, 1;
5, 4, 4, 2, 1;
8, 8, 7, 5, 3, 1;
13, 12, 12, 9, 7, 3, 1;
21, 21, 20 17, 14, 8, 4, 1;
...

Cf. A131334, A131332, A131333, A131335, A131337.

A131334 * A000012, (partial row sums of A131334 starting from the right).

A131410 A127647 * A000012.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 3, 3, 3, 3, 5, 5, 5, 5, 5, 8, 8, 8, 8, 8, 8, 13, 13, 13, 13, 13, 13, 13, 21, 21, 21, 21, 21, 21, 21, 21, 34, 34, 34, 34, 34, 34, 34, 34, 34, 55, 55, 55, 55, 55, 55, 55, 55, 55, 55, 89, 89, 89, 89, 89, 89, 89, 89, 89, 89, 89, 144, 144, 144

Offset: 1

Gary W. Adamson, Jul 08 2007

Row sums = A045925, n*Fib(n): (1, 2, 6, 12, 25, 48,...).

A104763 = A000012 * A127647.

			First few rows of the triangle are:
1;
1, 1;
2, 2, 2;
3, 3, 3, 3;
5, 5, 5, 5, 5;
8, 8, 8, 8, 8, 8;
...

Reinhard Zumkeller, Rows n = 1..120 of triangle, flattened

Cf. A127647, A045925, A104763, A000045.

a131410 n k = a131410_tabl !! (n-1) !! (n-1)
a131410_row n = a131410_tabl !! (n-1)
a131410_tabl = zipWith replicate [1..] $ tail a000045_list
-- Reinhard Zumkeller, Oct 07 2012

Table[Fibonacci[n], {n, 15}, {n}] // Flatten (* Vincenzo Librandi, Jan 28 2017 *)

A127647 * A000012 as infinite lower triangular matrices.
Partial sums of A127647 starting from the right, read by rows.
By rows, F(n) occurs n times.

A131437 (A000012 * A131436) + (A131436 * A000012) - A000012.

Original entry on oeis.org

1, 3, 5, 7, 9, 13, 15, 17, 21, 29, 31, 33, 37, 45, 61, 63, 65, 69, 77, 93, 125, 127, 129, 133, 141, 157, 189, 253, 255, 257, 261, 269, 285, 317, 381, 509, 511, 513, 517, 525, 541, 573, 637, 765, 1021, 1023, 1025, 1029, 1037, 1053, 1085, 1149, 1277, 1533, 2045

Offset: 1

Gary W. Adamson, Jul 11 2007

Left column = 2^n - 1; right border = A036563, 2^(n+1) - 3: (1, 5, 13, 29, 61, 125, ...). Row sums = A131438: (1, 8, 29, 82, 207, 492, 1129, ...).

			First few rows of the triangle are:
1;
3, 5;
7, 9, 13;
15, 17, 21, 29;
31, 33, 37, 45, 61;
63, 65, 69, 77, 93, 125;
...

Cf. A036563, A131436, A131438, A131439.

A000012 := proc(n,k)
        1 ;
end proc:
A131436 := proc(n,k)
        if k = n then
                2^n-1 ;
        else
                0;
        end if;
end proc:
A131437 := proc(n,k)
        add( A000012(n,i)*A131436(i,k) + A131436(n,i)*A000012(i,k),i=k..n) -1 ;
end proc:
seq(seq(A131437(n,k),k=1..n),n=1..15) ; # R. J. Mathar, Sep 24 2011

(A000012 * A131436) + (A131436 * A000012) - A000012; as infinite lower triangular matrices.

Corrected by R. J. Mathar, Sep 24 2011

A131843 Triangle read by rows: 2*A131821 - A000012.

Original entry on oeis.org

1, 3, 3, 5, 1, 5, 7, 1, 1, 7, 9, 1, 1, 1, 9, 11, 1, 1, 1, 1, 11, 13, 1, 1, 1, 1, 1, 13, 15, 1, 1, 1, 1, 1, 1, 15, 17, 1, 1, 1, 1, 1, 1, 1, 17, 19, 1, 1, 1, 1, 1, 1, 1, 1, 19, 21, 1, 1, 1, 1, 1, 1, 1, 1, 1, 21, 23, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 23, 25, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 25, 27, 1, 1, 1, 1, 1

Offset: 0

Gary W. Adamson, Jul 21 2007

Row sums = 5*n + 1, A016861: (1, 6, 11, 16, 21, ...).

			First few rows of the triangle are:
   1;
   3,  3;
   5,  1,  5;
   7,  1,  1,  7;
   9,  1,  1,  1,  9;
  11,  1,  1,  1,  1, 11;
  ...

B. D. Swan, Table of n, a(n) for n = 0..100000

Cf. A016861, A131821.

2*A131821 - A000012 as an infinite lower triangular matrix. 2*n - 1, (1, 3, 5, 7, ...) as right and left borders, with the rest zeros.

Corrected and edited by B. D. Swan (bdswan(AT)gmail.com), Dec 20 2008

cp's OEIS Frontend

A131333 A131332 * A000012.

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A131334 A000012(signed) * A065941.

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A131335 A000012 * A131334.

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A132823 A007318 + 2*A103451 - 2*A000012.

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A058393 A square array based on 1^n (A000012) with each term being the sum of 2 consecutive terms in the previous row.

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A130303 A130296 * A000012.

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A131336 A131334 * A000012.

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A131410 A127647 * A000012.

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Haskell

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A131437 (A000012 * A131436) + (A131436 * A000012) - A000012.

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A132823 A007318 + 2A103451 - 2A000012.