cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A125781 Rectangular table, read by antidiagonals, defined by the following rule: start with all 1's in row zero; from then on, row n+1 equals the partial sums of row n excluding terms in columns k = m*(m+1)/2 - 2 (m>=2).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 9, 8, 4, 1, 1, 24, 23, 14, 5, 1, 1, 77, 76, 52, 21, 6, 1, 1, 295, 294, 217, 91, 29, 7, 1, 1, 1329, 1328, 1033, 433, 141, 39, 8, 1, 1, 6934, 6933, 5604, 2307, 739, 216, 50, 9, 1, 1, 41351, 41350, 34416, 13804, 4276, 1274, 306, 62, 10, 1, 1
Offset: 0

Views

Author

Paul D. Hanna, Dec 09 2006

Keywords

Comments

Generated by a method similar to Moessner's factorial triangle (A125714).

Examples

			Rows are partial sums excluding terms in columns k = {1,4,8,13,...}:
row 2 = partial sums of [1, 3,4, 6,7,8, 10,11,12,13, ...];
row 3 = partial sums of [1, 8,14, 29,39,50, 75,90,106,123, ...];
row 4 = partial sums of [1, 23,52, 141,216,306, 535,695,876,1079,...].
The terms that are excluded in the partial sums are shown enclosed in
parenthesis in the table below. Rows of this table begin:
1,(1), 1, 1,(1), 1, 1, 1,(1), 1, 1, 1, 1,(1), 1, 1, 1, ...;
1,(2), 3, 4,(5), 6, 7, 8,(9), 10, 11, 12, 13,(14), 15, 16, 17, ...;
1,(4), 8, 14,(21), 29, 39, 50,(62), 75, 90, 106, 123,(141), 160, 181,.;
1,(9), 23, 52,(91), 141, 216, 306,(412), 535, 695, 876, 1079,(1305),..;
1,(24), 76, 217,(433), 739, 1274, 1969,(2845), 3924, 5479, 7335,...;
1,(77), 294, 1033,(2307), 4276, 8200, 13679,(21014), 30534, 45528,...;
1,(295), 1328, 5604,(13804), 27483, 58017, 103545,(167868), 255305,...;
1,(1329), 6933, 34416,(92433), 195978, 451283, 855463,(1454823),...;
1,(6934), 41350, 237328,(688611), 1544074, 3847960, 7700971,...;
1,(41351), 278679, 1822753,(5670713), 13371684, 35818351, 75299744,...;
1,(278680), 2101433, 15473117,(51291468), 126591212, 362337006,...;
1,(2101434), 17574551, 144165763,(506502769), 1303252476,...;
1,(17574552), 161740315, 1464992791,(5430460072), 14517950305,...;
Column 1 of this table equals column 1 of triangle A091351;
triangle A091351 begins:
1;
1, 1;
1, 2, 1;
1, 4, 3, 1;
1, 9, 9, 4, 1;
1, 24, 30, 16, 5, 1;
1, 77, 115, 70, 25, 6, 1;
1, 295, 510, 344, 135, 36, 7, 1;
1, 1329, 2602, 1908, 805, 231, 49, 8, 1; ...
where column k of A091351 = row sums of matrix power A091351^k for k>=0.

Crossrefs

Cf. A091351, A091352; columns: A125782, A125783, A125784, A125785, A125786; diagonals: A125787, A125788; A125789 (antidiagonal sums), A125714.

Programs

  • PARI
    {T(n,k)=local(A=0,b=2,c=0,d=0);if(n==0,A=1, until(d>k,if(c==b*(b+1)/2-2,b+=1,A+=T(n-1,c);d+=1);c+=1));A}

Formula

Surprisingly, column 1 equals A091352 = column 1 of triangle A091351, in which column k equals row sums of the matrix power A091351^k. Column 3 of this table also equals column 1 of matrix power A091351^2.

A125782 Column 2 of table A125781.

Original entry on oeis.org

1, 3, 8, 23, 76, 294, 1328, 6933, 41350, 278679, 2101433, 17574551, 161740315, 1626733107, 17771416520, 209739328923, 2661301094007, 36148700652162, 523597247829866, 8059284921781891, 131408547139817540
Offset: 0

Views

Author

Paul D. Hanna, Dec 09 2006

Keywords

Crossrefs

Cf. A091351; other columns: A091352, A125783, A125784, A125785, A125786.

Formula

a(n) = A091352(n+1) - 1.

A125783 Column 3 of table A125781; also, equals column 1 of matrix power A091351^2.

Original entry on oeis.org

1, 4, 14, 52, 217, 1033, 5604, 34416, 237328, 1822753, 15473117, 144165763, 1464992791, 16144683412, 191967912402, 2451561765083, 33487399558154, 487448547177703, 7535687673952024, 123349262218035648
Offset: 0

Views

Author

Paul D. Hanna, Dec 09 2006

Keywords

Comments

Column k of triangle A091351 = row sums of matrix power A091351^k for k>=0.

Crossrefs

Cf. A091351; other columns: A091352, A125782, A125784, A125785, A125786.

Formula

a(n) = A091352(n+2) - A091352(n+1) - 1.

A125785 Column 5 of table A125781.

Original entry on oeis.org

1, 6, 29, 141, 739, 4276, 27483, 195978, 1544074, 13371684, 126591212, 1303252476, 14517950305, 174196495882, 2241822436160, 30826098464147, 451299846525541, 7012090426122158, 115289977296253757, 2000463474160276658
Offset: 0

Views

Author

Paul D. Hanna, Dec 09 2006

Keywords

Comments

A091352 equals column 1 of both table A125781 and triangle A091351.

Examples

			A091352 begins: [1, 2, 4, 9, 24, 77, 295, 1329, 6934, 41351, ...];
a(5) = A091352(8) - 2*A091352(7) = 6934 - 2*1329 = 4276.

Crossrefs

Cf. A125781; other columns: A091352, A125782, A125783, A125784, A125786.

Formula

a(n) = A091352(n+2) - 2*A091352(n+1). a(n) = A125782(n+1) - A125783(n+1).

A125786 Column 6 of table A125781.

Original entry on oeis.org

1, 7, 39, 216, 1274, 8200, 58017, 451283, 3847960, 35818351, 362337006, 3965467281, 46749441514, 591291743032, 7993582141984, 115104783083605, 1759853074058289, 28485332959460764, 486811835886953020
Offset: 0

Views

Author

Paul D. Hanna, Dec 09 2006

Keywords

Comments

Column k of triangle A091351 = row sums of matrix power A091351^k for k>=0.

Examples

			a(n) = A125784(n+1) - A125783(n+1) for n>0:
A125784 begins: 1, 5, 21, 91, 433, 2307, 13804, 92433, 688611, ...;
A125783 begins: 1, 4, 14, 52, 217, 1033, 5604, 34416, 237328, ...;
term-by-term differences form this sequence.
This sequence can also be derived from the matrix square A091351^2:
1;
2, 1;
4, 4, [1];
9, 14, [6, 1];
24, 52, [30, 8, 1];
77, 217, [153, 52, 10, 1];
295, 1033, [845, 336, 80, 12, 1];
1329, 5604, [5152, 2294, 625, 114, 14, 1]; ...
the terms enclosed in square barackets sum to equal this sequence.

Crossrefs

Cf. A125781; other columns: A091352, A125782, A125783, A125784, A125785.

Formula

a(n) = Sum_{k=0..n} [A091351^2](n+2,k+2) where A091351^2 is the matrix square of A091351.
Showing 1-5 of 5 results.

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