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 1-10 of 13 results. Next

A350529 Square array read by antidiagonals downwards: T(n,k) is the number of sequences of length n with terms in 1..k such that no iterated difference is zero, n, k >= 0.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 0, 0, 1, 3, 2, 0, 0, 1, 4, 6, 2, 0, 0, 1, 5, 12, 10, 2, 0, 0, 1, 6, 20, 32, 16, 2, 0, 0, 1, 7, 30, 72, 86, 26, 2, 0, 0, 1, 8, 42, 138, 256, 232, 42, 2, 0, 0, 1, 9, 56, 234, 624, 906, 622, 68, 2, 0, 0
Offset: 0

Views

Author

Pontus von Brömssen, Jan 03 2022

Keywords

Comments

For fixed n, T(n,k) is a quasi-polynomial of degree n in k. For example, T(4,k) = k^4 - (116/27)*k^3 + (25/3)*k^2 + b(k)*k + c(k), where b and c are periodic with period 6.

Examples

			  n\k|  0  1  2   3     4      5       6        7         8         9         10
  ---+--------------------------------------------------------------------------
   0 |  1  1  1   1     1      1       1        1         1         1          1
   1 |  0  1  2   3     4      5       6        7         8         9         10
   2 |  0  0  2   6    12     20      30       42        56        72         90
   3 |  0  0  2  10    32     72     138      234       368       544        770
   4 |  0  0  2  16    86    256     624     1278      2370      4030       6462
   5 |  0  0  2  26   232    906    2790     6900     15096     29536      53678
   6 |  0  0  2  42   622   3180   12366    36964     95494    215146     443464
   7 |  0  0  2  68  1662  11116   54572   197294    601986   1562274    3652850
   8 |  0  0  2 110  4426  38754  240278  1051298   3788268  11325490   30041458
   9 |  0  0  2 178 11774 134902 1056546  5595236  23814458  82024662  246853482
  10 |  0  0  2 288 31316 469306 4643300 29762654 149631992 593798912 2027577296
For n = 4 and k = 3, the following T(4,3) = 16 sequences are counted: 1212, 1213, 1312, 1313, 1323, 2121, 2131, 2132, 2312, 2313, 2323, 3121, 3131, 3132, 3231, 3232.
		

Crossrefs

Rows: A000012 (n=0), A001477 (n=1), A002378 (n=2), A055232 (terms of row n=3 divided by 2).
Columns: A000007 (k=0), A019590 (k=1), A040000 (k=2), A054886 (k=3).

Programs

  • Python
    def A350529_col(k,nmax):
        d = []
        c = [0]*(nmax+1)
        while 1:
            if not d or all(d[-1]):
                c[len(d)] += 1 + (bool(d) and 2*d[0][0] != k+1)
                if len(d) < nmax:
                    d.append([0])
                    for i in range(len(d)-1):
                        d[-1].append(d[-1][-1]-d[-2][i])
            while d and d[-1][0] == k:
                d.pop()
            if not d or len(d) == 1 and 2*d[0][0] >= k: return c
            for i in range(len(d)):
                d[-1][i] += 1

A200155 Number of 0..n arrays x(0..3) of 4 elements with zero 3rd differences.

Original entry on oeis.org

4, 9, 22, 41, 66, 107, 158, 219, 304, 403, 516, 661, 824, 1005, 1226, 1469, 1734, 2047, 2386, 2751, 3172, 3623, 4104, 4649, 5228, 5841, 6526, 7249, 8010, 8851, 9734, 10659, 11672, 12731, 13836, 15037, 16288, 17589, 18994, 20453, 21966, 23591, 25274, 27015
Offset: 1

Views

Author

R. H. Hardin, Nov 13 2011

Keywords

Comments

Row 4 of A200154.

Examples

			Some solutions for n=6:
..3....4....2....0....0....4....4....3....3....3....2....2....6....3....5....3
..3....4....6....4....1....1....5....1....0....0....2....2....1....4....1....4
..4....3....6....6....3....0....4....1....0....1....3....2....1....4....0....3
..6....1....2....6....6....1....1....3....3....6....5....2....6....3....2....0
		

Crossrefs

Cf. A200154.

Formula

Empirical: a(n) = 2*a(n-1) - a(n-2) + 2*a(n-3) - 4*a(n-4) + 2*a(n-5) - a(n-6) + 2*a(n-7) - a(n-8).
Empirical g.f.: x*(4 + x + 8*x^2 - 2*x^3 + 4*x^4 + 2*x^6 - x^7) / ((1 - x)^4*(1 + x + x^2)^2). - Colin Barker, Mar 02 2018

A350530 Square array read by antidiagonals downwards: T(n,k) is the number of sequences of length n with terms in 0..k such that the (n-1)-st difference is zero, but no earlier iterated difference is zero, n, k >= 1.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 0, 0, 1, 3, 0, 0, 0, 1, 4, 2, 0, 0, 0, 1, 5, 4, 0, 0, 0, 0, 1, 6, 8, 0, 0, 0, 0, 0, 1, 7, 12, 4, 0, 0, 0, 0, 0, 1, 8, 18, 12, 8, 4, 0, 0, 0, 0, 1, 9, 24, 28, 36, 28, 4, 0, 0, 0, 0, 1, 10, 32, 52, 84, 116, 48, 16, 0, 0, 0, 0
Offset: 1

Views

Author

Pontus von Brömssen, Jan 03 2022

Keywords

Comments

For fixed n, T(n,k) is a quasi-polynomial of degree n-1 in k. For example, T(4,k) = (8/27)*k^3 - 2*k^2 + b(k)*k + c(k), where b and c are periodic with period 3.

Examples

			Array begins:
  n\k|  0  1  2  3  4   5    6     7     8      9     10
  ---+--------------------------------------------------
   1 |  1  1  1  1  1   1    1     1     1      1      1
   2 |  0  1  2  3  4   5    6     7     8      9     10
   3 |  0  0  0  2  4   8   12    18    24     32     40
   4 |  0  0  0  0  0   4   12    28    52     84    132
   5 |  0  0  0  0  0   8   36    84   176    332    568
   6 |  0  0  0  0  4  28  116   308   704   1396   2548
   7 |  0  0  0  0  4  48  232   728  2104   4940  11008
   8 |  0  0  0  0 16 100  556  1936  7092  19908  49364
   9 |  0  0  0  0 12 176 1348  6588 23356  74228 202504
  10 |  0  0  0  0  8 268 2492 15544 72820 259800 842688
For n = 4 and k = 6, the following T(4,6) = 12 sequences are counted: 1454, 1564, 2125, 2565, 3126, 3236, 4541, 4651, 5212, 5652, 6213, 6323.
		

Crossrefs

Rows: A000012 (n=1), A001477 (n=2), A007590 (n=3).
Columns: A000007 (k=0), A019590 (k=1), A130706 (k=2).

Programs

  • Python
    def A350530_col(k,nmax):
        d = []
        c = [0]*nmax
        while 1:
            if not d or all(d[-1][:-1]):
                if d and d[-1][-1] == 0:
                    c[len(d)-1] += 1 + (0 != 2*d[0][0] != k+1)
                elif len(d) < nmax:
                    d.append([-1])
                    for i in range(len(d)-1):
                        d[-1].append(d[-1][-1]-d[-2][i])
            while d and d[-1][0] == k:
                d.pop()
            if not d or len(d) == 1 and 2*d[0][0] >= k: return c
            for i in range(len(d)):
                d[-1][i] += 1

A200153 Number of 0..7 arrays x(0..n-1) of n elements with zero n-1st difference.

Original entry on oeis.org

1, 8, 32, 158, 636, 3000, 11370, 52166, 194818, 841122, 3338446, 18901856, 57842348, 378064896, 1073497924, 3840517750, 15453709184, 133757694784, 361144763380, 2500726090816, 6776168788838, 17399866594576, 65044240487692
Offset: 1

Views

Author

R. H. Hardin Nov 13 2011

Keywords

Comments

Column 7 of A200154

Examples

			Some solutions for n=6
..5....1....5....1....5....6....4....4....6....0....2....3....7....5....5....5
..0....7....7....0....4....5....0....2....1....0....1....6....5....0....2....3
..3....4....3....0....7....1....3....3....6....3....0....7....5....0....4....5
..6....2....2....0....7....0....4....3....6....3....2....5....4....2....4....6
..5....3....4....1....4....3....2....2....0....0....5....2....3....4....1....4
..0....1....0....6....5....6....4....4....1....0....2....3....7....5....0....0
		

A200156 Number of 0..n arrays x(0..4) of 5 elements with zero 4th difference.

Original entry on oeis.org

2, 15, 40, 103, 202, 381, 636, 1033, 1550, 2287, 3212, 4451, 5946, 7869, 10140, 12969, 16238, 20211, 24744, 30147, 36222, 43349, 51296, 60493, 70646, 82267, 95016, 109467, 125206, 142897, 162076, 183477, 206546, 232123, 259596, 289879, 322262
Offset: 1

Views

Author

R. H. Hardin, Nov 13 2011

Keywords

Comments

Row 5 of A200154.

Examples

			Some solutions for n=6:
..6....4....1....0....2....6....2....2....4....6....6....0....6....6....2....4
..4....5....1....6....0....1....2....3....6....2....5....6....6....5....3....4
..1....3....1....6....2....0....0....2....3....0....2....6....4....4....2....5
..0....1....2....4....4....2....0....1....0....0....1....3....3....4....2....5
..4....2....5....4....2....6....6....2....2....2....6....0....6....6....6....2
		

Crossrefs

Cf. A200154.

Formula

Empirical: a(n) = a(n-1) +a(n-2) +a(n-4) -3*a(n-5) -a(n-6) +a(n-8) +3*a(n-9) -a(n-10) -a(n-12) -a(n-13) +a(n-14).
Empirical g.f.: x*(2 + 13*x + 23*x^2 + 48*x^3 + 57*x^4 + 67*x^5 + 60*x^6 + 48*x^7 + 26*x^8 + 11*x^9 + x^10 - x^12 + x^13) / ((1 - x)^5*(1 + x)^3*(1 + x^2)^2*(1 + x + x^2)). - Colin Barker, May 17 2018

A200157 Number of 0..n arrays x(0..5) of 6 elements with zero 5th difference.

Original entry on oeis.org

8, 39, 112, 275, 730, 1589, 3000, 5181, 8350, 13871, 21588, 32019, 45682, 63225, 88580, 120521, 159996, 208123, 266020, 342493, 433532, 540855, 666180, 811455, 992870, 1201725, 1440468, 1711817, 2018490, 2387707, 2803024, 3268059, 3786430
Offset: 1

Views

Author

R. H. Hardin Nov 13 2011

Keywords

Comments

Row 6 of A200154

Examples

			Some solutions for n=6
..0....2....1....6....5....1....3....2....6....5....4....5....1....2....6....6
..5....4....3....1....3....0....3....4....3....4....4....5....1....0....6....1
..5....5....5....4....5....3....2....3....1....0....2....4....1....0....2....0
..4....4....5....4....4....4....1....4....3....1....3....3....2....1....0....0
..4....2....4....0....0....2....1....6....6....6....6....2....3....2....1....1
..5....2....6....1....0....1....3....2....1....5....4....0....1....2....1....6
		

Formula

Empirical: a(n) = 2*a(n-1) -a(n-2) +2*a(n-5) -4*a(n-6) +2*a(n-7) +a(n-10) -2*a(n-11) +a(n-12) -4*a(n-15) +8*a(n-16) -4*a(n-17) +a(n-20) -2*a(n-21) +a(n-22) +2*a(n-25) -4*a(n-26) +2*a(n-27) -a(n-30) +2*a(n-31) -a(n-32)

A200158 Number of 0..n arrays x(0..6) of 7 elements with zero 6th difference.

Original entry on oeis.org

2, 45, 182, 685, 2036, 5153, 11370, 23035, 43284, 76523, 129052, 208597, 324900, 491519, 723994, 1041095, 1467042, 2029069, 2759166, 3697749, 4888510, 6381185, 8237500, 10523625, 13313648, 16697587, 20769202, 25633157, 31415424, 38246327
Offset: 1

Views

Author

R. H. Hardin Nov 13 2011

Keywords

Comments

Row 7 of A200154

Examples

			Some solutions for n=6
..1....3....1....4....0....2....1....5....0....2....2....1....5....5....2....0
..6....0....1....6....6....5....5....6....3....1....0....6....3....0....0....6
..5....1....6....3....3....3....0....0....2....0....3....6....0....3....2....2
..2....1....5....4....1....2....0....1....2....2....1....3....3....5....1....1
..0....1....1....6....2....3....2....4....2....3....0....2....5....4....1....4
..1....3....0....4....4....4....1....2....1....1....5....5....1....2....5....6
..6....5....0....1....5....2....5....3....4....5....3....5....4....2....3....2
		

Formula

Empirical: a(n) = 2*a(n-1) -a(n-2) +2*a(n-6) -4*a(n-7) +2*a(n-8) -a(n-12) +2*a(n-13) -a(n-14) +2*a(n-15) -4*a(n-16) +2*a(n-17) +a(n-20) -6*a(n-21) +9*a(n-22) -4*a(n-23) -2*a(n-26) +6*a(n-27) -6*a(n-28) +2*a(n-29) -a(n-30) +2*a(n-31) -2*a(n-33) +a(n-34) -2*a(n-35) +6*a(n-36) -6*a(n-37) +2*a(n-38) +4*a(n-41) -9*a(n-42) +6*a(n-43) -a(n-44) -2*a(n-47) +4*a(n-48) -2*a(n-49) +a(n-50) -2*a(n-51) +a(n-52) -2*a(n-56) +4*a(n-57) -2*a(n-58) +a(n-62) -2*a(n-63) +a(n-64)

A200147 Number of 0..1 arrays x(0..n-1) of n elements with zero n-1st difference.

Original entry on oeis.org

1, 2, 2, 4, 2, 8, 2, 16, 6, 32, 2, 64, 2, 144, 14, 256, 2, 512, 2, 1024, 6, 2048, 2, 4096, 50, 8192, 6, 16384, 2, 34816, 2, 66176, 6, 147456, 130, 262168, 2, 524288, 38, 1048576, 2, 2127872, 2, 4194304, 134, 8388608, 2, 17825792, 4098, 33554432, 6, 67108864
Offset: 1

Views

Author

R. H. Hardin, Nov 13 2011

Keywords

Comments

Column 1 of A200154.

Examples

			All solutions for n=6
..0....0....0....1....1....0....1....1
..0....1....0....1....0....1....0....1
..1....1....0....0....1....0....0....1
..1....1....0....0....1....0....0....1
..0....1....0....1....0....1....0....1
..0....0....0....1....1....0....1....1
		

Crossrefs

Cf. A200154.

Extensions

a(38)-a(52) added from Ionascu link by Michel Marcus, Oct 10 2016

A200148 Number of 0..2 arrays x(0..n-1) of n elements with zero n-1st difference.

Original entry on oeis.org

1, 3, 5, 9, 15, 39, 45, 129, 149, 243, 369, 1365, 2061, 5901, 5951, 7785, 10519, 42519, 59473, 213561, 157559, 278835, 503513, 1553481, 1051639, 2491707, 2981747, 6622209, 10056593, 34139241
Offset: 1

Views

Author

R. H. Hardin Nov 13 2011

Keywords

Comments

Column 2 of A200154

Examples

			Some solutions for n=6
..0....0....1....1....2....2....2....1....2....2....2....0....2....0....1....1
..2....1....0....2....2....0....2....2....1....0....0....0....0....0....0....0
..1....2....0....0....1....1....2....1....0....0....0....0....2....0....0....1
..1....2....0....0....0....1....2....0....0....0....1....0....2....1....1....1
..2....1....0....2....0....0....2....0....1....0....2....0....0....2....2....0
..0....0....1....1....2....2....2....1....2....2....2....0....2....0....1....1
		

A200149 Number of 0..3 arrays x(0..n-1) of n elements with zero n-1st difference.

Original entry on oeis.org

1, 4, 8, 22, 40, 112, 182, 688, 844, 2090, 2478, 21056, 20762, 112864, 113870, 204796, 141246, 2255872, 1610768, 18946704, 15066098, 20640826, 12378248, 332333696, 279217826, 531539936, 618937930, 1715570198, 330283576, 20131457680
Offset: 1

Views

Author

R. H. Hardin Nov 13 2011

Keywords

Comments

Column 3 of A200154

Examples

			Some solutions for n=6
..0....1....3....0....3....3....0....2....0....1....0....1....3....3....2....2
..3....3....1....2....3....3....2....3....3....2....3....3....3....2....3....0
..3....2....1....3....1....3....0....1....1....3....0....3....1....1....0....2
..2....2....2....2....1....2....0....1....0....3....0....3....0....0....0....3
..1....3....3....0....3....1....2....3....1....2....3....3....1....0....3....2
..0....1....3....0....3....3....0....2....0....1....0....1....3....3....2....2
		
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