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.

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A223592 T(n,k) = Number of n X k 0..1 arrays with every row having the same least squares slope fit to a straight line, and every column the same least squares slope fit to a straight line, with a single point array taken as having zero slope.

Original entry on oeis.org

2, 4, 4, 8, 6, 8, 16, 10, 10, 16, 32, 18, 32, 18, 32, 64, 34, 44, 44, 34, 64, 128, 66, 192, 56, 192, 66, 128, 256, 130, 296, 212, 212, 296, 130, 256, 512, 258, 1744, 316, 1664, 316, 1744, 258, 512, 1024, 514, 2838, 1788, 2008, 2008, 1788, 2838, 514, 1024, 2048, 1026
Offset: 1

Views

Author

R. H. Hardin, Mar 22 2013

Keywords

Comments

Table starts
....2....4......8.....16.......32.......64........128........256........512
....4....6.....10.....18.......34.......66........130........258........514
....8...10.....32.....44......192......296.......1744.......2838......18736
...16...18.....44.....56......212......316.......1788.......2878......18844
...32...34....192....212.....1664.....2008......30512......38758.....812928
...64...66....296....316.....2008.....2352......31192......39542.....804584
..128..130...1744...1788....30512....31192....1123232.....998326...70377392
..256..258...2838...2878....38758....39542.....998326.....929038...50098958
..512..514..18736..18844...812928...804584...70377392...50098958.9405893184
.1024.1026..32016..32116..1167112..1161184...67719592...52979558.6185200888
.2048.2050.221344.221652.27692576.27446296.7247150528.4583585270
.4096.4098.388830.389110.42393166.42224950.7292561038

Examples

			Some solutions for n=3, k=4
..1..0..1..1....0..0..0..0....0..0..0..0....1..0..0..1....1..1..0..1
..0..0..1..0....1..1..1..1....0..1..1..0....0..0..0..0....1..1..0..1
..1..0..1..1....1..1..1..1....0..0..0..0....1..0..0..1....1..1..0..1

Links

Crossrefs

Column 1 is A000079.
Column 2 is A052548.

A254030 a(n) = 1*4^n + 2*3^n + 3*2^n + 4*1^n.

Original entry on oeis.org

10, 20, 50, 146, 470, 1610, 5750, 21146, 79430, 303050, 1169750, 4554746, 17852390, 70322090, 278050550, 1102537946, 4381257350, 17438542730, 69495104150, 277204002746, 1106488342310, 4418973508970, 17654960746550
Offset: 0

Views

Author

Luciano Ancora, Jan 26 2015

Keywords

Comments

This is the sequence of fourth terms of "second partial sums of m-th powers".

Links

Crossrefs

Cf. A052548, A254028, A254031, A254144, A254145, A254146.

Programs

  • Maple
    seq(add(i*(5 - i)^n, i = 1..4), n = 0..20); # Peter Bala, Jan 31 2017
  • Mathematica
    Table[3 2^n + 2^(2 n) + 2 3^n + 4, {n, 0, 25}] (* Bruno Berselli, Jan 27 2015 *)
LinearRecurrence[{10,-35,50,-24},{10,20,50,146},30] (* Harvey P. Dale, Jun 06 2020 *)
  • PARI
    Vec(-2*(77*x^3-100*x^2+40*x-5)/((x-1)*(2*x-1)*(3*x-1)*(4*x-1))  + O(x^100)) \\ Colin Barker, Jan 26 2015

Formula

G.f.: -2*(77*x^3-100*x^2+40*x-5) / ((x-1)*(2*x-1)*(3*x-1)*(4*x-1)). - Colin Barker, Jan 26 2015
From Peter Bala, Jan 31 2016: (Start)
a(n) = (x + 1)*( Bernoulli(n + 1, x + 1) - Bernoulli(n + 1, 1) )/(n + 1) - ( Bernoulli(n + 2, x + 1) - Bernoulli(n + 2, 1) )/(n + 2) at x = 4.
a(n) = 1/3!*Sum_{k = 0..n} (-1)^(k+n)*(k + 5)!*Stirling2(n,k)/
((k + 1)*(k + 2)). (End)
E.g.f.: exp(x)*(4 + 3*exp(x) + 2*exp(2*x) + exp(3*x)). - Stefano Spezia, May 19 2025

A254031 a(n) = 1*5^n + 2*4^n + 3*3^n + 4*2^n + 5*1^n.

Original entry on oeis.org

15, 35, 105, 371, 1449, 6035, 26265, 117971, 542409, 2538515, 12044025, 57756371, 279305769, 1359736595, 6654800985, 32708239571, 161307227529, 797687136275, 3953299529145, 19626731023571, 97576919443689, 485664640673555
Offset: 0

Views

Author

Luciano Ancora, Jan 26 2015

Keywords

Comments

This is the sequence of fifth terms of "second partial sums of m-th powers".

Links

Crossrefs

Cf. A052548, A254028, A254030, A254144, A254145, A254146.

Programs

  • Maple
    seq(add(i*(6 - i)^n, i = 1..5), n = 0..20); # Peter Bala, Jan 31 2017
  • Mathematica
    Table[2^(n + 2) + 2^(2 n + 1) + 3^(n + 1) + 5^n + 5, {n, 0, 25}] (* Bruno Berselli, Jan 27 2015 *)
LinearRecurrence[{15,-85,225,-274,120},{15,35,105,371,1449},30] (* Harvey P. Dale, Jan 24 2022 *)
  • PARI
    Vec(-(1044*x^4-1604*x^3+855*x^2-190*x+15)/((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)) + O(x^100)) \\ Colin Barker, Jan 26 2015

Formula

G.f.: -(1044*x^4 - 1604*x^3 + 855*x^2 - 190*x + 15) / ((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)). - Colin Barker, Jan 26 2015
From Peter Bala, Jan 31 2016: (Start)
a(n) = (x + 1)*( Bernoulli(n + 1, x + 1) - Bernoulli(n + 1, 1) )/(n + 1) - ( Bernoulli(n + 2, x + 1) - Bernoulli(n + 2, 1) )/(n + 2) at x = 5.
a(n) = (1/4!)*Sum_{k = 0..n} (-1)^(k+n)*(k + 6)!*Stirling2(n,k)/
((k + 1)*(k + 2)). (End)

A254144 a(n) = 1*6^n + 2*5^n + 3*4^n + 4*3^n + 5*2^n + 6*1^n.

Original entry on oeis.org

21, 56, 196, 812, 3724, 18236, 93436, 494732, 2685004, 14851676, 83384476, 473755052, 2717541484, 15709845116, 91395715516, 534498925772, 3139343105164, 18504595174556, 109397060622556, 648335998054892, 3850205790608044
Offset: 0

Views

Author

Luciano Ancora, Jan 26 2015

Keywords

Comments

This is the sequence of sixth terms of "second partial sums of m-th powers".

Links

Crossrefs

Cf. A052548, A254028, A254030, A254031, A254145, A254146.

Programs

  • Maple
    seq(add(i*(7 - i)^n, i = 1..6), n = 0..20); # Peter Bala, Jan 31 2017
  • Mathematica
    Table[5 2^n + 3 4^n + 4 3^n + 2 5^n + 6^n + 6, {n, 0, 25}] (* Bruno Berselli, Jan 27 2015 *)
  • PARI
    Vec(-(8028*x^5-13916*x^4+8939*x^3-2695*x^2+385*x-21)/((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)*(6*x-1)) + O(x^100)) \\ Colin Barker, Jan 26 2015

Formula

G.f.: -(8028*x^5 - 13916*x^4 + 8939*x^3 - 2695*x^2 + 385*x - 21) / ((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)*(6*x-1)). - Colin Barker, Jan 26 2015
From Peter Bala, Jan 31 2016: (Start)
a(n) = (x + 1)*( Bernoulli(n + 1, x + 1) - Bernoulli(n + 1, 1) )/(n + 1) - ( Bernoulli(n + 2, x + 1) - Bernoulli(n + 2, 1) )/(n + 2) at x = 6.
a(n) = (1/5!)*Sum_{k = 0..n} (-1)^(k+n)*(k + 7)!*Stirling2(n,k)/ ((k + 1)*(k + 2)). (End)

A254145 a(n) = 1*7^n + 2*6^n + 3*5^n + 4*4^n + 5*3^n + 6*2^n + 7*1^n.

Original entry on oeis.org

28, 84, 336, 1596, 8400, 47244, 278256, 1695036, 10592400, 67518444, 437200176, 2867080476, 18997064400, 126948964044, 854359702896, 5783851121916, 39350309552400, 268842017200044, 1843254419626416, 12675940450459356
Offset: 0

Views

Author

Luciano Ancora, Jan 26 2015

Keywords

Comments

This is the sequence of seventh terms of "second partial sums of m-th powers".

Links

Crossrefs

Cf. A052548, A254028, A254030, A254031, A254144, A254146.

Programs

  • Maple
    seq(add(i*(8 - i)^n, i = 1..7), n = 0..20); # Peter Bala, Jan 31 2017
  • Mathematica
    Table[6 2^n + 4 4^n + 5 3^n + 2 6^n + 3 5^n + 7^n + 7, {n, 0, 25}] (*  *)
LinearRecurrence[{28,-322,1960,-6769,13132,-13068,5040},{28,84,336,1596,8400,47244,278256},30] (* or *) Table[Total[ Range[ 7]Range[ 7,1,-1]^n],{n,0,20}] (* Harvey P. Dale, Jun 21 2016 *)
  • PARI
    Vec(-4*(17316*x^6 -32926*x^5 +24199*x^4 -8911*x^3 +1750*x^2 -175*x +7) / ((x -1)*(2*x -1)*(3*x -1)*(4*x -1)*(5*x -1)*(6*x -1)*(7*x -1))  + O(x^100)) \\ Colin Barker, Jan 26 2015

Formula

G.f.: -4*(17316*x^6 - 32926*x^5 + 24199*x^4 - 8911*x^3 + 1750*x^2 - 175*x + 7) / ((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)*(6*x-1)*(7*x-1)). - Colin Barker, Jan 26 2015
From Peter Bala, Jan 31 2016: (Start)
a(n) = (x + 1)*( Bernoulli(n + 1, x + 1) - Bernoulli(n + 1, 1) )/(n + 1) - ( Bernoulli(n + 2, x + 1) - Bernoulli(n + 2, 1) )/(n + 2) at x = 7.
a(n) = (1/6!)*Sum_{k = 0..n} (-1)^(k+n)*(k + 8)!*Stirling2(n,k)/ ((k + 1)*(k + 2)). (End)

A254146 a(n) = 1*8^n + 2*7^n + 3*6^n + 4*5^n + 5*4^n + 6*3^n + 7*2^n + 8*1^n.

Original entry on oeis.org

36, 120, 540, 2892, 17172, 109020, 725220, 4992492, 35277012, 254402940, 1864757700, 13850340492, 103996064052, 787943896860, 6015370201380, 46217575406892, 357036252710292, 2770979252910780, 21591510288112260, 168818732978719692, 1323861500735007732
Offset: 0

Views

Author

Luciano Ancora, Jan 27 2015

Keywords

Comments

This is the sequence of eighth terms of "second partial sums of m-th powers".

Links

Crossrefs

Cf. A052548, A254028, A254030, A254031, A254144, A254145.

Programs

  • Magma
    [7*2^n+5*4^n+8^n+6*3^n+3*6^n+4*5^n+2*7^n+8: n in [0..30]]; // Vincenzo Librandi, Jan 28 2015
  • Maple
    seq(add(i*(9 - i)^n, i = 1..8), n = 0..20); # Peter Bala, Jan 31 2017
  • Mathematica
    Table[7 2^n + 5 4^n + 8^n + 6 3^n + 3 6^n + 4 5^n + 2 7^n + 8, {n, 0, 30}] (* Vincenzo Librandi, Jan 28 2015 *)
LinearRecurrence[{36,-546,4536,-22449,67284,-118124,109584,-40320},{36,120,540,2892,17172,109020,725220,4992492},30] (* Harvey P. Dale, Mar 02 2022 *)
  • PARI
    vector(30, n, n--; 7*2^n + 5*4^n + 8^n + 6*3^n + 3*6^n + 4*5^n + 2*7^n + 8) \\ Colin Barker, Jan 28 2015

Formula

G.f.: -12*(55308*x^7 - 113262*x^6 + 92327*x^5 - 39312*x^4 + 9527*x^3 - 1323*x^2 + 98*x -3) / ((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)*(6*x-1)*(7*x-1)*(8*x-1)). - Colin Barker, Jan 28 2015
From Peter Bala, Jan 31 2016: (Start)
a(n) = (x + 1)*( Bernoulli(n + 1, x + 1) - Bernoulli(n + 1, 1) )/(n + 1) - ( Bernoulli(n + 2, x + 1) - Bernoulli(n + 2, 1) )/(n + 2) at x = 8.
a(n) = 1/7!*Sum_{k = 0..n} (-1)^(k+n)*(k + 9)!*Stirling2(n,k)/ ((k + 1)*(k + 2)). (End)
a(n) = 36*a(n-1)-546*a(n-2)+4536*a(n-3)-22449*a(n-4)+67284*a(n-5)-118124*a(n-6)+109584*a(n-7)-40320*a(n-8). - Wesley Ivan Hurt, May 24 2021

A334622 A(n,k) is the sum of the k-th powers of the descent set statistics for permutations of [n]; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 2, 4, 1, 1, 2, 6, 8, 1, 1, 2, 10, 24, 16, 1, 1, 2, 18, 88, 120, 32, 1, 1, 2, 34, 360, 1216, 720, 64, 1, 1, 2, 66, 1576, 14460, 24176, 5040, 128, 1, 1, 2, 130, 7224, 190216, 994680, 654424, 40320, 256, 1, 1, 2, 258, 34168, 2675100, 46479536, 109021500, 23136128, 362880, 512
Offset: 0

Views

Author

Alois P. Heinz, Sep 09 2020

Keywords

Examples

			Square array A(n,k) begins:
   1,   1,     1,      1,        1,          1,            1, ...
   1,   1,     1,      1,        1,          1,            1, ...
   2,   2,     2,      2,        2,          2,            2, ...
   4,   6,    10,     18,       34,         66,          130, ...
   8,  24,    88,    360,     1576,       7224,        34168, ...
  16, 120,  1216,  14460,   190216,    2675100,     39333016, ...
  32, 720, 24176, 994680, 46479536, 2368873800, 128235838496, ...
  ...

Links

Crossrefs

Columns k=0-4 give: A011782, A000142, A060350, A291902, A291903.
Rows n=0+1, 2-3 give: A000012, A007395(k+1), A052548(k+1).
Main diagonal gives A334623.
Cf. A060351, A335545.

Programs

  • Maple
    b:= proc(u, o, t) option remember; expand(`if`(u+o=0, 1,
      add(b(u-j, o+j-1, t+1)*x^floor(2^(t-1)), j=1..u)+
      add(b(u+j-1, o-j, t+1), j=1..o)))
    end:
A:= (n, k)-> (p-> add(coeff(p, x, i)^k, i=0..degree(p)))(b(n, 0$2)):
seq(seq(A(n, d-n), n=0..d), d=0..10);
  • Mathematica
    b[u_, o_, t_] := b[u, o, t] = Expand[If[u + o == 0, 1,
    Sum[b[u - j, o + j - 1, t + 1] x^Floor[2^(t - 1)], {j, 1, u}] +
    Sum[b[u + j - 1, o - j, t + 1], {j, 1, o}]]];
A[n_, k_] := Function[p, Sum[Coefficient[p, x, i]^k, {i, 0, Exponent[p, x]}]][b[n, 0, 0]];
Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Dec 20 2020, after Alois P. Heinz *)

Formula

A(n,k) = Sum_{j=0..ceiling(2^(n-1))-1} A060351(n,j)^k.

A250742 T(n,k)=Number of (n+1)X(k+1) 0..1 arrays with nondecreasing x(i,j)-x(i,j-1) in the i direction and nonincreasing x(i,j)-x(i-1,j) in the j direction.

Original entry on oeis.org

6, 10, 10, 18, 14, 18, 34, 22, 22, 34, 66, 38, 30, 38, 66, 130, 70, 46, 46, 70, 130, 258, 134, 78, 62, 78, 134, 258, 514, 262, 142, 94, 94, 142, 262, 514, 1026, 518, 270, 158, 126, 158, 270, 518, 1026, 2050, 1030, 526, 286, 190, 190, 286, 526, 1030, 2050, 4098, 2054, 1038
Offset: 1

Views

Author

R. H. Hardin, Nov 27 2014

Keywords

Comments

Table starts
....6...10...18...34...66..130..258..514.1026.2050.4098..8194.16386.32770.65538
...10...14...22...38...70..134..262..518.1030.2054.4102..8198.16390.32774.65542
...18...22...30...46...78..142..270..526.1038.2062.4110..8206.16398.32782.65550
...34...38...46...62...94..158..286..542.1054.2078.4126..8222.16414.32798.65566
...66...70...78...94..126..190..318..574.1086.2110.4158..8254.16446.32830.65598
..130..134..142..158..190..254..382..638.1150.2174.4222..8318.16510.32894.65662
..258..262..270..286..318..382..510..766.1278.2302.4350..8446.16638.33022.65790
..514..518..526..542..574..638..766.1022.1534.2558.4606..8702.16894.33278.66046
.1026.1030.1038.1054.1086.1150.1278.1534.2046.3070.5118..9214.17406.33790.66558
.2050.2054.2062.2078.2110.2174.2302.2558.3070.4094.6142.10238.18430.34814.67582

Examples

			Some solutions for n=4 k=4
..0..0..0..0..0....1..0..1..0..1....0..1..0..0..1....1..1..1..1..1
..0..0..0..0..0....1..0..1..0..1....0..1..0..0..1....1..1..1..1..1
..0..0..0..0..0....1..0..1..0..1....0..1..0..0..1....0..0..0..0..0
..1..1..1..1..1....1..0..1..0..1....0..1..0..0..1....0..0..0..0..0
..1..1..1..1..1....1..0..1..0..1....0..1..0..0..1....1..1..1..1..1

Links

Crossrefs

Column 1 is A052548(n+1)
Column 2 is A153972(n+1)
Diagonal is A000918(n+2)

Formula

The constraints apparently result in horizontally or vertically banded arrays, hence:
Empirical: T(n,k) = 2^(k+1)+2^(n+1)-2
Empirical for column k:
k=1: a(n) = 3*a(n-1) -2*a(n-2); a(n) = 2^(n+1) +2
k=2: a(n) = 3*a(n-1) -2*a(n-2); a(n) = 2^(n+1) +6
k=3: a(n) = 3*a(n-1) -2*a(n-2); a(n) = 2^(n+1) +14
k=4: a(n) = 3*a(n-1) -2*a(n-2); a(n) = 2^(n+1) +30
k=5: a(n) = 3*a(n-1) -2*a(n-2); a(n) = 2^(n+1) +62
k=6: a(n) = 3*a(n-1) -2*a(n-2); a(n) = 2^(n+1) +126
k=7: a(n) = 3*a(n-1) -2*a(n-2); a(n) = 2^(n+1) +254

A003311 Write down the numbers from 3 to infinity. Take next number, M say, that has not been crossed off. Counting through the numbers that have not yet been crossed off after that M, cross off the first, (M+1)st, (2M+1)st, (3M+1)st, etc. Repeat. The numbers that are left form the sequence.

Original entry on oeis.org

3, 5, 8, 11, 15, 18, 23, 27, 32, 38, 42, 47, 53, 57, 63, 71, 75, 78, 90, 93, 98, 105, 113, 117, 123, 132, 137, 140, 147, 161, 165, 168, 176, 183, 188, 197, 206, 212, 215, 227, 233, 237, 243, 252, 258, 267, 278, 282, 287, 293, 303, 312, 317, 323
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Examples

			The first few sieving stages are as follows:
  3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 ...
  3 X 5 6 X 8 9 XX 11 12 XX 14 15 XX 17 18 XX 20 ...
  3 X 5 X X 8 9 XX 11 12 XX XX 15 XX 17 18 XX 20 ...
  3 X 5 X X 8 X XX 11 12 XX XX 15 XX 17 18 XX 20 ...
  3 X 5 X X 8 X XX 11 XX XX XX 15 XX 17 18 XX 20 ...
  3 X 5 X X 8 X XX 11 XX XX XX 15 XX XX 18 XX 20 ...

References

  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A003309, A003310, A100464, A003312, A100562, A100585, A052548.

Programs

  • Haskell
    a003311 n = a003311_list !! (n-1)
a003311_list = f [3..] where
   f (x:xs) = x : f (g xs) where
     g zs = us ++ g vs where (_:us, vs) = splitAt x zs
-- Reinhard Zumkeller, Nov 12 2014

Extensions

Entry revised Nov 29 2004

A056469 Number of elements in the continued fraction for Sum_{k=0..n} 1/2^2^k.

Original entry on oeis.org

2, 3, 4, 6, 10, 18, 34, 66, 130, 258, 514, 1026, 2050, 4098, 8194, 16386, 32770, 65538, 131074, 262146, 524290, 1048578, 2097154, 4194306, 8388610, 16777218, 33554434, 67108866, 134217730, 268435458, 536870914, 1073741826, 2147483650
Offset: 0

Views

Author

Benoit Cloitre, Dec 07 2002

Keywords

Comments

Let f_1(x) := 1 - sqrt(1 - x^2) = 2*x^2 + 2*x^4 + 4*x^6 + ... and for n>1 let f_n(x) := f_{n-1}(f_1(x)) = x^(2^n)*(2 + 2^n*x^2 + 2^n*a(n-1)*x^4 + ...). - Michael Somos, Jun 29 2023

Examples

			G.f. = 2 + 3*x + 4*x^2 + 6*x^3 + 10*x^4 + 18*x^5 + 34*x^6 + ... - _Michael Somos_, Jun 29 2023

Links

Crossrefs

Cf. A007400. Apart from initial term, same as A052548. See also A089985.

Programs

  • Magma
    [Floor(2^(n-1)+2): n in [0..60]]; // Vincenzo Librandi, Sep 21 2011
  • Mathematica
    LinearRecurrence[{3,-2},{2,3,4},40] (* Harvey P. Dale, Apr 23 2015 *)
a[ n_] := If[n < 0, 0, Floor[2^n/2] + 2]; (* Michael Somos, Jun 29 2023 *)
  • PARI
    {a(n) = if(n<0, 0, 2^n\2 + 2)}; /* Michael Somos, Jun 29 2023 */
  • Sage
    [floor(gaussian_binomial(n,1,2)+3) for n in range(-1,32)] # Zerinvary Lajos, May 31 2009

Formula

a(0)=2; for n > 0, a(n) = 2^(n-1) + 2 = A052548(n-1) + 2.
a(n) = floor(2^(n-1) + 2). - Vincenzo Librandi, Sep 21 2011
From Colin Barker, Mar 22 2013: (Start)
a(n) = 3*a(n-1) - 2*a(n-2) for n > 2.
G.f.: -(x^2+3*x-2) / ((x-1)*(2*x-1)). (End)
E.g.f.: exp(x)*(2 + sinh(x)). - Stefano Spezia, Oct 19 2023
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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