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.

Previous Showing 11-18 of 18 results.

A229277 Number of ascending runs in {1,...,3}^n.

Original entry on oeis.org

0, 3, 15, 63, 243, 891, 3159, 10935, 37179, 124659, 413343, 1358127, 4428675, 14348907, 46235367, 148272039, 473513931, 1506635235, 4778186031, 15109399071, 47652720147, 149931729243, 470715894135, 1474909801623, 4613015762523, 14403906360531, 44906296300479
Offset: 0

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Author

Alois P. Heinz, Sep 18 2013

Keywords

Links

Crossrefs

Column k=3 of A229079.
Cf. A081038.

Programs

  • Maple
    a:= n-> `if`(n=0, 0, 3^(n-1)*(2*n+1)):
seq(a(n), n=0..30);
  • Mathematica
    a[0] = 0; a[n_] := 3^(n - 1)*(2*n + 1); Array[a, 30, 0] (* Amiram Eldar, May 17 2022 *)

Formula

G.f.: -3*(x-1)*x/(3*x-1)^2.
a(n) = 3^(n-1)*(2*n+1) for n>0, a(0) = 0.
a(n) = 3*A081038(n-1) for n>0.
From Amiram Eldar, May 17 2022: (Start)
Sum_{n>=1} 1/a(n) = 3*(sqrt(3)*arctanh(1/sqrt(3)) - 1).
Sum_{n>=1} (-1)^(n+1)/a(n) = 3 - sqrt(3)*Pi/2. (End)

A229278 Number of ascending runs in {1,...,4}^n.

Original entry on oeis.org

0, 4, 26, 144, 736, 3584, 16896, 77824, 352256, 1572864, 6946816, 30408704, 132120576, 570425344, 2449473536, 10468982784, 44560285696, 188978561024, 798863917056, 3367254360064, 14156212207616, 59373627899904, 248489627877376, 1037938976620544
Offset: 0

Views

Author

Alois P. Heinz, Sep 18 2013

Keywords

Links

Crossrefs

Column k=4 of A229079.

Programs

  • Maple
    a:= n-> `if`(n=0, 0, 2^(2*n-3)*(5*n+3)):
seq(a(n), n=0..30);

Formula

G.f.: -2*(3*x-2)*x/(4*x-1)^2.
a(n) = 2^(2*n-3)*(5*n+3) for n>0, a(0) = 0.

A229279 Number of ascending runs in {1,...,5}^n.

Original entry on oeis.org

0, 5, 40, 275, 1750, 10625, 62500, 359375, 2031250, 11328125, 62500000, 341796875, 1855468750, 10009765625, 53710937500, 286865234375, 1525878906250, 8087158203125, 42724609375000, 225067138671875, 1182556152343750, 6198883056640625, 32424926757812500
Offset: 0

Views

Author

Alois P. Heinz, Sep 18 2013

Keywords

Links

Crossrefs

Column k=5 of A229079.

Programs

  • Maple
    a:= n-> `if`(n=0, 0, 5^(n-1)*(3*n+2)):
seq(a(n), n=0..30);

Formula

G.f.: -5*(2*x-1)*x/(5*x-1)^2.
a(n) = 5^(n-1)*(3*n+2) for n>0, a(0) = 0.

A229280 Number of ascending runs in {1,...,6}^n.

Original entry on oeis.org

0, 6, 57, 468, 3564, 25920, 182736, 1259712, 8538048, 57106944, 377913600, 2479113216, 16144468992, 104485552128, 672625741824, 4310029025280, 27505821597696, 174908814262272, 1108696193630208, 7007637010120704, 44178581150760960, 277868041444786176
Offset: 0

Views

Author

Alois P. Heinz, Sep 18 2013

Keywords

Links

Crossrefs

Column k=6 of A229079.

Programs

  • Maple
    a:= n-> `if`(n=0, 0, 6^(n-1)*(7*n+5)/2):
seq(a(n), n=0..30);

Formula

G.f.: -3*(5*x-2)*x/(6*x-1)^2.
a(n) = 6^(n-1)*(7*n+5)/2 for n>0, a(0) = 0.

A229281 Number of ascending runs in {1,...,7}^n.

Original entry on oeis.org

0, 7, 77, 735, 6517, 55223, 453789, 3647119, 28824005, 224827239, 1735205101, 13276336703, 100843663893, 761270796055, 5716451614013, 42728053589487, 318086621166181, 2359538070441671, 17447288549040525, 128644674234925471, 946108300385970869
Offset: 0

Views

Author

Alois P. Heinz, Sep 18 2013

Keywords

Links

Crossrefs

Column k=7 of A229079.

Programs

  • Maple
    a:= n-> `if`(n=0, 0, 7^(n-1)*(4*n+3)):
seq(a(n), n=0..30);

Formula

G.f.: -7*(3*x-1)*x/(7*x-1)^2.
a(n) = 7^(n-1)*(4*n+3) for n>0, a(0) = 0.

A229282 Number of ascending runs in {1,...,8}^n.

Original entry on oeis.org

0, 8, 100, 1088, 11008, 106496, 999424, 9175040, 82837504, 738197504, 6509559808, 56908316672, 493921239040, 4260607557632, 36558761623552, 312261302288384, 2656420092706816, 22517998136852480, 190277084256403456, 1603281467343896576, 13474770085092524032
Offset: 0

Views

Author

Alois P. Heinz, Sep 18 2013

Keywords

Links

Crossrefs

Column k=8 of A229079.

Programs

  • Maple
    a:= n-> `if`(n=0, 0, 2^(3*n-4)*(9*n+7)):
seq(a(n), n=0..30);

Formula

G.f.: -4*(7*x-2)*x/(8*x-1)^2.
a(n) = 2^(3*n-4)*(9*n+7) for n>0, a(0) = 0.

A229283 Number of ascending runs in {1,...,9}^n.

Original entry on oeis.org

0, 9, 126, 1539, 17496, 190269, 2007666, 20726199, 210450636, 2109289329, 20920706406, 205720279659, 2008387814976, 19487638017189, 188098071296346, 1807266603941919, 17294855095950516, 164918796807813849, 1567655079768657486, 14859368894402912979
Offset: 0

Views

Author

Alois P. Heinz, Sep 18 2013

Keywords

Links

Crossrefs

Column k=9 of A229079.

Programs

  • Maple
    a:= n-> `if`(n=0, 0, 9^(n-1)*(5*n+4)):
seq(a(n), n=0..30);

Formula

G.f.: -9*(4*x-1)*x/(9*x-1)^2.
a(n) = 9^(n-1)*(5*n+4) for n>0, a(0) = 0.
a(n) = 18*a(n-1) - 81*a(n-2). - Wesley Ivan Hurt, May 24 2024

A229284 Number of ascending runs in {1,...,10}^n.

Original entry on oeis.org

0, 10, 155, 2100, 26500, 320000, 3750000, 43000000, 485000000, 5400000000, 59500000000, 650000000000, 7050000000000, 76000000000000, 815000000000000, 8700000000000000, 92500000000000000, 980000000000000000, 10350000000000000000, 109000000000000000000
Offset: 0

Views

Author

Alois P. Heinz, Sep 18 2013

Keywords

Links

Crossrefs

Column k=10 of A229079.

Programs

  • Maple
    a:= n-> `if`(n=0, 0, 10^(n-1)*(11*n+9)/2):
seq(a(n), n=0..30);

Formula

G.f.: -5*(9*x-2)*x/(10*x-1)^2.
a(n) = 10^(n-1)*(11*n+9)/2 for n>0, a(0) = 0.
Previous Showing 11-18 of 18 results.

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