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 1-6 of 6 results.

A244812 The hexagonal spiral of Champernowne, read along the 300-degree ray.

Original entry on oeis.org

1, 1, 0, 6, 0, 8, 1, 3, 5, 3, 4, 4, 6, 1, 5, 9, 9, 1, 2, 6, 2, 1, 7, 7, 1, 2, 3, 7, 6, 2, 9, 6, 7, 3, 7, 6, 4, 4, 6, 5, 7, 5, 5, 3, 6, 6, 6, 1, 1, 7, 7, 9, 2, 8, 0, 6, 9, 0, 3, 5, 0, 4, 1, 7, 0, 2, 9, 3, 3, 6, 3, 4, 4, 0, 3, 5, 9, 6, 8, 2, 7, 4, 8, 7, 1, 9, 9, 0, 8, 2, 1, 4, 2, 9, 4, 3, 9, 4, 2, 7, 6, 4, 7, 7, 2
Offset: 1

Views

Author

Robert G. Wilson v, Jul 06 2014

Keywords

Examples

			see A244807 example section for its diagram.

Crossrefs

Cf. A007376, A244804, A244807, A244808, A244809, A244810, A244811, A244813, A244814, A244815, A244816, A244817, A244818.

Programs

  • Mathematica
    almostNatural[n_, b_] := Block[{m = 0, d = n, i = 1, l, p}, While[m <= d, l = m; m = (b - 1) i*b^(i - 1) + l; i++]; i--; p = Mod[d - l, i]; q = Floor[(d - l)/i] + b^(i - 1); If[p != 0, IntegerDigits[q, b][[p]], Mod[q - 1, b]]]; f[n_] := 12n^2 - 23n + 12 (* see formula section of A244807 *); Array[ almostNatural[ f@#, 10] &, 105]

Formula

(12n^2 - 23n + 12)th almost natural number (A033307), Also see formula section of A056105.

A033577 a(n) = (3*n+1) * (4*n+1).

Original entry on oeis.org

1, 20, 63, 130, 221, 336, 475, 638, 825, 1036, 1271, 1530, 1813, 2120, 2451, 2806, 3185, 3588, 4015, 4466, 4941, 5440, 5963, 6510, 7081, 7676, 8295, 8938, 9605, 10296, 11011, 11750, 12513, 13300, 14111, 14946, 15805, 16688, 17595, 18526, 19481, 20460, 21463
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Also the 120ยบ spoke (or ray) of a hexagonal spiral of Ulam. - Robert G. Wilson v, Jul 06 2014
If two independent real random variables x and y are distributed according to the same exponential distribution with pdf(x) = lambda * exp(-lambda * x) for some lambda > 0, then the probability that 3 <= x/(n*y) < 4 is given by n/a(n) for n>1. - Andres Cicuttin, Dec 11 2016

Examples

			See A056105 example section for hexagonal spiral of Ulam diagram. - _Robert G. Wilson v_, Jul 06 2014

Links

Crossrefs

Subsequence of A281333.
Cf. A003215, A056105, A056106, A056107, A056108, A056109, A244802, A244803, A244804, A244805, A244806.

Programs

Formula

From Colin Barker, Dec 12 2016: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>2.
G.f.: (1 + 17*x + 6*x^2)/(1-x)^3. (End)
E.g.f.: (1 + 19*x + 12*x^2)*exp(x). - G. C. Greubel, Oct 12 2019

Extensions

More terms from Wesley Ivan Hurt, Jul 06 2014

A244805 The 240-degree spoke (or ray) of a hexagonal spiral of Ulam.

Original entry on oeis.org

1, 16, 55, 118, 205, 316, 451, 610, 793, 1000, 1231, 1486, 1765, 2068, 2395, 2746, 3121, 3520, 3943, 4390, 4861, 5356, 5875, 6418, 6985, 7576, 8191, 8830, 9493, 10180, 10891, 11626, 12385, 13168, 13975, 14806, 15661, 16540, 17443, 18370, 19321, 20296, 21295, 22318, 23365, 24436, 25531
Offset: 1

Views

Author

Robert G. Wilson v, Jul 06 2014

Keywords

Comments

Numbers of the form 1 + k/2 + k^2/3 (associated k are in A008588). - Bruno Berselli, Jan 20 2017

Examples

			See A056105 example section for its diagram.

Links

Crossrefs

Cf. A056105, A244802, A056106, A244803, A056107, A244804, A056108, A056109, A244806, A003215, A033577.
Cf. A281333 (1 + floor(n/2) + floor(n^2/3)).

Programs

  • Magma
    [12*n^2-21*n+10: n in [1..50]]; // Wesley Ivan Hurt, Jul 06 2014
  • Maple
    A244805:=n->12*n^2 - 21*n + 10: seq(A244805(n), n=1..50); # Wesley Ivan Hurt, Jul 06 2014
  • Mathematica
    f[n_] := 12 n^2 - 21 n + 10; Array[f, 47]
  • PARI
    vector(50, n, 12*n^2 - 21*n + 10) \\ Michel Marcus, Jul 06 2014
  • PARI
    Vec(x*(1 + 13*x + 10*x^2) / (1 - x)^3 + O(x^50)) \\ Colin Barker, Dec 12 2016

Formula

a(n) = 12*n^2 - 21*n + 10 (see A056105).
From Colin Barker, Dec 12 2016: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>3.
G.f.: x*(1 + 13*x + 10*x^2) / (1 - x)^3.
(End)

A244806 The 180-degree spoke (or ray) of a hexagonal spiral of Ulam.

Original entry on oeis.org

1, 18, 59, 124, 213, 326, 463, 624, 809, 1018, 1251, 1508, 1789, 2094, 2423, 2776, 3153, 3554, 3979, 4428, 4901, 5398, 5919, 6464, 7033, 7626, 8243, 8884, 9549, 10238, 10951, 11688, 12449, 13234, 14043, 14876, 15733, 16614, 17519, 18448, 19401, 20378, 21379, 22404, 23453, 24526, 25623
Offset: 1

Views

Author

Robert G. Wilson v, Jul 06 2014

Keywords

Examples

			See A056105 example section for its diagram.

Links

Crossrefs

Cf. A056105, A244802, A056106, A244803, A056107, A244804, A056108, A244805, A056109, A003215, A033577.

Programs

  • Magma
    [12*n^2 - 19*n + 8 : n in [1..50]]; // Wesley Ivan Hurt, Jul 06 2014
  • Maple
    A244806:=n->12*n^2 - 19*n + 8: seq(A244806(n), n=1..50); # Wesley Ivan Hurt, Jul 06 2014
  • Mathematica
    f[n_] := 12n^2 - 19n + 8; Array[f, 47]
  • PARI
    vector(50, n, 12*n^2 - 19*n + 8) \\ Michel Marcus, Jul 06 2014
  • PARI
    Vec(x*(1 + 15*x + 8*x^2) / (1 - x)^3 + O(x^50)) \\ Colin Barker, Dec 12 2016

Formula

a(n) = 12*n^2 - 19*n + 8.
See A056105 example section for its formula.
From Colin Barker, Dec 12 2016: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>3.
G.f.: x*(1 + 15*x + 8*x^2) / (1 - x)^3.
(End)

A244802 The 60-degree spoke (or ray) of a hexagonal spiral of Ulam.

Original entry on oeis.org

1, 10, 43, 100, 181, 286, 415, 568, 745, 946, 1171, 1420, 1693, 1990, 2311, 2656, 3025, 3418, 3835, 4276, 4741, 5230, 5743, 6280, 6841, 7426, 8035, 8668, 9325, 10006, 10711, 11440, 12193, 12970, 13771, 14596, 15445, 16318, 17215, 18136, 19081, 20050, 21043, 22060, 23101, 24166, 25255
Offset: 1

Views

Author

Robert G. Wilson v, Jul 06 2014

Keywords

Examples

			See A056105 example section for a diagram.

Links

Crossrefs

Cf. A056105, A056106, A244803, A056107, A244804, A056108, A244805, A056109, A244806, A003215.

Programs

  • Magma
    [12*n^2-27*n+16 : n in [1..50]]; // Wesley Ivan Hurt, Jul 06 2014
  • Maple
    A244802:=n->12*n^2-27*n+16: seq(A244802(n), n=1..50); # Wesley Ivan Hurt, Jul 06 2014
  • Mathematica
    f[n_] := 12n^2 - 27n + 16; Array[f, 47]
  • PARI
    vector(50, n, 12*n^2 - 27*n + 16) \\ Michel Marcus, Jul 06 2014
  • PARI
    Vec(x*(1 + 7*x + 16*x^2) / (1 - x)^3 + O(x^50)) \\ Colin Barker, Dec 12 2016

Formula

See A056105 example section for a formula.
From Colin Barker, Dec 12 2016: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>3.
G.f.: x*(1 + 7*x + 16*x^2) / (1 - x)^3.
(End)

A244803 The 360 degree spoke (or ray) of a hexagonal spiral of Ulam.

Original entry on oeis.org

1, 12, 47, 106, 189, 296, 427, 582, 761, 964, 1191, 1442, 1717, 2016, 2339, 2686, 3057, 3452, 3871, 4314, 4781, 5272, 5787, 6326, 6889, 7476, 8087, 8722, 9381, 10064, 10771, 11502, 12257, 13036, 13839, 14666, 15517, 16392, 17291, 18214, 19161, 20132, 21127, 22146, 23189, 24256, 25347
Offset: 1

Views

Author

Robert G. Wilson v, Jul 06 2014

Keywords

Examples

			See A056105 example section for a diagram.

Links

Crossrefs

Cf. A056105, A244802, A056106, A056107, A244804, A056108, A244805, A056109, A244806, A003215, A033577.

Programs

  • Magma
    [12*n^2-25*n+14 : n in [1..50]]; // Wesley Ivan Hurt, Jul 06 2014
  • Maple
    A244803:=n->12*n^2-25*n+14: seq(A244803(n), n=1..50); # Wesley Ivan Hurt, Jul 06 2014
  • Mathematica
    f[n_] := 12n^2 - 25n + 14; Array[f, 47]
  • PARI
    vector(50, n, 12*n^2 - 25*n + 14) \\ Michel Marcus, Jul 06 2014
  • PARI
    Vec(x*(1 + 2*x)*((1 + 7*x) / (1 - x)^3) + O(x^50)) \\ Colin Barker, Dec 12 2016

Formula

See A056105 example section for a formula.
From Colin Barker, Dec 12 2016: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>3.
G.f.: x*(1 + 2*x)*((1 + 7*x) / (1 - x)^3).
(End)
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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