A244805 -id:A244805 - cp's OEIS frontend

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

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

N. J. A. Sloane

Also the 120º 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

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

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Subsequence of A281333.

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

List([0..50], n-> (3*n+1)*(4*n+1)); # G. C. Greubel, Oct 12 2019

[(3*n+1)*(4*n+1) : n in [0..50]]; // Wesley Ivan Hurt, Jul 06 2014

A033577:=n->(3*n+1)*(4*n+1): seq(A033577(n), n=0..50); # Wesley Ivan Hurt, Jul 06 2014

f[n_] := (3n + 1)(4n + 1); Array[f, 50, 0] (* Robert G. Wilson v, Jul 06 2014 *)
LinearRecurrence[{3,-3,1},{1,20,63},50] (* Harvey P. Dale, Jul 16 2020 *)

vector(50, m, 12*m^2 - 17*m + 6) \\ Michel Marcus, Jul 06 2014

Vec((1 + 17*x + 6*x^2) / (1 - x)^3 + O(x^50)) \\ Colin Barker, Dec 12 2016

[(3*n+1)*(4*n+1) for n in range(50)] # G. C. Greubel, Oct 12 2019

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

More terms from Wesley Ivan Hurt, Jul 06 2014

A244814 The hexagonal spiral of Champernowne, read along the 240-degree ray.

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v, Jul 06 2014

			see A244807 example section for its diagram.

Cf. A007376, A244805, A244807, A244808, A244809, A244810, A244811, A244812, A244813, A244815, A244816, A244817, A244818.

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 - 21n + 10 (* see formula section of A244807 *); Array[ almostNatural[ f@#, 10] &, 105]

(12n^2 - 21n + 10)th almost natural number (A033307), Also see formula section of A056105.

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

Robert G. Wilson v, Jul 06 2014

			See A056105 example section for its diagram.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

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

[12*n^2 - 19*n + 8 : n in [1..50]]; // Wesley Ivan Hurt, Jul 06 2014

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]
```

vector(50, n, 12*n^2 - 19*n + 8) \\ Michel Marcus, Jul 06 2014

Vec(x*(1 + 15*x + 8*x^2) / (1 - x)^3 + O(x^50)) \\ Colin Barker, Dec 12 2016

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

Robert G. Wilson v, Jul 06 2014

			See A056105 example section for a diagram.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

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

[12*n^2-27*n+16 : n in [1..50]]; // Wesley Ivan Hurt, Jul 06 2014

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]
```

vector(50, n, 12*n^2 - 27*n + 16) \\ Michel Marcus, Jul 06 2014

Vec(x*(1 + 7*x + 16*x^2) / (1 - x)^3 + O(x^50)) \\ Colin Barker, Dec 12 2016

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

Robert G. Wilson v, Jul 06 2014

			See A056105 example section for a diagram.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

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

[12*n^2-25*n+14 : n in [1..50]]; // Wesley Ivan Hurt, Jul 06 2014

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]
```

vector(50, n, 12*n^2 - 25*n + 14) \\ Michel Marcus, Jul 06 2014

Vec(x*(1 + 2*x)*((1 + 7*x) / (1 - x)^3) + O(x^50)) \\ Colin Barker, Dec 12 2016

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)

A244804 The 300-degree spoke (or ray) of a hexagonal spiral of Ulam.

Original entry on oeis.org

1, 14, 51, 112, 197, 306, 439, 596, 777, 982, 1211, 1464, 1741, 2042, 2367, 2716, 3089, 3486, 3907, 4352, 4821, 5314, 5831, 6372, 6937, 7526, 8139, 8776, 9437, 10122, 10831, 11564, 12321, 13102, 13907, 14736, 15589, 16466, 17367, 18292, 19241, 20214, 21211, 22232, 23277, 24346, 25439

Offset: 1

Robert G. Wilson v, Jul 06 2014

			See A056105 example section for its diagram.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A056105, A244802, A056106, A244803, A056107, A056108, A244805, A056109, A244806, A003215, A033577.

[ 12*n^2 - 23*n + 12 : n in [1..50] ]; // Wesley Ivan Hurt, Jul 06 2014

A244804:=n->12*n^2 - 23*n + 12: seq(A244804(n), n=1..50); # Wesley Ivan Hurt, Jul 06 2014

Mathematica
```
f[n_] := 12n^2 - 23n + 12; Array[f, 47]
```

vector(50, n, 12*n^2 - 23*n + 12) \\ Michel Marcus, Jul 06 2014

Vec(x*(1 + 11*x + 12*x^2) / (1 - x)^3 + O(x^50)) \\ Colin Barker, Dec 12 2016

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 + 11*x + 12*x^2) / (1 - x)^3.
(End)

A281333 a(n) = 1 + floor(n/2) + floor(n^2/3).

Original entry on oeis.org

1, 1, 3, 5, 8, 11, 16, 20, 26, 32, 39, 46, 55, 63, 73, 83, 94, 105, 118, 130, 144, 158, 173, 188, 205, 221, 239, 257, 276, 295, 316, 336, 358, 380, 403, 426, 451, 475, 501, 527, 554, 581, 610, 638, 668, 698, 729, 760, 793, 825, 859, 893, 928, 963, 1000, 1036, 1074, 1112, 1151, 1190

Offset: 0

Bruno Berselli, Jan 20 2017

Bruno Berselli, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,1,0,-1,-1,1).

Subsequences: A033577, A244805 (numbers of the form 1 + k/2 + k^2/3), A212978 (second bisection).
Cf. A236771: n + floor(n/2) + floor(n^2/3).
Cf. A008619: 1 + floor(n/2); A087483: 1 + floor(n^2/3).
Cf. A000212, A004526.

[1 + n div 2 + n^2 div 3: n in [0..60]];

A281333:=n->1 + floor(n/2) + floor(n^2/3): seq(A281333(n), n=0..100); # Wesley Ivan Hurt, Feb 09 2017

Table[1 + Floor[n/2] + Floor[n^2/3], {n, 0, 60}]
LinearRecurrence[{1,1,0,-1,-1,1},{1,1,3,5,8,11},80] (* Harvey P. Dale, Sep 29 2024 *)

makelist(1+floor(n/2)+floor(n^2/3), n, 0, 60);

vector(60, n, n--; 1+floor(n/2)+floor(n^2/3))

[1+int(n/2)+int(n**2/3) for n in range(60)]

[1+floor(n/2)+floor(n^2/3) for n in range(60)]

G.f.: (1 + x^2 + x^3 + x^4)/((1 + x)*(1 + x + x^2)*(1 - x)^3).
a(n) = a(n-1) + a(n-2) - a(n-4) - a(n-5) + a(n-6).
a(n) = 1 + floor(n/2 + n^2/3).
a(n) = (12*n^2 + 18*n + 4*(-1)^(2*n/3) + 4*(-1)^(-2*n/3) + 9*(-1)^n + 19)/36.
a(n) - n = a(-n).
a(6*k+r) = 12*k^2 + (4*r+3)*k + a(r), where 0 <= r <= 5. Particular cases:
a(6*k) = A244805(k+1), a(6*k+1) = A033577(k).
a(n+2) -  a(n) =   A004773(n+2).
a(n+3) -  a(n) =   A014601(n+2).
a(n+4) -  a(n) =   A047480(n+3).
a(n) - a(-n+3) = 2*A001651(n-1).
a(n) + a(-n+3) = 2*A097922(n-1).
a(n) = 1 + A004526(n) + A000212(n) = A008619(n) + A000212(n). - Omar E. Pol, Dec 23 2020

cp's OEIS Frontend

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

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A244814 The hexagonal spiral of Champernowne, read along the 240-degree ray.

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

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

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

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A244804 The 300-degree spoke (or ray) of a hexagonal spiral of Ulam.

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