A056106 -id:A056106 - cp's OEIS frontend

A053698 a(n) = n^3 + n^2 + n + 1.

1, 4, 15, 40, 85, 156, 259, 400, 585, 820, 1111, 1464, 1885, 2380, 2955, 3616, 4369, 5220, 6175, 7240, 8421, 9724, 11155, 12720, 14425, 16276, 18279, 20440, 22765, 25260, 27931, 30784, 33825, 37060, 40495, 44136, 47989, 52060, 56355, 60880

Offset: 0

Henry Bottomley, Mar 23 2000

a(n) = 1111 in base n.

n^3 + n^2 + n + 1 = (n^2 + 1)*(n + 1), therefore a(n) is never prime. - Alonso del Arte, Apr 22 2014

			a(2) = 15 because 2^3 + 2^2 + 2 + 1 = 8 + 4 + 2 + 1 = 15.
a(3) = 40 because 3^3 + 3^2 + 3 + 1 = 27 + 9 + 3 + 1 = 40.
a(4) = 85 because 4^3 + 4^2 + 4 + 1 = 64 + 16 + 4 + 1 = 85.
From _Bruno Berselli_, Jan 02 2017: (Start)
The terms of the sequence are provided by the row sums of the following triangle (see the seventh formula in the previous section):
.   1;
.   3,   1;
.   9,   5,   1;
.  19,  13,   7,   1;
.  33,  25,  17,   9,   1;
.  51,  41,  31,  21,  11,   1;
.  73,  61,  49,  37,  25,  13,  1;
.  99,  85,  71,  57,  43,  29, 15,  1;
. 129, 113,  97,  81,  65,  49, 33, 17,  1;
. 163, 145, 127, 109,  91,  73, 55, 37, 19,  1;
. 201, 181, 161, 141, 121, 101, 81, 61, 41, 21, 1;
...
Columns from the first to the fifth, respectively: A058331, A001844, A056220 (after -1), A059993, A161532. Also, eighth column is A161549.
(End)

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A237627 (subset of semiprimes).
Cf. A056106 (first differences).
Cf. A062158, A027444, A104878, A055129.

[n^3+n^2+n+1: n in [0..50]]; // Vincenzo Librandi, May 01 2011

A053698:=n->n^3 + n^2 + n + 1; seq(A053698(n), n=0..50); # Wesley Ivan Hurt, Apr 22 2014

Table[n^3 + n^2 + n + 1, {n, 0, 39}] (* Alonso del Arte, Apr 22 2014 *)
FromDigits["1111", Range[0, 50]] (* Paolo Xausa, May 11 2024 *)

Vec((1 + 5*x^2) / (1 - x)^4 + O(x^50)) \\ Colin Barker, Jan 02 2017

def a(n): return (n**3+n**2+n+1) # Torlach Rush, May 08 2024

For n >= 2, a(n) = (n^4-1)/(n-1) = A024002(n)/A024000(n) = A002522(n)*(n+1) = A002061(n+1) + A000578(n).
G.f.: (1+5*x^2) / (1-x)^4. - Colin Barker, Jan 06 2012
a(n) = -A062158(-n). - Bruno Berselli, Jan 26 2016
a(n) = Sum_{i=0..n} 2*n*(n-i)+1. - Bruno Berselli, Jan 02 2017
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n > 3. - Colin Barker, Jan 02 2017
a(n) = A104878(n+3,n) = A055129(4,n) for n > 0. - Mathew Englander, Jan 06 2021
E.g.f.: exp(x)*(x^3+4*x^2+3*x+1). - Nikolaos Pantelidis, Feb 06 2023

A244807 The hexagonal spiral of Champernowne, read along the East (or 90-degree) ray.

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v, Jul 06 2014

Inspired by Stanislaw M. Ulam's hexagonal spiral, circa 1963. See example section of A056105.

When A056105, A056106, A056107, A056108, A056109 & A003215 were submitted, the offsets were 0. Here the offset is 1.

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

Cf. A007376, A054552, A056105, A244808, A244809, A244810, A244811, A244812, A244813, A244814, 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_] := 3n^2- 8n +6 (* see formula section of A244807 *); Array[ almostNatural[ f@#, 10] &, 105]

For each 30 degrees of the compass, the corresponding spoke (or ray) has a generating formula as follows:
 3n^2- 8n +6
12n^2-27n+16
 3n^2- 7n+ 5
12n^2-25n+14
 3n^2 -6n +4
12n^2-23n+12
 3n^2 -5n +3
12n^2-21n+10
 3n^2 -4n +2
12n^2-19n +8
 3n^2 -3n +1
12n^2-17n+ 6
Also see formula section of A056105.

A033577 a(n) = (3n+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

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

A244809 The hexagonal spiral of Champernowne, read along the 30-degree ray.

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v, Jul 06 2014

			See A244807 example section for its diagram.

Cf. A007376, A056106, A244807, A244808, A244810, A244811, A244812, A244813, A244814, 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_] := 3n^2- 7n+ 5 (* see formula section of A244807 *); Array[ almostNatural[ f@#, 10] &, 105]

(3n^2-7n+5)th almost natural number (A033307), Also see formula section of A056105.

A096777 a(n) = a(n-1) + Sum_{k=1..n-1}(a(k) mod 2), a(1) = 1.

Original entry on oeis.org

1, 2, 3, 5, 8, 11, 15, 20, 25, 31, 38, 45, 53, 62, 71, 81, 92, 103, 115, 128, 141, 155, 170, 185, 201, 218, 235, 253, 272, 291, 311, 332, 353, 375, 398, 421, 445, 470, 495, 521, 548, 575, 603, 632, 661, 691, 722, 753, 785, 818, 851, 885, 920, 955, 991, 1028

Offset: 1

Reinhard Zumkeller, Jul 09 2004

a(n) = a(n-1) + (number of odd terms so far in the sequence). Example: 15 is 11 + 4 odd terms so far in the sequence (they are 1,3,5,11). See A007980 for the same construction with even integers. - Eric Angelini, Aug 05 2007

A016789 and A032766 give positions where even and odd terms occur; a(3*n)=A056106(n); a(3*n-1)=A077588(n); a(3*n-2)=A056108(n). - Reinhard Zumkeller, Dec 29 2007

			G.f. = x + 2*x^2 + 3*x^3 + 5*x^4 + 8*x^5 + 11*x^6 + 15*x^7 + 20*x^8 + ... - _Michael Somos_, Apr 18 2020

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
J.-L. Baril, T. Mansour, A. Petrossian, Equivalence classes of permutations modulo excedances, 2014.
Eric Weisstein's World of Mathematics, Odd Number
Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).

Cf. A004396, A007980, A016789, A032766, A056106, A056108, A061347, A077588, A097602, A131093.

a096777 n = a096777_list !! (n-1)
a096777_list = 1 : zipWith (+) a096777_list
                               (scanl1 (+) (map (`mod` 2) a096777_list))
-- Reinhard Zumkeller, Mar 11 2014

[Floor((n-2)^2/3)+n: n in [1..60]]; // Vincenzo Librandi, Dec 27 2015

A096777:=n->n + floor((n-2)^2/3); seq(A096777(n), n=1..100); # Wesley Ivan Hurt, Mar 06 2014

Table[n + Floor[(n-2)^2/3], {n, 100}] (* Wesley Ivan Hurt, Mar 06 2014 *)

a(n)=(n-2)^2\3+n \\ Charles R Greathouse IV, Mar 06 2014

a(n+1) - a(n) = A004396(n).
a(n) = floor(n/3) * (3*floor(n/3) + 2*(n mod 3) - 1) + n mod 3 + 0^(n mod 3). - Reinhard Zumkeller, Dec 29 2007
a(n) = floor((n-2)^2/3) + n. - Christopher Hunt Gribble, Mar 06 2014
G.f.: -x*(x^4+1) / ((x-1)^3*(x^2+x+1)). - Colin Barker, Mar 07 2014
Euler transform of finite sequence [2, 0, 1, 1, 0, 0, 0, -1]. - Michael Somos, Apr 18 2020
a(n) = (10 + 3*n*(n - 1) - A061347(n+1))/9. - Stefano Spezia, Sep 22 2022

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

Robert G. Wilson v, Jul 06 2014

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

			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, A056109, A244806, A003215, A033577.

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

[12*n^2-21*n+10: n in [1..50]]; // Wesley Ivan Hurt, Jul 06 2014

A244805:=n->12*n^2 - 21*n + 10: seq(A244805(n), n=1..50); # Wesley Ivan Hurt, Jul 06 2014

f[n_] := 12 n^2 - 21 n + 10; Array[f, 47]

vector(50, n, 12*n^2 - 21*n + 10) \\ Michel Marcus, Jul 06 2014

Vec(x*(1 + 13*x + 10*x^2) / (1 - x)^3 + O(x^50)) \\ Colin Barker, Dec 12 2016

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

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)

cp's OEIS Frontend

A053698 a(n) = n^3 + n^2 + n + 1.

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A244807 The hexagonal spiral of Champernowne, read along the East (or 90-degree) ray.

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

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A244809 The hexagonal spiral of Champernowne, read along the 30-degree ray.

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A096777 a(n) = a(n-1) + Sum_{k=1..n-1}(a(k) mod 2), a(1) = 1.

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Haskell

Magma

Maple

Mathematica

PARI

Formula

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

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Magma

Maple

Mathematica

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