A281333 -id:A281333 - 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

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)

A212978 Number of (w,x,y) with all terms in {0,...,n} and range = 2*n-w-x.

Original entry on oeis.org

1, 5, 11, 20, 32, 46, 63, 83, 105, 130, 158, 188, 221, 257, 295, 336, 380, 426, 475, 527, 581, 638, 698, 760, 825, 893, 963, 1036, 1112, 1190, 1271, 1355, 1441, 1530, 1622, 1716, 1813, 1913, 2015, 2120, 2228, 2338, 2451, 2567, 2685, 2806, 2930

Offset: 0

Clark Kimberling, Jun 03 2012

For a guide to related sequences, see A212959.

Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).

Cf. A212959.

Second bisection of A281333.

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[Max[w, x, y] - Min[w, x, y] == 2 n - w - x,
  s = s + 1],
{w, 0, n}, {x, 0, n}, {y, 0, n}]; s)]];
m = Map[t[#] &, Range[0, 60]]   (* A212978 *)
LinearRecurrence[{2,-1,1,-2,1},{1,5,11,20,32},50] (* Harvey P. Dale, Sep 30 2017 *)

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

G.f.: (1 + 3*x + 2*x^2 + 2*x^3)/((1 - x)^3*(1 + x + x^2)). [corrected by Bruno Berselli, Jan 23 2017]

A236771 a(n) = n + floor(n/2 + n^2/3).

Original entry on oeis.org

0, 1, 4, 7, 11, 15, 21, 26, 33, 40, 48, 56, 66, 75, 86, 97, 109, 121, 135, 148, 163, 178, 194, 210, 228, 245, 264, 283, 303, 323, 345, 366, 389, 412, 436, 460, 486, 511, 538, 565, 593, 621, 651, 680, 711, 742, 774, 806, 840, 873, 908, 943, 979

Offset: 0

Bruno Berselli, Feb 06 2014

If a(k) is prime then k == 3, 4 or 8 (mod 12). The primes are 7, 11, 97, 109, 163, 283, 389, 593, 1129, 1987, 2039, 2713, ... .

This sequence is between A042965 and A236773.

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).

Cf. A004772; A032766: n+floor(n/2).
Cf. A042965: n+floor(1/2+n/3); A236773: n+floor(n^2/2+n^3/3).
Cf. A281333: 1+floor(n/2)+floor(n^2/3).

Magma
```
[n+Floor(n/2+n^2/3): n in [0..60]];
```

Table[n + Floor[n/2 + n^2/3], {n, 0, 60}]

G.f.: x*(1 + 3*x + 2*x^2 - 2*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) = (2*n*(2*n+9) - 2*(-1)^floor(2*(n-1)/3) + 3*(-1)^n - 5)/12.
a(n+2) - a(n) = A004772(n+4).
Also: a(n) = n + floor(n/2) + floor(n^2/3).

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

PARI

Sage

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

PARI

Formula

A212978 Number of (w,x,y) with all terms in {0,...,n} and range = 2*n-w-x.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A236771 a(n) = n + floor(n/2 + n^2/3).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Formula

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