A094159 -id:A094159 - cp's OEIS frontend

A227720 Round(1/s(n)), where s(n) = n*log(1+1/n) - (2n-1)/(2n).

5, 16, 34, 57, 86, 121, 163, 210, 263, 322, 388, 459, 536, 619, 709, 804, 905, 1012, 1126, 1245, 1370, 1501, 1639, 1782, 1931, 2086, 2248, 2415, 2588, 2767, 2953, 3144, 3341, 3544, 3754, 3969, 4190, 4417, 4651, 4890, 5135, 5386, 5644, 5907, 6176, 6451, 6733

Offset: 1

Clark Kimberling, Jul 22 2013

That s(n) > 0 for n >=1 follows from the chain 1 < log 2 < 3/4 < 2 log 3/2 < 5/6 < 3 log 4/3 < 7/8 < 4 log 5/4 < ... ; i.e., n log((n+1)/n) - (2n-1)/(2n) > 0 and (2n+1)/(2n+2) - n log((n+1)/n) > 0. For the first, closeness to 0 is indicated by A227719 and A227720, and for the second, by A227721 and a sequence which possibly equals A094159. Conjecture: the four sequences are linearly recurrent.

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A227719, A227721, A094159.

s[n_] := n*Log[1 + 1/n] - (2 n - 1)/(2 n);
Table[Floor[1/s[n]], {n, 1, 100}]  (* A227719 *)
Table[Round[1/s[n]], {n, 1, 100}]  (* A227720 *)

a(n) = -2*a(n-1) - a(n-2) + a(n-4) - 2*a(n-5) + a(n-6) (conjectured).

G.f.: (-5 - 6 x - 7 x^2 - 5 x^3 - x^4)/((-1 + x)^3 (1 + x + x^2 + x^3)) (conjectured).

A227721 Floor(1/s(n)), where s(n) = (2n+1)/(2n+2) - n*log((n+1)/n).

Original entry on oeis.org

17, 44, 83, 134, 197, 272, 359, 458, 569, 692, 827, 974, 1133, 1304, 1487, 1682, 1889, 2108, 2339, 2582, 2837, 3104, 3383, 3674, 3977, 4292, 4619, 4958, 5309, 5672, 6047, 6434, 6833, 7244, 7667, 8102, 8549, 9008, 9479, 9962, 10457, 10964, 11483, 12014, 12557

Offset: 1

Clark Kimberling, Jul 22 2013

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A227719, A227720, A094159.

s[n_] := s[n] = (2 n + 1)/(2 n + 2) - n*Log[1 + 1/n]
Table[Floor[1/s[n]], {n, 1, 100}] (* A227721 *)
Table[Round[1/s[n]], {n, 1, 100}] (* conjecture: A094159 *)

a(n) = 2 + 9*n + 6*n^2 (conjectured).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) (conjectured).
G.f.:  (-17 + 7 x - 2 x^2)/(-1 + x)^3 (conjectured).

A008730 Molien series 1/((1-x)^2(1-x^12)) for 3-dimensional group [2,n] = 22n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 66, 69, 72, 76, 80, 84, 88, 92, 96, 100, 104, 108, 112, 116, 120, 125, 130, 135, 140, 145, 150, 155, 160, 165, 170, 175, 180, 186, 192, 198, 204

Offset: 0

N. J. A. Sloane

			..1....2....3....4....5....6....7....8....9...10...11...12
.14...16...18...20...22...24...26...28...30...32...34...36
.39...42...45...48...51...54...57...60...63...66...69...72
.76...80...84...88...92...96..100..104..108..112..116..120
125..130..135..140..145..150..155..160..165..170..175..180
186..192..198..204..210..216..222..228..234..240..246..252
259..266..273..280..287..294..301..308..315..322..329..336
344..352..360..368..376..384..392..400..408..416..424..432
441..450..459..468..477..486..495..504..513..522..531..540
550..560..570..580..590..600..610..620..630..640..650..660
...
The columns are: A051866, A139267, A094159, A033579, A049452, A033581, A049453, A033580, A195319, A202804, A211014, A049598
- _Philippe Deléham_, Apr 03 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 195
Index entries for Molien series
Index entries for linear recurrences with constant coefficients, signature (2, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -2, 1).

Cf. A001840, A001972, A008724, A008725, A008726, A008727, A008728, A008729, A008732. - Vladimir Joseph Stephan Orlovsky, Mar 14 2010

Cf. A221912

R:=PowerSeriesRing(Integers(), 70); Coefficients(R!( 1/((1-x)^2*(1-x^12)) )); // G. C. Greubel, Jul 30 2019

seq(coeff(series(1/(1-x)^2/(1-x^12), x, n+1), x, n), n=0..80);

CoefficientList[Series[1/((1-x)^2*(1-x^12)), {x,0,70}], x] (* Vincenzo Librandi, Jun 11 2013 *)
LinearRecurrence[{2,-1,0,0,0,0,0,0,0,0,0,1,-2,1},{1,2,3,4,5,6,7,8,9,10,11,12,14,16},70] (* Harvey P. Dale, Jan 01 2024 *)

my(x='x+O('x^70)); Vec(1/((1-x)^2*(1-x^12))) \\ G. C. Greubel, Jul 30 2019

(1/((1-x)^2*(1-x^12))).series(x, 70).coefficients(x, sparse=False) # G. C. Greubel, Jul 30 2019

G.f. 1/( (1-x)^3 * (1+x) *(1+x+x^2) *(1-x+x^2) * (1+x^2) *(1-x^2+x^4)). - R. J. Mathar, Aug 11 2021
From Mitch Harris, Sep 08 2008: (Start)
a(n) = Sum_{j=0..n+12} floor(j/12).
a(n-12) = (1/2)*floor(n/12)*(2*n - 10 - 12*floor(n/12)). (End)
a(n) = A221912(n+12). - Philippe Deléham, Apr 03 2013

More terms from Vladimir Joseph Stephan Orlovsky, Mar 14 2010

A386477 a(0) = 1; thereafter a(n) = 2(6n^2 - 3*n + 1).

Original entry on oeis.org

1, 8, 38, 92, 170, 272, 398, 548, 722, 920, 1142, 1388, 1658, 1952, 2270, 2612, 2978, 3368, 3782, 4220, 4682, 5168, 5678, 6212, 6770, 7352, 7958, 8588, 9242, 9920, 10622, 11348, 12098, 12872, 13670, 14492, 15338, 16208, 17102, 18020, 18962, 19928, 20918, 21932, 22970, 24032, 25118, 26228, 27362, 28520, 29702, 30908, 32138, 33392, 34670

Offset: 0

Scott R. Shannon and N. J. A. Sloane, Jul 22 2025

Definition: A regular hexagram of radius R is formed by placing six equally-spaced points P_0 .. P_5 around the boundary of a circle of radius R, and drawing line segments P_0 - P_2 - P_4 - P_0 and P_1 - P_3 - P_5 - P_1.
Theorem 1: a(n) is the maximum number of regions that can be formed in the plane by drawing n regular hexagrams with the same radius and the same center.
Conjecture 2: a(n) is the maximum number of regions that can be formed in the plane by drawing n regular hexagrams with any radii and any centers.
The following construction works for any n >= 1. Take 6*n equally-spaced points P_i around a circle, and draw hexagrams starting at P_i for i = 0, ..., n-1.
The resulting planar graph decomposes into 6*n triangular cells each with 2*n-1 cells (see the red triangle in the "Three pentagons" illustration), plus the interior and exterior regions, for a total of 12*n^2 - 6*n + 2 regions. There are 12*n^2 vertices, for n>0.

Paolo Xausa, Table of n, a(n) for n = 0..10000
Scott R. Shannon, Illustration for a(1) = 8 [Note that the cell counts shown on these five figures do not include the black exterior region, so the totals are off by 1]
Scott R. Shannon, Illustration for a(2) = 38
Scott R. Shannon, Illustration for a(3) = 92
Scott R. Shannon, Illustration for a(8) = 722
Scott R. Shannon, Illustration for a(10) = 1142
N. J. A. Sloane, Sketch to illustrate a(2) = 38. The two hexagrams are colored red and black, respectively.
N. J. A. Sloane, Sketch to illustrate a(3) = 92. The three hexagrams are colored red, blue, and black, respectively.
N. J. A. Sloane, Analogous illustration for three pentagrams
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A094159, A152746.

See A077588, A069894, and A383466 for analogous sequences based on triangles, squares, and pentagrams.

A386477[n_] := If[n == 0, 1, 6*n*(2*n - 1) + 2]; Array[A386477, 50, 0] (* or *)
Join[{1}, 6*PolygonalNumber[6, Range[49]] + 2] (* or *)
LinearRecurrence[{3, -3, 1}, {1, 8, 38, 92}, 50] (* Paolo Xausa, Jul 24 2025 *)

From Stefano Spezia, Jul 23 2025: (Start)
G.f.: (1 + 5*x + 17*x^2 + x^3)/(1 - x)^3.
E.g.f.: 2*exp(x)*(1 + 3*x + 6*x^2) - 1. (End)
a(n) = A152746(n) + 2, for n >= 1. - Paolo Xausa, Jul 24 2025

A194713 13 times hexagonal numbers: a(n) = 13n(2*n-1).

Original entry on oeis.org

0, 13, 78, 195, 364, 585, 858, 1183, 1560, 1989, 2470, 3003, 3588, 4225, 4914, 5655, 6448, 7293, 8190, 9139, 10140, 11193, 12298, 13455, 14664, 15925, 17238, 18603, 20020, 21489, 23010, 24583, 26208, 27885, 29614, 31395, 33228, 35113, 37050, 39039, 41080, 43173

Offset: 0

Omar E. Pol, Oct 02 2011

Sequence found by reading the line from 0, in the direction 0, 13, ..., in the square spiral whose vertices are the generalized 15-gonal numbers.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000384, A002939, A094159, A143698, A152745, A154617, A195320.

[13*n*(2*n-1): n in [0..50]]; // Vincenzo Librandi, Oct 03 2011

a(n)=13*n*(2*n-1) \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 26*n^2 - 13*n = 13*A000384(n).
a(n) = a(n-1) + 52*n - 39, a(0)=0. - Vincenzo Librandi, Oct 03 2011
From Elmo R. Oliveira, Dec 15 2024: (Start)
G.f.: 13*x*(1 + 3*x)/(1 - x)^3.
E.g.f.: 13*x*(1 + 2*x)*exp(x).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n >= 3. (End)

cp's OEIS Frontend

A227720 Round(1/s(n)), where s(n) = n*log(1+1/n) - (2n-1)/(2n).

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A227721 Floor(1/s(n)), where s(n) = (2n+1)/(2n+2) - n*log((n+1)/n).

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A008730 Molien series 1/((1-x)^2*(1-x^12)) for 3-dimensional group [2,n] = *22n.

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A386477 a(0) = 1; thereafter a(n) = 2*(6*n^2 - 3*n + 1).

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A194713 13 times hexagonal numbers: a(n) = 13*n*(2*n-1).

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Author

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Programs

Magma

PARI

Formula

A008730 Molien series 1/((1-x)^2(1-x^12)) for 3-dimensional group [2,n] = 22n.

A386477 a(0) = 1; thereafter a(n) = 2(6n^2 - 3*n + 1).

A194713 13 times hexagonal numbers: a(n) = 13n(2*n-1).