A015518 -id:A015518 - cp's OEIS frontend

A271116 Integers n such that round(3^n/12) is divisible by n.

1, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 97, 101, 103, 107, 109, 113, 121, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307

Offset: 1

Altug Alkan, Mar 31 2016

In other words, numbers n such that A015518(n-1) is divisible by n.
This sequence generates prime numbers except 2 and 3.
The first few composite terms in this sequence are 91, 121, 671, 703, 949. Note that they are pseudoprimes to base 3.
Also, numbers k such that 3^(k-1) mod (4*k) = 1. In the first million terms only 908 terms are nonprimes. - David A. Corneth, Oct 02 2020

			5 is a term because round(3^5/12) = 20 is divisible by 5.
6 is not a term because round(3^6/12) = 61 which is not divisible by 6. - _David A. Corneth_, Oct 02 2020

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A005935, A015518.

Select[Range@ 308, Divisible[Round[3^#/12], #] &] (* Michael De Vlieger, Mar 31 2016 *)

f(n) = round(3^n/12);
for(n=1, 1e3, if(f(n) % n == 0, print1(n, ", ")));

is(n) = lift(Mod(3, 4*n)^(n-1))==1 \\ David A. Corneth, Oct 02 2020

A360036 Expansion of e.g.f. xexp(x)(sinh(x))^2.

Original entry on oeis.org

0, 0, 0, 6, 24, 100, 360, 1274, 4368, 14760, 49200, 162382, 531432, 1727180, 5580120, 17936130, 57395616, 182948560, 581130720, 1840247318, 5811307320, 18305618100, 57531942600, 180441092746, 564859072944, 1765184603000, 5507375961360, 17157594341214, 53379182394888

Offset: 0

Enrique Navarrete, Jan 22 2023

a(n) is the number of ordered set partitions of an n-set into 3 sets such that the first and second sets have an odd number of elements and an element is selected from the third.

			The first 4 cases are shown below for a(4)=24 (where the element selected from the third set is in parenthesis):
{1}, {2}, {(3), 4}
{1}, {2}, {3, (4)}
{2}, {1}, {(3), 4}
{2}, {1}, {3, (4)}.

Index entries for linear recurrences with constant coefficients, signature (6,-7,-12,17,6,-9).

Cf. A081251, A015518, A360023, A360035.

With[{nn=30},CoefficientList[Series[x Exp[x]Sinh[x]^2,{x,0,nn}],x] Range[0,nn]!] (* or *) LinearRecurrence[{6,-7,-12,17,6,-9},{0,0,0,6,24,100},30] (* Harvey P. Dale, Aug 17 2025 *)

a(n) = n*A081251(n-2) for n >= 3.
a(n) = n*(3^(n-1) + (-1)^(n-1) - 2)/4.
G.f.: 2*x^3*(3 - 6*x - x^2)/((1 - x)^2*(1 + x)^2*(1 - 3*x)^2). - Stefano Spezia, Jan 23 2023

A015609 a(n) = 11a(n-1) + 12a(n-2).

Original entry on oeis.org

0, 1, 11, 133, 1595, 19141, 229691, 2756293, 33075515, 396906181, 4762874171, 57154490053, 685853880635, 8230246567621, 98762958811451, 1185155505737413, 14221866068848955, 170662392826187461

Offset: 0

Olivier Gérard

Number of walks of length n between any two distinct nodes of the complete graph K_13. Example: a(2)=11 because the walks of length 2 between the nodes A and B of the complete graph ABCDEFGHIJKLM are ACB, ADB, AEB, AFB, AGB, AHB, AIB, AJB, AKB, ALB and AMB. - Emeric Deutsch, Apr 01 2004

General form: k=12^n-k. Also: A001045, A078008, A097073, A115341, A015518, A054878, A015521, A109499, A015531, A109500, A109501, A015552, A093134, A015565, A015577, A015585. - Vladimir Joseph Stephan Orlovsky, Dec 11 2008

Vincenzo Librandi, Table of n, a(n) for n = 0..900
Index entries for linear recurrences with constant coefficients, signature (11,12).

Cf. A001045, A078008, A097073, A115341, A015518, A054878, A015521, A109499, A015531, A109500, A109501, A015552, A093134, A015565, A015577, A015585, A015592. - Vladimir Joseph Stephan Orlovsky, Dec 11 2008

[(1/13)*(12^n-(-1)^n): n in [0..20]]; // Vincenzo Librandi, Oct 11 2011

CoefficientList[Series[x/(1-11*x-12*x^2), {x, 0, 50}], x] (* or *) LinearRecurrence[{11,12}, {0,1}, 30] (* G. C. Greubel, Dec 30 2017 *)

x='x+O('x^30); concat([0], Vec(x/(1-11*x-12*x^2))) \\ G. C. Greubel, Dec 30 2017

[lucas_number1(n,11,-12) for n in range(0, 18)] # Zerinvary Lajos, Apr 27 2009

[abs(gaussian_binomial(n,1,-12)) for n in range(0,18)] # Zerinvary Lajos, May 28 2009

From Emeric Deutsch, Apr 01 2004: (Start)
a(n) = 12^(n-1) - a(n-1).
G.f.: x/(1 - 11*x - 12*x^2). (End)
E.g.f.: exp(-x)*(exp(13*x) - 1)/13. - Stefano Spezia, Mar 11 2020

A053535 Expansion of 1/((1+3x)(1-9*x)).

Original entry on oeis.org

1, 6, 63, 540, 4941, 44226, 398763, 3586680, 32286681, 290560446, 2615103063, 23535750420, 211822285221, 1906398972666, 17157595536963, 154418345483760, 1389765152400561, 12507886242464886, 112570976569604463

Offset: 0

Barry E. Williams, Jan 15 2000

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,27).

Cf. A015518.

a:=[1,6];; for n in [3..30] do a[n]:=6*a[n-1]+27*a[n-2]; od; a; # G. C. Greubel, May 16 2019

R:=PowerSeriesRing(Integers(), 20); Coefficients(R!( 1/((1+3*x)*(1-9*x)) )); // G. C. Greubel, May 16 2019

LinearRecurrence[{6,27}, {1,6}, 20] (* G. C. Greubel, May 16 2019 *)

my(x='x+O('x^20)); Vec(1/((1+3*x)*(1-9*x))) \\ G. C. Greubel, May 16 2019

(1/((1+3*x)*(1-9*x))).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, May 16 2019

a(n) = (3^n/4)*(3^(n+1) + (-1)^n).
a(n) = 6*a(n-1) + 27*a(n-2), with a(0)=1, a(1)=6.
E.g.f.: (3*exp(9*x) + exp(-3*x))/4. - G. C. Greubel, May 16 2019

A053536 Expansion of 1/((1+4x)(1-12*x)).

Original entry on oeis.org

1, 8, 112, 1280, 15616, 186368, 2240512, 26869760, 322502656, 3869769728, 46438285312, 557255229440, 6687079530496, 80244887257088, 962938915520512, 11555265912504320, 138663195245019136, 1663958325760360448, 19967499977843802112, 239609999459247718400

Offset: 0

Barry E. Williams, Jan 15 2000

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

G. C. Greubel, Table of n, a(n) for n = 0..920
Index entries for linear recurrences with constant coefficients, signature (8,48).

Cf. A015518.

a:=[1,8];; for n in [3..30] do a[n]:=8*a[n-1]+48*a[n-2]; od; a; # G. C. Greubel, May 16 2019

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( 1/((1+4*x)*(1-12*x)) )); // G. C. Greubel, May 16 2019

LinearRecurrence[{8,48}, {1,8}, 30] (* G. C. Greubel, May 16 2019 *)

Vec(1/((1+4*x)*(1-12*x)) + O(x^30)) \\ Michel Marcus, Dec 03 2014

(1/((1+4*x)*(1-12*x))).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, May 16 2019

a(n) = (4^n/4)*(3^(n+1) + (-1)^n).
a(n) = 8*a(n-1) + 48*a(n-2), with a(0)=1, a(1)=8.
E.g.f.: (3*exp(12*x) + exp(-4*x))/4. - G. C. Greubel, May 16 2019
a(n) = 2^n*A053524(n+1). - R. J. Mathar, Mar 08 2021

Terms a(12) onward added by G. C. Greubel, May 16 2019

A053537 Expansion of 1/((1+5x)(1-15*x)).

Original entry on oeis.org

1, 10, 175, 2500, 38125, 568750, 8546875, 128125000, 1922265625, 28832031250, 432490234375, 6487304687500, 97309814453125, 1459645996093750, 21894696044921875, 328420410156250000, 4926306304931640625, 73894593811035156250, 1108418910980224609375

Offset: 0

Barry E. Williams, Jan 15 2000

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

G. C. Greubel, Table of n, a(n) for n = 0..845
Index entries for linear recurrences with constant coefficients, signature (10,75).

Cf. A015518.

a:=[1,10];; for n in [3..30] do a[n]:=10*a[n-1]+75*a[n-2]; od; a; # G. C. Greubel, May 16 2019

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( 1/((1+5*x)*(1-15*x)) )); // G. C. Greubel, May 16 2019

LinearRecurrence[{10, 75}, {1, 10}, 30] (* G. C. Greubel, May 16 2019 *)
CoefficientList[Series[1/((1+5x)(1-15x)),{x,0,20}],x] (* Harvey P. Dale, Jun 15 2022 *)

Vec(1/((1+5*x)*(1-15*x)) + O(x^30)) \\ Michel Marcus, Dec 03 2014

(1/((1+5*x)*(1-15*x))).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, May 16 2019

a(n) = (5^n/4)*(3^(n+1) + (-1)^n).
a(n) = 10*a(n-1) + 75*a(n-2), with a(0)=1, a(1)=10.
E.g.f.: (3*exp(15*x) + exp(-5*x))/4. - G. C. Greubel, May 16 2019

Terms a(11) onward added by G. C. Greubel, May 16 2019

A109447 Binomial coefficients C(n,k) with n-k odd, read by rows.

Original entry on oeis.org

1, 2, 1, 3, 4, 4, 1, 10, 5, 6, 20, 6, 1, 21, 35, 7, 8, 56, 56, 8, 1, 36, 126, 84, 9, 10, 120, 252, 120, 10, 1, 55, 330, 462, 165, 11, 12, 220, 792, 792, 220, 12, 1, 78, 715, 1716, 1287, 286, 13, 14, 364, 2002, 3432, 2002, 364, 14, 1, 105, 1365, 5005, 6435, 3003, 455, 15

Offset: 1

Philippe Deléham, Aug 27 2005

The same as A119900 without 0's. A reflected version of A034867 or A202064. - Alois P. Heinz, Feb 07 2014
From Vladimir Shevelev, Feb 07 2014: (Start)
Also table of coefficients of polynomials  P_1(x)=1, P_2(x)=2, for n>=2, P_(n+1)(x) = 2*P_n(x)+(x-1)* P_(n-1)(x).  The polynomials P_n(x)/2^(n-1) are connected with sequences A000045 (x=5), A001045 (x=9), A006130 (x=13), A006131 (x=17), A015440 (x=21), A015441 (x=25), A015442 (x=29), A015443 (x=33), A015445 (x=37), A015446 (x=41), A015447 (x=45), A053404 (x=49); also the polynomials P_n(x) are connected with sequences A000129, A002605, A015518, A063727, A085449, A002532, A083099, A015519, A003683, A002534, A083102, A015520. (End)

			Starred terms in Pascal's triangle (A007318), read by rows:
1;
1*, 1;
1, 2*, 1;
1*, 3, 3*, 1;
1, 4*, 6, 4*, 1;
1*, 5, 10*, 10, 5*, 1;
1, 6*, 15, 20*, 15, 6*, 1;
1*, 7, 21*, 35, 35*, 21, 7*, 1;
1, 8*, 28, 56*, 70, 56*, 28, 8*, 1;
1*, 9, 36*, 84, 126*, 126, 84*, 36, 9*, 1;
Triangle T(n,k) begins:
1;
2;
1,    3;
4,    4;
1,   10,  5;
6,   20,  6;
1,   21,  35,   7;
8,   56,  56,   8;
1,   36, 126,  84,  9;
10, 120, 252, 120, 10;

Alois P. Heinz, Rows n = 1..200, flattened

Cf. A109446.

T:= (n, k)-> binomial(n, 2*k+1-irem(n, 2)):
seq(seq(T(n, k), k=0..ceil((n-2)/2)), n=1..20);  # Alois P. Heinz, Feb 07 2014

Flatten[ Table[ If[ OddQ[n - k], Binomial[n, k], {}], {n, 0, 15}, {k, 0, n}]] (* Robert G. Wilson v *)

More terms from Robert G. Wilson v, Aug 30 2005

Corrected offset by Alois P. Heinz, Feb 07 2014

A124137 A signed aerated and skewed version of A038137.

Original entry on oeis.org

1, 0, 1, -1, 0, 2, 0, -2, 0, 3, 1, 0, -5, 0, 5, 0, 3, 0, -10, 0, 8, -1, 0, 9, 0, -20, 0, 13, 0, -4, 0, 22, 0, -38, 0, 21, 1, 0, -14, 0, 51, 0, -71, 0, 34, 0, 5, 0, -40, 0, 111, 0, -130, 0, 55, -1, 0, 20, 0, -105, 0, 233, 0, -235, 0, 89

Offset: 0

Philippe Deléham, Nov 30 2006

			Triangle begins:
1;
0, 1;
-1, 0, 2;
0, -2, 0, 3;
1, 0, -5, 0, 5;
0, 3, 0, -10, 0, 8;
-1, 0, 9, 0, -20, 0, 13;
0, -4, 0, 22, 0, -38, 0, 21;
1, 0, -14, 0, 51, 0, -71, 0, 34;
0, 5, 0, -40, 0, 111, 0, -130, 0, 55;

G. C. Greubel, Rows n=0..100 of triangle, flattened

Cf. A038137, A208336, A000045, A001629, A001628, A001872, A001873

T[0, 0]:= 1; T[n_, n_]:= Fibonacci[n + 1]; T[n_, k_]:= T[n, k] = If[k < 0 || n < k, 0, T[n - 1, k - 1] + T[n - 2, k - 2] - T[n - 2, k]]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten  (* G. C. Greubel, May 27 2018 *)

{T(n,k) = if(n==0 && k==0, 1, if(k==n, fibonacci(n+1), if(k<0 || nG. C. Greubel, May 27 2018

T(n,k) = T(n-1,k-1) + T(n-2,k-2) - T(n-2,k), T(0,0)=1, T(n,k)=0 if k<0 or if nA000045(n+1).

Sum_{0<=k<=n} x^k*T(n,k)= A014983(n+1), A033999(n), A056594(n), A000012(n), A015518(n+1), A015525(n+1) for x=-2, -1, 0, 1, 2, 3 respectively.

Corrected and extended by Philippe Deléham, Apr 05 2012

A160444 Expansion of g.f.: x^2(1 + x - x^2)/(1 - 2x^2 - 2*x^4).

Original entry on oeis.org

0, 1, 1, 1, 2, 4, 6, 10, 16, 28, 44, 76, 120, 208, 328, 568, 896, 1552, 2448, 4240, 6688, 11584, 18272, 31648, 49920, 86464, 136384, 236224, 372608, 645376, 1017984, 1763200, 2781184, 4817152, 7598336, 13160704, 20759040, 35955712, 56714752

Offset: 1

Willibald Limbrunner (w.limbrunner(AT)gmx.de), May 14 2009

This sequence is the case k=3 of a family of sequences with recurrences a(2*n+1) = a(2*n) + a(2*n-1), a(2*n+2) = k*a(2*n-1) + a(2*n), a(1)=0, a(2)=1. Values of k, for k >= 0, are given by A057979 (k=0), A158780 (k=1), A002965 (k=2), this sequence (k=3). See "Family of sequences for k" link for other connected sequences.
It seems that the ratio of two successive numbers with even, or two successive numbers with odd, indices approaches sqrt(k) for these sequences as n-> infinity.
This algorithm can be found in a historical figure named "Villardsche Figur" of the 13th century. There you can see a geometrical interpretation.

G. C. Greubel, Table of n, a(n) for n = 1..1000
W. Beinert, Villardscher Teilungskanon, Lexikon der Typographie
W. Limbrunner, Das Quadrat, ein Wunder der Geometrie. (in German)
Willibald Limbrunner, Family of sequences for k
M-T. Zenner, Villard de Honnecourt and Euclidean Geoometry, Nexus Network Journal 4 (2002) 65-78.
Index entries for linear recurrences with constant coefficients, signature (0,2,0,2).

Cf. A002532, A002533, A002534, A002535, A002605, A002965, A003665.
Cf. A003683, A015518, A015519, A026150, A046717, A057979, A063727.
Cf. A083098, A083099, A083100, A084057, A158780.

I:=[0,1,1,1]; [n le 4 select I[n] else 2*(Self(n-2) +Self(n-4)): n in [1..40]]; // G. C. Greubel, Feb 18 2023

LinearRecurrence[{0,2,0,2}, {0,1,1,1}, 40] (* G. C. Greubel, Feb 18 2023 *)

@CachedFunction
def a(n): # a = A160444
    if (n<5): return ((n+1)//3)
    else: return 2*(a(n-2) + a(n-4))
[a(n) for n in range(1, 41)] # G. C. Greubel, Feb 18 2023

a(n) = 2*a(n-2) + 2*a(n-4).
a(2*n+1) = A002605(n).
a(2*n) = A026150(n-1).

Edited by R. J. Mathar, May 14 2009

A226493 Closed walks of length n in K_4 graph.

Original entry on oeis.org

0, 12, 24, 84, 240, 732, 2184, 6564, 19680, 59052, 177144, 531444, 1594320, 4782972, 14348904, 43046724, 129140160, 387420492, 1162261464, 3486784404, 10460353200, 31381059612, 94143178824, 282429536484, 847288609440, 2541865828332, 7625597484984, 22876792454964

Offset: 1

Gustavo Gordillo, Jun 09 2013

Essentially the same as A218034.

Mike Krebs and Tony Shaheen, Expander Families and Cayley Graphs, Oxford University Press, Inc. 2011

K. Böhmová, C. Dalfó, and C. Huemer, On cyclic Kautz digraphs, Preprint 2016.
Cristina Dalfó, From subKautz digraphs to cyclic Kautz digraphs, arXiv:1709.01882 [math.CO], 2017.
C. Dalfó, The spectra of subKautz and cyclic Kautz digraphs, Linear Algebra and its Applications, 531 (2017), p. 210-219.
Carlos I. Perez-Sanchez, The Spectral Action on quivers, arXiv:2401.03705 [math.RT], 2024.
Index entries for linear recurrences with constant coefficients, signature (2,3).

Column k=4 of A106512.

Cf. A218034.

Mathematica
```
Table[3 (-1)^k + 3^k, {k, 30}]
```

a(n) = { 3*(-1)^n + 3^n } \\ Andrew Howroyd, Sep 11 2019

a(n) = 3*(-1)^n + 3^n = 12*A015518(n-1).

G.f.: 12*x^2 / ( (1+x)*(1-3*x) ). - R. J. Mathar, Jun 29 2013

cp's OEIS Frontend

A271116 Integers n such that round(3^n/12) is divisible by n.

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A360036 Expansion of e.g.f. x*exp(x)*(sinh(x))^2.

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A015609 a(n) = 11*a(n-1) + 12*a(n-2).

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A053535 Expansion of 1/((1+3*x)*(1-9*x)).

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A053536 Expansion of 1/((1+4*x)*(1-12*x)).

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A053537 Expansion of 1/((1+5*x)*(1-15*x)).

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A360036 Expansion of e.g.f. xexp(x)(sinh(x))^2.

A015609 a(n) = 11a(n-1) + 12a(n-2).

A053535 Expansion of 1/((1+3x)(1-9*x)).

A053536 Expansion of 1/((1+4x)(1-12*x)).

A053537 Expansion of 1/((1+5x)(1-15*x)).

A160444 Expansion of g.f.: x^2(1 + x - x^2)/(1 - 2x^2 - 2*x^4).