A053545 -id:A053545 - cp's OEIS frontend

A005893 Number of points on surface of tetrahedron; coordination sequence for sodalite net (equals 2*n^2+2 for n > 0).

1, 4, 10, 20, 34, 52, 74, 100, 130, 164, 202, 244, 290, 340, 394, 452, 514, 580, 650, 724, 802, 884, 970, 1060, 1154, 1252, 1354, 1460, 1570, 1684, 1802, 1924, 2050, 2180, 2314, 2452, 2594, 2740, 2890, 3044, 3202, 3364, 3530, 3700, 3874, 4052, 4234

Offset: 0

N. J. A. Sloane

Number of n-matchings of the wheel graph W_{2n} (n > 0). Example: a(2)=10 because in the wheel W_4 (rectangle ABCD and spokes OA,OB,OC,OD) we have the 2-matchings: (AB, OC), (AB, OD), (BC, OA), (BC,OD), (CD,OA), (CD,OB), (DA,OB), (DA,OC), (AB,CD) and (BC,DA). - Emeric Deutsch, Dec 25 2004
For n > 0 a(n) is the difference of two tetrahedral (or pyramidal) numbers: binomial(n+3, 3) = (n+1)(n+2)(n+3)/6. a(n) = A000292(n+1) - A000292(n-3) = (n+1)(n+2)(n+3)/6 - (n-3)(n-2)(n-1)/6. - Alexander Adamchuk, May 20 2006; updated by Peter Munn, Aug 25 2017 due to changed offset in A000292
Equals binomial transform of [1, 3, 3, 1, -1, 1, -1, 1, -1, 1, ...]. Binomial transform of A005893 = nonzero terms of A053545: (1, 5, 19, 63, 191, ...). - Gary W. Adamson, Apr 28 2008
Disregarding the terms < 10, the sums of four consecutive triangular numbers (A000217). - Rick L. Shepherd, Sep 30 2009
Use a set of n concentric circles where n >= 0 to divide the plane. a(n) is the maximal number of regions after the 2nd division. - Frank M Jackson, Sep 07 2011
Euler transform of length 4 sequence [4, 0, 0, -1]. - Michael Somos, May 14 2014
Also, growth series for affine Coxeter group (or affine Weyl group) A_3 or D_3. - N. J. A. Sloane, Jan 11 2016
For n > 2 the generalized Pell's equation x^2 - 2*(a(n) - 2)y^2 = (a(n) - 4)^2 has a finite number of positive integer solutions. - Muniru A Asiru, Apr 19 2016
Union of A188896, A277449, {1,4}. - Muniru A Asiru, Nov 25 2016
Interleaving of A008527 and A108099. - Bruce J. Nicholson, Oct 14 2019

			G.f. = 1 + 4*x + 10*x^2 + 20*x^3 + 34*x^4 + 52*x^5 + 74*x^6 + 100*x^7 + ...

N. Bourbaki, Groupes et Algèbres de Lie, Chap. 4, 5 and 6, Hermann, Paris, 1968. See Chap. VI, Section 4, Problem 10b, page 231, W_a(t).
H. S. M. Coxeter, "Polyhedral numbers," in R. S. Cohen et al., editors, For Dirk Struik. Reidel, Dordrecht, 1974, pp. 25-35.
B. Grünbaum, Uniform tilings of 3-space, Geombinatorics, 4 (1994), 49-56. See tiling #28.
R. W. Marks and R. B. Fuller, The Dymaxion World of Buckminster Fuller. Anchor, NY, 1973, p. 46.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (pdf).
Steven Edwards and William Griffiths, On Generalized Delannoy Numbers, J. Int. Seq., Vol. 23 (2020), Article 20.3.6.
J. M. Grau, C. Miguel, and A. M. Oller-Marcén, Generalized Quaternion Rings over Z/nZ for an odd n, arXiv:1706.04760 [math.RA], 2017. See Theorem 1, p. 10.
Guo-Niu Han, Enumeration of Standard Puzzles. [Cached copy.]
Milan Janjić, Hessenberg Matrices and Integer Sequences, J. Int. Seq. 13 (2010), Article 10.7.8.
M. O'Keeffe, N-dimensional diamond, sodalite and rare sphere packings, Acta Cryst., A 47 (1991), 749-753.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
Reticular Chemistry Structure Resource, sod.
Aditya Sivakumar and Dmitri Tymoczko, Intuitive Musical Homotopy, 2018.
B. K. Teo and N. J. A. Sloane, Magic numbers in polygonal and polyhedral clusters, Inorgan. Chem. 24 (1985),4545-4558. DOI: 10.1021/ic00220a025.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000217, A000292, A053545, A206399.
Cf. similar sequences listed in A255843.
The growth series for the affine Coxeter groups D_3 through D_12 are A005893 and A266759-A266767.
For partial sums see A005894.
The 28 uniform 3D tilings: cab: A299266, A299267; crs: A299268, A299269; fcu: A005901, A005902; fee: A299259, A299265; flu-e: A299272, A299273; fst: A299258, A299264; hal: A299274, A299275; hcp: A007899, A007202; hex: A005897, A005898; kag: A299256, A299262; lta: A008137, A299276; pcu: A005899, A001845; pcu-i: A299277, A299278; reo: A299279, A299280; reo-e:  A299281, A299282; rho: A008137, A299276; sod: A005893, A005894; sve: A299255, A299261; svh: A299283, A299284; svj: A299254, A299260; svk: A010001, A063489; tca: A299285, A299286; tcd: A299287, A299288; tfs: A005899, A001845; tsi: A299289, A299290; ttw: A299257, A299263; ubt: A299291, A299292; bnn: A007899, A007202. See the Proserpio link in A299266 for overview.

[2*n^2-0^n+2: n in [0..60]]; // Vincenzo Librandi, Sep 27 2011

A005893:=-(z+1)*(1+z^2)/(z-1)^3; # Simon Plouffe in his 1992 dissertation

Join[{1}, Table[2*(n + 1)^2 + 2, {n, 0, 200}]] (* Vladimir Joseph Stephan Orlovsky, Jul 10 2011 *)
Join[{1},LinearRecurrence[{3,-3,1},{4,10,20},50]] (* Harvey P. Dale, Feb 26 2012 *)
a[ n_] := SeriesCoefficient[ (1 - x^4) / (1 - x)^4, {x, 0, Abs@n}]; (* Michael Somos, May 14 2014 *)
a[ n_] := 2 n^2 + 2 - Boole[n == 0]; (* Michael Somos, May 14 2014 *)

a(n)=2*n^2-0^n+2 \\ Charles R Greathouse IV, Sep 24 2015

G.f.: (1 - x^4)/(1-x)^4.
a(n) = A071619(n-1) + A071619(n) + A071619(n+1), n > 0. - Ralf Stephan, Apr 26 2003
a(n) = binomial(n+3, 3) - binomial(n-1, 3) for n >= 1. - Mitch Harris, Jan 08 2008
a(n) = (n+1)^2 + (n-1)^2. - Benjamin Abramowitz, Apr 14 2009
a(n) = A000217(n-2) + A000217(n-1) + A000217(n) + A000217(n+1) for n >= 2. - Rick L. Shepherd, Sep 30 2009
a(n) = 2*n^2 - 0^n + 2. - Vincenzo Librandi, Sep 27 2011
a(0)=1, a(1)=4, a(2)=10, a(3)=20, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Feb 26 2012
a(n) = A228643(n+1,2) for n > 0. - Reinhard Zumkeller, Aug 29 2013
a(n) = a(-n) for all n in Z. - Michael Somos, May 14 2014
For n >= 2: a(n) = a(n-1) + 4*n - 2. - Bob Selcoe, Mar 22 2016
E.g.f.: -1 + 2*(1 + x + x^2)*exp(x). - Ilya Gutkovskiy, Apr 19 2016
a(n) = 2*A002522(n), n>0. - R. J. Mathar, May 30 2022
From Amiram Eldar, Sep 16 2022: (Start)
Sum_{n>=0} 1/a(n) = (coth(Pi)*Pi + 3)/4.
Sum_{n>=0} (-1)^n/a(n) = (cosech(Pi)*Pi + 3)/4. (End)
Empirical: Integral_{u=-oo..+oo} sigmoid(u)*log(sigmoid(n * u)) du = -Pi^2*a(n) / (24*n), where sigmoid(x) = 1/(1+exp(-x)). Also works for non-integer n>0. - Carlo Wood, Dec 04 2023
Let P(k,n) be the n-th k-gonal number. Then P(a(k),n) = (k*n-k+1)^2 + (k-1)^2*(n-1). - Charlie Marion, May 15 2024

A053730 a(n) = 2^(n-2)*(n^2 - n + 4).

Original entry on oeis.org

1, 2, 6, 20, 64, 192, 544, 1472, 3840, 9728, 24064, 58368, 139264, 327680, 761856, 1753088, 3997696, 9043968, 20316160, 45350912, 100663296, 222298112, 488636416, 1069547520, 2332033024, 5066719232, 10972299264, 23689428992

Offset: 0

N. J. A. Sloane, Mar 24 2000

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-12,8).

Cf. A053545.

List([0..30], n-> 2^(n-2)*(n^2 -n +4)); # G. C. Greubel, Sep 06 2019

I:=[1, 2, 6]; [n le 3 select I[n] else 6*Self(n-1)-12*Self(n-2) +8*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Apr 28 2012

seq(2^(n-2)*(n^2 -n +4), n=0..30); # G. C. Greubel, Sep 06 2019

CoefficientList[Series[(1-4*x+6*x^2)/(1-2*x)^3,{x,0,30}],x] (* Vincenzo Librandi, Apr 28 2012 *)
LinearRecurrence[{6,-12,8}, {1,2,6}, 30] (* G. C. Greubel, Sep 06 2019 *)

vector(30, n, 2^(n-3)*(n^2 -3*n +6)) \\ G. C. Greubel, Sep 06 2019

[2^(n-2)*(n^2 -n +4) for n in (0..30)] # G. C. Greubel, Sep 06 2019

G.f.: (1-4*x+6*x^2)/(1-2*x)^3. - Colin Barker, Apr 01 2012
a(n) = 6*a(n-1) - 12*a(n-2) + 8*a(n-3). - Vincenzo Librandi, Apr 28 2012
a(n) = Sum_{k=0..n} binomial(n,k) * A077028(n,k), where A077028(n,k) = (n-k)*k + 1. - Paul D. Hanna, Oct 11 2015

A196508 a(n) = 2^n*(n^2 - n + 4)-4.

Original entry on oeis.org

0, 4, 20, 76, 252, 764, 2172, 5884, 15356, 38908, 96252, 233468, 557052, 1310716, 3047420, 7012348, 15990780, 36175868, 81264636, 181403644, 402653180, 889192444, 1954545660, 4278190076, 9328132092, 20266876924, 43889197052

Offset: 0

R. J. Mathar, Oct 03 2011

Jolley, Summation of Series, Dover (1961), eq. (46) on page 8.

Vincenzo Librandi, Table of n, a(n) for n = 0..3000
Index entries for linear recurrences with constant coefficients, signature (7,-18,20,-8)

[2^n*(n^2-n+4)-4: n in [0..30]]; // Vincenzo Librandi, Oct 05 2011

a(n) = 2*2 + 4*4 + 7*8 + 11*16 + 16*32 + ... (n terms of A000124*A000079).
a(n) = 4*A053545(n).
G.f.: 4*x*(1 - 2*x + 2*x^2) / ( (x-1)*(2*x-1)^3 ).

A289714 Triangle T(n,k) read by rows: the number of semigroups of orientation-preserving partial transformations on n element with right waist k.

Original entry on oeis.org

1, 1, 1, 1, 3, 5, 1, 7, 19, 34, 1, 15, 63, 135, 235, 1, 31, 191, 471, 911, 1556, 1, 63, 543, 1503, 3183, 5883, 9969, 1, 127, 1471, 4495, 10319, 20483, 37031, 62602, 1, 255, 3839, 12799, 31615, 67007, 128607, 229743, 388343, 1, 511, 9727, 35071, 92671, 208735, 423583, 796687, 1412863, 2389768, 1

Offset: 0

R. J. Mathar, Sep 02 2017

			1 ;
1 1 ;
1 3 5 ;
1 7 19 34 ;
1 15 63 135 235;
1 31 191 471 911 1556 ;
1 63 543 1503 3183 5883 9969 ;
1 127 1471 4495 10319 20483 37031 62602 ;
1 255 3839 12799 31615 67007 128607 229743 388343 ;
1 511 9727 35071 92671 208735 423583 796687 1412863 2389768 ;
1 1023 24063 93183 262143 625023 1336383 2638143 4894623 8637363 14621533 ;

A. Umar, Combinatorial Results for Semigroups of Orientation-Preserving Partial Transformations, J. Int. Seq. 14 (2011) # 11.7.5

Cf. A289713 (row sums), A000225 (column 1), A053545 (column 2)

A289714 := proc(n,k)
    if k = 0 then
        1;
    else
        n*add(binomial(n-1,r-1)*binomial(r+k-2,r-1),r=1..n)-(n-2)*2^(n-1)-1 ;
    end if ;
end proc:

cp's OEIS Frontend

A005893 Number of points on surface of tetrahedron; coordination sequence for sodalite net (equals 2*n^2+2 for n > 0).

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A053730 a(n) = 2^(n-2)*(n^2 - n + 4).

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A196508 a(n) = 2^n*(n^2 - n + 4)-4.

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A289714 Triangle T(n,k) read by rows: the number of semigroups of orientation-preserving partial transformations on n element with right waist k.

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Maple