A108099 -id:A108099 - 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

A187046 T(n,k)=Number of n-step one or two space at a time bishop's tours on a kXk board summed over all starting positions.

Original entry on oeis.org

1, 4, 0, 9, 4, 0, 16, 20, 0, 0, 25, 52, 28, 0, 0, 36, 100, 136, 24, 0, 0, 49, 164, 360, 272, 8, 0, 0, 64, 244, 696, 1084, 456, 0, 0, 0, 81, 340, 1144, 2660, 2896, 584, 0, 0, 0, 100, 452, 1704, 5032, 9216, 6952, 400, 0, 0, 0, 121, 580, 2376, 8164, 20648, 29500, 14024, 80, 0

Offset: 1

R. H. Hardin Mar 02 2011

Table starts
.1.4..9..16....25......36......49......64......81.....100.....121....144....169
.0.4.20..52...100.....164.....244.....340.....452.....580.....724....884...1060
.0.0.28.136...360.....696....1144....1704....2376....3160....4056...5064...6184
.0.0.24.272..1084....2660....5032....8164...12056...16708...22120..28292..35224
.0.0..8.456..2896....9216...20648...37472...59524...86656..118868.156160.198532
.0.0..0.584..6952...29500...79088..162320..281320..435436..623508.845084
.0.0..0.400.14024...86608..285048..668176.1272720.2110944.3180724
.0.0..0..80.22200..226864..961728.2625640.5543744.9907248
.0.0..0...0.26672..522608.3009408.9817240
.0.0..0...0.22448.1050048.8701592

			Some n=4 solutions for 4X4
..0..2..0..0....0..4..0..0....0..0..0..0....4..0..0..0....0..3..0..0
..1..0..3..0....0..0..0..0....0..0..3..0....0..3..0..0....0..0..2..0
..0..0..0..4....0..1..0..3....0..0..0..2....0..0..1..0....0..1..0..4
..0..0..0..0....0..0..2..0....4..0..1..0....0..0..0..2....0..0..0..0

R. H. Hardin, Table of n, a(n) for n = 1..144

Row 2 is A108099(n-2)

Empirical: Row n is a polynomial of degree 2 for k>2n-3

A164013 3 times centered triangular numbers: 9n(n+1)/2 + 3.

Original entry on oeis.org

3, 12, 30, 57, 93, 138, 192, 255, 327, 408, 498, 597, 705, 822, 948, 1083, 1227, 1380, 1542, 1713, 1893, 2082, 2280, 2487, 2703, 2928, 3162, 3405, 3657, 3918, 4188, 4467, 4755, 5052, 5358, 5673, 5997, 6330, 6672, 7023, 7383, 7752

Offset: 0

Omar E. Pol, Nov 07 2009

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A005448, A045943, A108099, A164015, A164016.

LinearRecurrence[{3,-3,1},{3,12,30},50] (* Harvey P. Dale, Mar 26 2015 *)

a(n)=9*binomial(n+1,2)+3 \\ Charles R Greathouse IV, Jul 17 2011

a(n) = a(n-1) + 9*n (with a(0)=3). - Vincenzo Librandi, Nov 30 2010
a(0)=3, a(1)=12, a(2)=30, a(n)=3*a(n-1)-3*a(n-2)+a(n-3). - Harvey P. Dale, Mar 26 2015
From G. C. Greubel, Sep 06 2017: (Start)
G.f.: 3*(1 + x + x^2)/(1 - x)^3.
E.g.f.: (3/2)*(2 + 6*x + 3*x^2)*exp(x). (End)

A164015 5 times centered pentagonal numbers: 5(5n^2 + 5*n + 2)/2.

Original entry on oeis.org

5, 30, 80, 155, 255, 380, 530, 705, 905, 1130, 1380, 1655, 1955, 2280, 2630, 3005, 3405, 3830, 4280, 4755, 5255, 5780, 6330, 6905, 7505, 8130, 8780, 9455, 10155, 10880, 11630, 12405, 13205, 14030, 14880, 15755, 16655, 17580, 18530

Offset: 0

Omar E. Pol, Nov 07 2009

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A005891, A152734, A164013, A108099, A164016.

Table[5(5n^2+5n+2)/2,{n,0,40}] (* or *) LinearRecurrence[{3,-3,1},{5,30,80},40] (* Harvey P. Dale, Oct 08 2011 *)

a(n)=25*n*(n+1)/2+5 \\ Charles R Greathouse IV, Jul 17 2011

a(n) = 5*A005891(n).
a(n) = a(n-1) + 25*n (with a(0)=5). - Vincenzo Librandi, Nov 30 2010
a(0)=5, a(1)=30, a(2)=80, a(n) = 3*a(n-1)-3*a(n-2)+a(n-3). - Harvey P. Dale, Oct 08 2011
G.f.: (5*(x*(x+3)+1))/(1-x)^3. - Harvey P. Dale, Oct 08 2011
E.g.f.: (5/2)*(2 + 10*x + 5*x^2)*exp(x). - G. C. Greubel, Sep 06 2017

A164016 6 times centered hexagonal numbers: 18n(n+1) + 6.

Original entry on oeis.org

6, 42, 114, 222, 366, 546, 762, 1014, 1302, 1626, 1986, 2382, 2814, 3282, 3786, 4326, 4902, 5514, 6162, 6846, 7566, 8322, 9114, 9942, 10806, 11706, 12642, 13614, 14622, 15666, 16746, 17862, 19014, 20202, 21426, 22686, 23982, 25314

Offset: 0

Omar E. Pol, Nov 07 2009

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A003215, A152746, A164013, A108099, A164015.

Table[18n(n+1)+6,{n,0,40}] (* or *) LinearRecurrence[{3,-3,1},{6,42,114},40] (* Harvey P. Dale, Dec 16 2012 *)

a(n)=18*n*(n+1)+6 \\ Charles R Greathouse IV, Jul 17 2011

a(n) = A003215(n)*6.
a(n) = a(n-1) + 36*n (with a(0)=6). - Vincenzo Librandi, Nov 30 2010
a(n) = 3*a(n-1)-3*a(n-2)+a(n-3) with a(0)=6, a(1)=42, a(2)=114. - Harvey P. Dale, Dec 16 2012
From G. C. Greubel, Sep 07 2017: (Start)
G.f.: 6*(1 + 4*x + x^2)/(1 - x)^3.
E.g.f.: 6*(1 + 6*x + 3*x^2)*exp(x). (End)

A171272 a(n) = 1 + 4n(1 + 2*n^2)/3.

Original entry on oeis.org

1, 5, 25, 77, 177, 341, 585, 925, 1377, 1957, 2681, 3565, 4625, 5877, 7337, 9021, 10945, 13125, 15577, 18317, 21361, 24725, 28425, 32477, 36897, 41701, 46905, 52525, 58577, 65077, 72041, 79485, 87425, 95877, 104857, 114381, 124465, 135125, 146377, 158237, 170721, 183845

Offset: 0

Paul Curtz, Dec 06 2009

Binomial transform of quasi-finite sequence 1,4,16,16,0,0,... (0 continued).

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

[1+4*n*(1+2*n^2)/3: n in [0..40]]; // Vincenzo Librandi, Aug 05 2011

LinearRecurrence[{4,-6,4,-1},{1,5,25,77},50] (* Harvey P. Dale, Nov 22 2011 *)

a(n)=4*n*(1+2*n^2)/3+1 \\ Charles R Greathouse IV, Jul 07 2011

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
First differences: a(n+1) - a(n) = A108099(n).
Second differences: a(n+2) - 2*a(n+1) + a(n) = A008598(n+1).
Third differences: a(n+3) - 3*a(n+2) + 3*a(n+1) - a(n) = 16.
a(n) = (A168574(n) + A168547(n))/2. - This formula is the link to the Janet table of the PSE.
G.f.: ( 1 + x + 11*x^2 + 3*x^3 ) / (x-1)^4. - R. J. Mathar, Jul 07 2011
E.g.f.: (3 +12*x +24*x^2 +8*x^3)*exp(x)/3. - G. C. Greubel, Nov 02 2018

A236461 Sum of two consecutive primes that is also sum of two consecutive even positive squares.

Original entry on oeis.org

52, 100, 340, 1460, 2452, 2740, 4420, 20404, 21220, 36452, 48052, 62660, 66980, 94180, 103060, 108580, 128020, 140452, 142580, 169364, 171700, 195940, 221780, 254900, 260644, 361252, 378452, 490052, 498004, 717604, 736900, 756452, 766324, 791284, 879140, 889780, 916660, 1016740, 1104100, 1164340, 1232452, 1283204

Offset: 1

Zak Seidov, Jan 26 2014

nonn

All values of (q - p) are multiples of 6.
m = p + q = x^2 + (x+2)^2; {m,p,q,x}: {52, 23, 29, 4}, {100, 47, 53, 6}, {340, 167,  173, 12}, {1460, 727, 733, 26}, {2452, 1223,  1229, 34}, {2740, 1367, 1373, 36}, {4420, 2207, 2213, 46}.
Intersection of A001043 and A108099. - Michel Marcus, Jan 27 2014

			52 = 23 + 29 = 4^2 + 6^2.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A001043, A108099.

count:= 0: R:= NULL:
for m from 1 while count < 100 do
  y:= 8*m^2+8*m+4;
  if prevprime(y/2) + nextprime(y/2)=y then
     count:= count+1;
     R:= R, y;
  fi
od:
R; # Robert Israel, Jan 07 2020

With[{nn=100000},Intersection[Total/@Partition[Prime[Range[nn]],2,1],Total/@ Partition[Range[2,2nn,2]^2,2,1]]] (* Harvey P. Dale, Jul 03 2021 *)

v=vector(1300000); pp=3; forprime(p=5,#v/2,v[p+pp]++;pp=p);forstep(k=2,sqrtint(#v/2)-1,2,v[2*(k^2+2*k+2)]++);for(k=1,#v,if(v[k]==2,print1(k,", "))) \\ Hugo Pfoertner, Jan 07 2020

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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A187046 T(n,k)=Number of n-step one or two space at a time bishop's tours on a kXk board summed over all starting positions.

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A164013 3 times centered triangular numbers: 9*n*(n+1)/2 + 3.

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A164015 5 times centered pentagonal numbers: 5*(5*n^2 + 5*n + 2)/2.

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A164016 6 times centered hexagonal numbers: 18*n*(n+1) + 6.

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A171272 a(n) = 1 + 4*n*(1 + 2*n^2)/3.

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A236461 Sum of two consecutive primes that is also sum of two consecutive even positive squares.

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A164013 3 times centered triangular numbers: 9n(n+1)/2 + 3.

A164015 5 times centered pentagonal numbers: 5(5n^2 + 5*n + 2)/2.

A164016 6 times centered hexagonal numbers: 18n(n+1) + 6.

A171272 a(n) = 1 + 4n(1 + 2*n^2)/3.