cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A299273 Partial sums of A299272.

Original entry on oeis.org

1, 7, 25, 62, 125, 224, 366, 555, 804, 1121, 1505, 1973, 2535, 3183, 3939, 4816, 5797, 6910, 8172, 9555, 11094, 12811, 14665, 16699, 18941, 21335, 23933, 26770, 29773, 33004, 36506, 40187, 44120, 48357, 52785, 57489, 62531, 67775, 73319, 79236, 85365, 91818, 98680, 105763, 113194, 121071, 129177
Offset: 0

Views

Author

N. J. A. Sloane, Feb 10 2018

Keywords

Links

Crossrefs

Cf. A299272.
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.
Cf. A049347, A099837.

Programs

  • Magma
    Q:=Rationals(); R:=PowerSeriesRing(Q, 40); Coefficients(R!((1+x)^3*(1+x^2)*(1+3*x+5*x^2+3*x^3+x^4)/((1-x)^4*(1+x+x^2)^3))); // G. C. Greubel, Feb 20 2018
  • Mathematica
    CoefficientList[Series[(1+x)^3*(1+x^2)*(1+3*x+5*x^2+3*x^3+x^4)/((1-x)^4*(1+x+x^2)^3), {x, 0, 50}], x] (* G. C. Greubel, Feb 20 2018 *)
  • PARI
    x='x+O('x^30); Vec((1+x)^3*(1+x^2)*(1+3*x+5*x^2+3*x^3+x^4)/((1-x)^4*(1+x+x^2)^3)) \\ G. C. Greubel, Feb 20 2018

Formula

Conjectures from Colin Barker, Feb 11 2018: (Start)
G.f.: (1 + x)^3*(1 + x^2)*(1 + 3*x + 5*x^2 + 3*x^3 + x^4) / ((1 - x)^4*(1 + x + x^2)^3).
a(n) = a(n-1) + 3*a(n-3) - 3*a(n-4) - 3*a(n-6) + 3*a(n-7) + a(n-9) - a(n-10) for n>9.
(End)
These conjectures are correct. - N. J. A. Sloane, Feb 12 2018
a(n) = (12*(2*n + 1)*(26*n*(n + 1) + 45) + (9*n^2 + 39*n - 54)*A099837(n+3)/2 + 3*(3*(n - 9)*n - 38)*A049347(n+2)/2)/486. - Stefano Spezia, Jun 06 2024

A299274 Coordination sequence for "hal" 3D uniform tiling.

Original entry on oeis.org

1, 4, 9, 18, 30, 47, 69, 91, 125, 160, 191, 238, 282, 331, 391, 448, 508, 582, 650, 709, 790, 877, 964, 1047, 1140, 1253, 1353, 1463, 1560, 1667, 1801, 1908, 2043, 2165, 2297, 2471, 2580, 2737, 2893, 3020, 3202, 3344, 3529, 3686, 3856, 4082, 4205, 4429, 4613, 4765, 5025, 5173, 5410
Offset: 0

Views

Author

N. J. A. Sloane, Feb 10 2018

Keywords

Comments

This tiling is the f-tCO-trille tiling, described on page 298 of Symmetries of Things. (Note that there is a typo in the figure caption - tO should be tT.)- Chaim Goodman-Strauss, Feb 13 2018.
Other names for this tiling: In RCSR, HL42, 4/3/c8, 6^2.8^2 F-RD. Wells (page 146), refers to it as the 4-connected Fm3m net or 96(k) or H and L 4_2 (cubic) net, with a reference to Andreini Fig 23.
First 120 terms computed by Davide M. Proserpio using ToposPro.

References

  • J. H. Conway, H. Burgiel and Chaim Goodman-Strauss, The Symmetries of Things, A K Peters, Ltd., 2008, ISBN 978-1-56881-220-5.
  • B. Grünbaum, Uniform tilings of 3-space, Geombinatorics, 4 (1994), 49-56. See tiling #21.
  • A. F. Wells, Three-dimensional Nets and Polyhedra, Wiley, 1977

Links

Crossrefs

See A299275 for partial sums.
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.

Formula

From N. J. A. Sloane, Feb 13 2018 (Start):
Based on the 120 terms computed from the definition by Davide M. Proserpio, and using gfun, it appears that the g.f. is p(x)/q(x), where p(x) and q(x) are respectively
6*x^43 + 12*x^42 + 26*x^41 + 38*x^40 + 47*x^39 + 45*x^38 + 31*x^37 + 9*x^36 - 14*x^35 - 30*x^34 - 35*x^33 - 10*x^32 + 50*x^31 + 173*x^30 + 368*x^29 + 645*x^28 + 1006*x^27 + 1426*x^26 + 1889*x^25 + 2367*x^24 + 2835*x^23 + 3267*x^22 + 3630*x^21 + 3887*x^20 + 4038*x^19 + 4040*x^18 + 3931*x^17 + 3695*x^16 + 3379*x^15 + 2992*x^14 + 2567*x^13 + 2127*x^12 + 1701*x^11 + 1308*x^10 + 964*x^9 + 680*x^8 + 453*x^7 + 285*x^6 + 166*x^5 + 87*x^4 + 41*x^3 + 16*x^2 + 5*x + 1
and
(x + 1)*(x^2 + 1)*(x^6 + x^3 + 1)*(x^2 + x + 1)^2*(x^4 - x^3 + x^2 - x + 1)^2*(1 - x)^3*(x^4 + x^3 + x^2 + x + 1)^3.
The denominator q(x) can also be written as
(1-x^3)*(1-x^4)*(1-x^5)*(1-x^9)*(1-x^10)^2/((1-x)^3*(1+x)^2).
However, this g.f. is so much more complicated than the g.f.s for any of the other 27 3D uniform tilings, at present I am only willing to state it as a conjecture.
It should not be used to extend the sequence beyond 120 terms. (End)

A299275 Partial sums of A299274.

Original entry on oeis.org

1, 5, 14, 32, 62, 109, 178, 269, 394, 554, 745, 983, 1265, 1596, 1987, 2435, 2943, 3525, 4175, 4884, 5674, 6551, 7515, 8562, 9702, 10955, 12308, 13771, 15331, 16998, 18799, 20707, 22750, 24915, 27212, 29683, 32263, 35000, 37893, 40913, 44115, 47459, 50988, 54674, 58530, 62612, 66817
Offset: 0

Views

Author

N. J. A. Sloane, Feb 10 2018

Keywords

Comments

First 120 terms computed by Davide M. Proserpio using ToposPro.

Links

Crossrefs

Cf. A299274.
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.

Formula

There is a conjectured g.f., see the g.f. for A299274 and divide by 1-x. Note: this should not be used to generate a b-file. - N. J. A. Sloane, Feb 13 2018

A299276 Partial sums of A008137.

Original entry on oeis.org

1, 5, 14, 31, 59, 101, 161, 242, 347, 479, 641, 837, 1070, 1343, 1659, 2021, 2433, 2898, 3419, 3999, 4641, 5349, 6126, 6975, 7899, 8901, 9985, 11154, 12411, 13759, 15201, 16741, 18382, 20127, 21979, 23941, 26017, 28210, 30523, 32959, 35521, 38213, 41038
Offset: 0

Views

Author

N. J. A. Sloane, Feb 10 2018

Keywords

Comments

Euler transform of length 6 sequence [5, -1, 1, -1, 1, -1]. - Michael Somos, Oct 03 2018

Examples

			G.f. = 1 + 5*x + 14*x^2 + 31*x^3 + 59*x^4 + 101*x^5 + 161*x^6 + ... - _Michael Somos_, Oct 03 2018

Links

Crossrefs

Cf. A008137.
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.

Programs

  • Mathematica
    a[ n_] := (8 n^3 + 12 n^2 + 40 n + 18 - {3, 3, 0, -3, -3, 3}[[Mod[n, 5] + 1]]) / 15; (* Michael Somos, Oct 03 2018 *)
  • PARI
    Vec((1 + x)^3*(1 - x + x^2)*(1 + x^2) / ((1 - x)^4*(1 + x + x^2 + x^3 + x^4)) + O(x^60)) \\ Colin Barker, Feb 11 2018
  • PARI
    {a(n) = (8*n^3 + 12*n^2 + 40*n + 18 - 3*(n%5<2) + 3*(n%5>2)) / 15}; /* Michael Somos, Oct 03 2018 */

Formula

From Colin Barker, Feb 11 2018: (Start)
G.f.: (1 + x)^3*(1 - x + x^2)*(1 + x^2) / ((1 - x)^4*(1 + x + x^2 + x^3 + x^4)).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + a(n-5) - 3*a(n-6) + 3*a(n-7) - a(n-8) for n>7.
(End)
a(n) = -a(-1-n) for all n in Z.

A299277 Coordination sequence for "pcu-i" 3D uniform tiling.

Original entry on oeis.org

1, 5, 13, 26, 46, 73, 104, 140, 187, 240, 292, 352, 417, 482, 567, 660, 740, 838, 944, 1031, 1150, 1290, 1399, 1531, 1677, 1787, 1944, 2130, 2261, 2431, 2624, 2750, 2941, 3180, 3334, 3538, 3777, 3920, 4149, 4440, 4610, 4852, 5144, 5297, 5560, 5910, 6097, 6373, 6717, 6881, 7182, 7590, 7787, 8101, 8504, 8672
Offset: 0

Views

Author

N. J. A. Sloane, Feb 10 2018

Keywords

Comments

First 80 terms computed by Davide M. Proserpio using ToposPro.

References

  • B. Grünbaum, Uniform tilings of 3-space, Geombinatorics, 4 (1994), 49-56. See tiling #20.

Links

Crossrefs

See A299278 for partial sums.
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.

Programs

  • Mathematica
    CoefficientList[Series[(x^16-x^15+x^14-2x^13+2x^12-x^11+4x^10+x^9+9x^8+12x^6-x^5+ 9x^4+4x^2+1)(x+1)^5/((1+x^2)(1-x^3)(1-x^6)^2),{x,0,60}],x] (* or *) LinearRecurrence[{ 2,-4,7,-10,14,-16,18,-18,16,-14,10,-7,4,-2,1},{1,5,13,26,46,73,104,140,187,240,292,352,417,482,567,660,740,838,944,1031},60] (* Harvey P. Dale, Mar 09 2024 *)
  • PARI
    Vec((x^16 - x^15 + x^14 - 2*x^13 + 2*x^12 - x^11 + 4*x^10 + x^9 + 9*x^8 + 12*x^6 - x^5 + 9*x^4 + 4*x^2 + 1) * (x + 1)^5 / ((1 + x^2)*(1 - x^3)*(1 - x^6)^2) + O(x^60)) \\ Colin Barker, Feb 14 2018

Formula

G.f.: (x^16 - x^15 + x^14 - 2*x^13 + 2*x^12 - x^11 + 4*x^10 + x^9 + 9*x^8 + 12*x^6 - x^5 + 9*x^4 + 4*x^2 + 1) * (x + 1)^5 / ((1 + x^2)*(1 - x^3)*(1 - x^6)^2). - N. J. A. Sloane, Feb 13 2018
a(n) = -a(n-2) + a(n-3) + a(n-5) + 2*a(n-6) + 2*a(n-8) - 2*a(n-9) - 2*a(n-11) - a(n-12) - a(n-14) + a(n-15) + a(n-17) for n>21. - Colin Barker, Feb 14 2018

A299278 Partial sums of A299277.

Original entry on oeis.org

1, 6, 19, 45, 91, 164, 268, 408, 595, 835, 1127, 1479, 1896, 2378, 2945, 3605, 4345, 5183, 6127, 7158, 8308, 9598, 10997, 12528, 14205, 15992, 17936, 20066, 22327, 24758, 27382, 30132, 33073, 36253, 39587, 43125, 46902, 50822, 54971, 59411, 64021, 68873, 74017, 79314, 84874
Offset: 0

Views

Author

N. J. A. Sloane, Feb 10 2018

Keywords

Comments

First 80 terms computed by Davide M. Proserpio using ToposPro.

Links

Crossrefs

Cf. A299277.
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.

Programs

  • PARI
    Vec((x^16 - x^15 + x^14 - 2*x^13 + 2*x^12 - x^11 + 4*x^10 + x^9 + 9*x^8 + 12*x^6 - x^5 + 9*x^4 + 4*x^2 + 1) * (x + 1)^5 / ((1 - x)*(1 + x^2)*(1 - x^3)*(1 - x^6)^2) + O(x^60)) \\ Colin Barker, Feb 14 2018

Formula

G.f.: (x^16 - x^15 + x^14 - 2*x^13 + 2*x^12 - x^11 + 4*x^10 + x^9 + 9*x^8 + 12*x^6 - x^5 + 9*x^4 + 4*x^2 + 1) * (x + 1)^5 / ((1 - x)*(1 + x^2)*(1 - x^3)*(1 - x^6)^2). - N. J. A. Sloane, Feb 13 2018
a(n) = a(n-1) - a(n-2) + 2*a(n-3) - a(n-4) + a(n-5) + a(n-6) - 2*a(n-7) + 2*a(n-8) - 4*a(n-9) + 2*a(n-10) - 2*a(n-11) + a(n-12) + a(n-13) - a(n-14) + 2*a(n-15) - a(n-16) + a(n-17) - a(n-18) for n>21. - Colin Barker, Feb 14 2018

A299280 Partial sums of A299279.

Original entry on oeis.org

1, 9, 39, 107, 233, 413, 699, 1047, 1557, 2129, 2927, 3779, 4929, 6117, 7683, 9263, 11309, 13337, 15927, 18459, 21657, 24749, 28619, 32327, 36933, 41313, 46719, 51827, 58097, 63989, 71187, 77919, 86109, 93737, 102983, 111563, 121929, 131517, 143067, 153719, 166517
Offset: 0

Views

Author

N. J. A. Sloane, Feb 10 2018

Keywords

Links

Crossrefs

Cf. A299279.
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.

Programs

  • Mathematica
    LinearRecurrence[{1,3,-3,-3,3,1,-1},{1,9,39,107,233,413,699,1047},50] (* Harvey P. Dale, Jul 22 2021 *)
  • PARI
    Vec((1 + 8*x + 27*x^2 + 44*x^3 + 39*x^4 - 3*x^6 + 4*x^7) / ((1 - x)^4*(1 + x)^3) + O(x^60)) \\ Colin Barker, Feb 11 2018

Formula

From Colin Barker, Feb 11 2018: (Start)
G.f.: (1 + 8*x + 27*x^2 + 44*x^3 + 39*x^4 - 3*x^6 + 4*x^7) / ((1 - x)^4*(1 + x)^3).
a(n) = (5*n^3 + 8*n^2 + 6*n - 6) / 2 for n>0 and even.
a(n) = (5*n^3 + 7*n^2 + 5*n + 1) / 2 for n odd.
a(n) = a(n-1) + 3*a(n-2) - 3*a(n-3) - 3*a(n-4) + 3*a(n-5) + a(n-6) - a(n-7) for n>7. (End)
E.g.f.: (8 - (6 - 17*x - 23*x^2 - 5*x^3)*cosh(x) + (1 + 19*x + 22*x^2 + 5*x^3)*sinh(x))/2. - Stefano Spezia, Jun 06 2024

A299281 Coordination sequence for "reo-e" 3D uniform tiling.

Original entry on oeis.org

1, 6, 19, 41, 72, 114, 166, 224, 288, 364, 454, 550, 648, 758, 886, 1020, 1152, 1296, 1462, 1634, 1800, 1978, 2182, 2392, 2592, 2804, 3046, 3294, 3528, 3774, 4054, 4340, 4608, 4888, 5206, 5530, 5832, 6146, 6502, 6864, 7200, 7548, 7942, 8342, 8712, 9094, 9526, 9964, 10368
Offset: 0

Views

Author

N. J. A. Sloane, Feb 10 2018

Keywords

Comments

First 20 terms computed by Davide M. Proserpio using ToposPro.

References

  • B. Grünbaum, Uniform tilings of 3-space, Geombinatorics, 4 (1994), 49-56. See tiling #9.

Links

Crossrefs

See A299282 for partial sums.
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.
Cf. A056594.

Programs

  • PARI
    Vec((1 + x)*(1 + x^2 + x^3)*(1 + 2*x + 3*x^2 + x^4 - 2*x^5 + x^6) / ((1 - x)^3*(1 + x^2)^2) + O(x^60)) \\ Colin Barker, Feb 14 2018

Formula

G.f.: (x+1)*(x^3+x^2+1)*(x^6-2*x^5+x^4+3*x^2+2*x+1) / ((x^2+1)^2*(1-x)^3). - N. J. A. Sloane, Feb 12 2018
a(n) = 3*a(n-1) - 5*a(n-2) + 7*a(n-3) - 7*a(n-4) + 5*a(n-5) - 3*a(n-6) + a(n-7) for n>8. - Colin Barker, Feb 14 2018
a(n) = (9*n^2 + 4*(1 - A056594(n)) - (n - 4)*A056594(n+1))/2 for n > 3. - Stefano Spezia, Apr 23 2023

Extensions

a(21)-a(40) from Davide M. Proserpio, Feb 12 2018

A299282 Partial sums of A299281.

Original entry on oeis.org

1, 7, 26, 67, 139, 253, 419, 643, 931, 1295, 1749, 2299, 2947, 3705, 4591, 5611, 6763, 8059, 9521, 11155, 12955, 14933, 17115, 19507, 22099, 24903, 27949, 31243, 34771, 38545, 42599, 46939, 51547, 56435, 61641, 67171, 73003, 79149, 85651, 92515, 99715, 107263, 115205
Offset: 0

Views

Author

N. J. A. Sloane, Feb 10 2018

Keywords

Links

Crossrefs

Cf. A056594, A299281.
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.

Programs

  • PARI
    Vec((1 + x)*(1 + x^2 + x^3)*(1 + 2*x + 3*x^2 + x^4 - 2*x^5 + x^6) / ((1 - x)^4*(1 + x^2)^2) + O(x^70)) \\ Colin Barker, Feb 14 2018

Formula

From Colin Barker, Feb 14 2018: (Start)
G.f.: (1 + x)*(1 + x^2 + x^3)*(1 + 2*x + 3*x^2 + x^4 - 2*x^5 + x^6) / ((1 - x)^4*(1 + x^2)^2).
a(n) = 4*a(n-1) - 8*a(n-2) + 12*a(n-3) - 14*a(n-4) + 12*a(n-5) - 8*a(n-6) + 4*a(n-7) - a(n-8) for n>8. (End)
a(n) = (n*(6*n^2 + 9*n + 11) - 12 + (n - 8)*A056594(n) - (n + 1)*A056594(n+1))/4 for n > 2. - Stefano Spezia, Apr 23 2023

A299283 Coordination sequence for "svh" 3D uniform tiling.

Original entry on oeis.org

1, 7, 22, 48, 84, 130, 186, 253, 330, 417, 514, 622, 740, 868, 1006, 1155, 1314, 1483, 1662, 1852, 2052, 2262, 2482, 2713, 2954, 3205, 3466, 3738, 4020, 4312, 4614, 4927, 5250, 5583, 5926, 6280, 6644, 7018, 7402, 7797, 8202, 8617, 9042, 9478, 9924, 10380
Offset: 0

Views

Author

N. J. A. Sloane, Feb 10 2018

Keywords

Comments

First 20 terms computed by Davide M. Proserpio using ToposPro.

References

  • B. Grünbaum, Uniform tilings of 3-space, Geombinatorics, 4 (1994), 49-56. See tiling #15.

Links

Crossrefs

See A299284 for partial sums.
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.

Programs

  • Mathematica
    LinearRecurrence[{2,-1,0,1,-2,1},{1,7,22,48,84,130,186},50] (* Harvey P. Dale, May 19 2019 *)
  • PARI
    Vec((1 + 5*x + 9*x^2 + 11*x^3 + 9*x^4 + 5*x^5 + x^6) / ((1 - x)^3*(1 + x)*(1 + x^2)) + O(x^60)) \\ Colin Barker, Feb 11 2018

Formula

G.f.: (x^6+5*x^5+9*x^4+11*x^3+9*x^2+5*x+1)/((x+1)*(x^2+1)*(1-x)^3).
a(n) = 2*a(n-1) - a(n-2) + a(n-4) - 2*a(n-5) + a(n-6) for n>6. - Colin Barker, Feb 11 2018
a(n) = (29 - (-1)^n + 82*n^2 + 4*A056594(n))/16 for n > 0. - Stefano Spezia, Jun 06 2024
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