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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A299279 Coordination sequence for "reo" 3D uniform tiling.

Original entry on oeis.org

1, 8, 30, 68, 126, 180, 286, 348, 510, 572, 798, 852, 1150, 1188, 1566, 1580, 2046, 2028, 2590, 2532, 3198, 3092, 3870, 3708, 4606, 4380, 5406, 5108, 6270, 5892, 7198, 6732, 8190, 7628, 9246, 8580, 10366, 9588, 11550, 10652, 12798, 11772, 14110, 12948, 15486, 14180
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 #7.

Links

Crossrefs

See A299280 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[{0, 3, 0, -3, 0, 1}, {1, 8, 30, 68, 126, 180, 286, 348}, 50] (* Paolo Xausa, Jan 16 2025 *)
  • PARI
    Vec((1 + 8*x + 27*x^2 + 44*x^3 + 39*x^4 - 3*x^6 + 4*x^7) / ((1 - x)^3*(1 + x)^3) + O(x^60)) \\ Colin Barker, Feb 11 2018

Formula

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

A299254 Coordination sequence for 3D uniform tiling formed by stacking parallel layers of the 3^4.6 2D tiling (cf. A250120).

Original entry on oeis.org

1, 7, 21, 45, 79, 122, 175, 237, 309, 391, 482, 583, 693, 813, 943, 1082, 1231, 1389, 1557, 1735, 1922, 2119, 2325, 2541, 2767, 3002, 3247, 3501, 3765, 4039, 4322, 4615, 4917, 5229, 5551, 5882, 6223, 6573, 6933, 7303, 7682, 8071, 8469, 8877, 9295, 9722, 10159, 10605, 11061, 11527, 12002
Offset: 0

Views

Author

N. J. A. Sloane, Feb 06 2018

Keywords

References

  • B. Grünbaum, Uniform tilings of 3-space, Geombinatorics, 4 (1994), 49-56. See tiling #17.

Links

Crossrefs

Cf. A040000, A250120.
Partial sums: A299260.
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, 0, 1, -2, 1}, {1, 7, 21, 45, 79, 122, 175, 237}, 50] (* Paolo Xausa, Jan 16 2025 *)
  • PARI
    Vec((1 + x)*(1 + x + x^2)*(1 + 3*x + 3*x^3 + x^4) / ((1 - x)^3*(1 + x + x^2 + x^3 + x^4)) + O(x^60)) \\ Colin Barker, Feb 07 2018

Formula

G.f.: (x^2+x+1)*(x^4+3*x^3+3*x+1)*(x+1) / ((x^4+x^3+x^2+x+1)*(1-x)^3). (This is the product of the g.f.'s for A250120 and A040000. - N. J. A. Sloane, Nov 10 2018)
a(n) = 2*a(n-1) - a(n-2) + a(n-5) - 2*a(n-6) + a(n-7) for n>7. - Colin Barker, Feb 07 2018
a(n) = 2*((sqrt(5) - 5)*(5 + 12*n^2) - (sqrt(5) - 1)*cos(2*n*Pi/5) + (sqrt(5) - 1)*cos(4*n*Pi/5))/(5*(sqrt(5) - 5)) for n > 0. - Stefano Spezia, Jun 06 2024

A299255 Coordination sequence for 3D uniform tiling formed by stacking parallel layers of the 3.3.4.3.4 2D tiling (cf. A219529).

Original entry on oeis.org

1, 7, 23, 50, 87, 135, 194, 263, 343, 434, 535, 647, 770, 903, 1047, 1202, 1367, 1543, 1730, 1927, 2135, 2354, 2583, 2823, 3074, 3335, 3607, 3890, 4183, 4487, 4802, 5127, 5463, 5810, 6167, 6535, 6914, 7303, 7703, 8114, 8535, 8967, 9410, 9863, 10327, 10802, 11287, 11783, 12290, 12807, 13335
Offset: 0

Views

Author

N. J. A. Sloane, Feb 07 2018

Keywords

References

  • B. Grünbaum, Uniform tilings of 3-space, Geombinatorics, 4 (1994), 49-56. See tiling #14.

Links

Crossrefs

Cf. A219529.
See A299261 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. A099837.

Programs

  • Mathematica
    LinearRecurrence[{2,-1,1,-2,1},{1,7,23,50,87,135},60] (* Harvey P. Dale, Apr 01 2018 *)
  • PARI
    Vec((1 + x)^5 / ((1 - x)^3*(1 + x + x^2)) + O(x^60)) \\ Colin Barker, Feb 09 2018

Formula

G.f.: (x + 1)^5 / ((x^2 + x + 1)*(1 - x)^3).
a(n) = 2*a(n-1) - a(n-2) + a(n-3) - 2*a(n-4) + a(n-5) for n>5. - Colin Barker, Feb 09 2018
a(n) = 2*(8 + 24*n^2 + A099837(n+3)/2)/9 for n > 0. - Stefano Spezia, Jun 06 2024

A299256 Coordination sequence for 3D uniform tiling formed by stacking parallel layers of the 3.6.3.6 2D tiling (cf. A008579).

Original entry on oeis.org

1, 6, 18, 40, 72, 112, 162, 220, 288, 364, 450, 544, 648, 760, 882, 1012, 1152, 1300, 1458, 1624, 1800, 1984, 2178, 2380, 2592, 2812, 3042, 3280, 3528, 3784, 4050, 4324, 4608, 4900, 5202, 5512, 5832, 6160, 6498, 6844, 7200, 7564, 7938, 8320, 8712, 9112, 9522, 9940, 10368, 10804, 11250, 11704
Offset: 0

Views

Author

N. J. A. Sloane, Feb 07 2018

Keywords

References

  • B. Grünbaum, Uniform tilings of 3-space, Geombinatorics, 4 (1994), 49-56. See tiling #18.

Links

Crossrefs

Cf. A008579.
For partial sums see A299262.
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

  • GAP
    a:=[18,40,72,112];; for n in [5..50] do a[n]:=2*a[n-1]-2*a[n-3]+a[n-4]; od; Concatenation([1,6],a); # Muniru A Asiru, Oct 26 2018
  • Magma
    [1, 6] cat [9*n^2 div 2: n in [2..50]]; // Vincenzo Librandi, Oct 26 2018
  • Maple
    seq(coeff(series((1+2*x)*(x^4-2*x^3-2*x^2-2*x-1)/((x-1)^3*(1+x)),x,n+1), x, n), n = 0 .. 35); # Muniru A Asiru, Oct 26 2018
  • Mathematica
    Join[{1, 6}, LinearRecurrence[{2, 0, -2, 1}, {18, 40, 72, 112}, 50]] (* Vincenzo Librandi, Oct 26 2018 *)
  • PARI
    Vec((1 + 2*x)*(1 + 2*x + 2*x^2 + 2*x^3 - x^4) / ((1 - x)^3*(1 + x)) + O(x^60)) \\ Colin Barker, Feb 09 2018

Formula

G.f.: (1 + 2*x)*(x^4 - 2*x^3 - 2*x^2 - 2*x - 1) / ((x - 1)^3*(x + 1)).
From Colin Barker, Feb 09 2018: (Start)
a(n) = 9*n^2 / 2 for n>1.
a(n) = (9*n^2 - 1) / 2 for n>1.
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4) for n>5. (End)
E.g.f.: (2 + 4*x + 9*x*(x + 1)*cosh(x) + (9*x^2 + 9*x - 1)*sinh(x))/2. - Stefano Spezia, Mar 14 2024

A299257 Coordination sequence for 3D uniform tiling formed by stacking parallel layers of the 3.12.12 2D tiling (cf. A250122).

Original entry on oeis.org

1, 5, 12, 22, 36, 56, 82, 111, 144, 183, 226, 272, 324, 382, 442, 505, 576, 653, 730, 810, 900, 996, 1090, 1187, 1296, 1411, 1522, 1636, 1764, 1898, 2026, 2157, 2304, 2457, 2602, 2750, 2916, 3088, 3250, 3415, 3600, 3791, 3970, 4152, 4356, 4566, 4762, 4961, 5184, 5413, 5626, 5842, 6084, 6332
Offset: 0

Views

Author

N. J. A. Sloane, Feb 07 2018

Keywords

References

  • B. Grünbaum, Uniform tilings of 3-space, Geombinatorics, 4 (1994), 49-56. See tiling #19.

Links

Crossrefs

Cf. A250122.
Partial sums: A299263.
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[{3, -5, 7, -7, 5, -3, 1}, {1, 5, 12, 22, 36, 56, 82, 111, 144, 183}, 60] (* Paolo Xausa, Jun 20 2024 *)
  • PARI
    Vec((1 + x)*(1 + x + x^2 + 3*x^3 - x^4 + 5*x^5 - 3*x^6 + 4*x^7 - 2*x^8) / ((1 - x)^3*(1 + x^2)^2) + O(x^60)) \\ Colin Barker, Feb 09 2018

Formula

G.f.: (2*x^8 - 4*x^7 + 3*x^6 - 5*x^5 + x^4 - 3*x^3 - x^2 - x - 1)*(x + 1) / ((x - 1)^3*(x^2 + 1)^2).
From Colin Barker, Feb 09 2018: (Start)
a(n) = (4 - (2+8*i)*(-i)^n - (2-8*i)*i^n + i*((-i)^n-i^n)*n + 18*n^2) / 8 for n>2, where i=sqrt(-1).
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>9. (End)
a(n) = 1/2 + 9*n^2/4 + (-1)^floor(n/2)*(A027656(n-1)/2 - A010699(n)/4). - R. J. Mathar, Feb 12 2021

A299258 Coordination sequence for 3D uniform tiling formed by stacking parallel layers of the 4.6.12 2D tiling (cf. A072154).

Original entry on oeis.org

1, 5, 13, 25, 41, 62, 89, 121, 157, 197, 242, 293, 349, 409, 473, 542, 617, 697, 781, 869, 962, 1061, 1165, 1273, 1385, 1502, 1625, 1753, 1885, 2021, 2162, 2309, 2461, 2617, 2777, 2942, 3113, 3289, 3469, 3653, 3842, 4037, 4237, 4441, 4649, 4862, 5081, 5305, 5533, 5765, 6002, 6245, 6493, 6745
Offset: 0

Views

Author

N. J. A. Sloane, Feb 07 2018

Keywords

References

  • B. Grünbaum, Uniform tilings of 3-space, Geombinatorics, 4 (1994), 49-56. See tiling #23.

Links

Crossrefs

Cf. A072154.
Partial sums: A299264.
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,-1,0,0,1,-2,1},{1,5,13,25,41,62,89,121},60] (* Harvey P. Dale, Mar 14 2023 *)
  • PARI
    Vec((1 + x)^3*(1 - x + x^2)*(1 + x + x^2) / ((1 - x)^3*(1 + x + x^2 + x^3 + x^4)) + O(x^60)) \\ Colin Barker, Feb 09 2018

Formula

G.f.: (x^2+x+1)*(x^2-x+1)*(x+1)^3 / ((x^4+x^3+x^2+x+1)*(1-x)^3).
a(n) = 2*a(n-1) - a(n-2) + a(n-5) - 2*a(n-6) + a(n-7) for n>7. - Colin Barker, Feb 09 2018
a(n) ~ 12*n^2/5. - Stefano Spezia, Jun 06 2024

A299259 Coordination sequence for 3D uniform tiling formed by stacking parallel layers of the 4.8.8 2D tiling (cf. A008576).

Original entry on oeis.org

1, 5, 13, 26, 45, 69, 98, 133, 173, 218, 269, 325, 386, 453, 525, 602, 685, 773, 866, 965, 1069, 1178, 1293, 1413, 1538, 1669, 1805, 1946, 2093, 2245, 2402, 2565, 2733, 2906, 3085, 3269, 3458, 3653, 3853, 4058, 4269, 4485, 4706, 4933, 5165, 5402, 5645, 5893, 6146, 6405, 6669, 6938, 7213, 7493
Offset: 0

Views

Author

N. J. A. Sloane, Feb 07 2018

Keywords

Links

Crossrefs

Cf. A008576, A061347.
Partial sums give A299265.
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

  • Magma
    I:=[13, 26, 45, 69, 98]; [1,5] cat [n le 5 select I[n] else 2*Self(n-1) - Self(n-2) + Self(n-3) - 2*Self(n-4) + Self(n-5): n in [1..30]]; // G. C. Greubel, Feb 20 2018
  • Mathematica
    CoefficientList[Series[(x+1)^3*(x^2+1)/((1-x)^3*(x^2+x+1)), {x, 0, 50}], x] (* G. C. Greubel, Feb 20 2018 *)
  • PARI
    Vec((1 + x)^3*(1 + x^2) / ((1 - x)^3*(1 + x + x^2)) + O(x^60)) \\ Colin Barker, Feb 09 2018

Formula

G.f.: (x + 1)^3*(x^2 + 1) / ((1 - x)^3*(x^2 + x + 1)).
a(n) = 2*a(n-1) - a(n-2) + a(n-3) - 2*a(n-4) + a(n-5) for n>5. - Colin Barker, Feb 09 2018
a(n) = (4*(5 + 6*n^2) + A061347(n))/9 for n > 0. - Stefano Spezia, Feb 17 2024

A299260 Partial sums of A299254.

Original entry on oeis.org

1, 8, 29, 74, 153, 275, 450, 687, 996, 1387, 1869, 2452, 3145, 3958, 4901, 5983, 7214, 8603, 10160, 11895, 13817, 15936, 18261, 20802, 23569, 26571, 29818, 33319, 37084, 41123, 45445, 50060, 54977, 60206, 65757, 71639, 77862, 84435, 91368, 98671, 106353, 114424
Offset: 0

Views

Author

N. J. A. Sloane, Feb 07 2018

Keywords

Links

Crossrefs

Cf. A299254.
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[{3, -3, 1, 0, 1, -3, 3, -1}, {1, 8, 29, 74, 153, 275, 450, 687}, 50] (* Paolo Xausa, Jan 16 2025 *)
  • PARI
    Vec((1 + x)*(1 + x + x^2)*(1 + 3*x + 3*x^3 + x^4) / ((1 - x)^4*(1 + x + x^2 + x^3 + x^4)) + O(x^60)) \\ Colin Barker, Feb 09 2018

Formula

From Colin Barker, Feb 09 2018: (Start)
G.f.: (1 + x)*(1 + x + x^2)*(1 + 3*x + 3*x^3 + x^4) / ((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) = (1/5)*(8*n^3 + 12*n^2 + 14*n + 5 + [n == 1 (mod 5)] - [n == 3 (mod 5)]). - Eric Simon Jacob, Feb 14 2023

A299261 Partial sums of A299255.

Original entry on oeis.org

1, 8, 31, 81, 168, 303, 497, 760, 1103, 1537, 2072, 2719, 3489, 4392, 5439, 6641, 8008, 9551, 11281, 13208, 15343, 17697, 20280, 23103, 26177, 29512, 33119, 37009, 41192, 45679, 50481, 55608, 61071, 66881, 73048, 79583, 86497, 93800, 101503, 109617
Offset: 0

Views

Author

N. J. A. Sloane, Feb 07 2018

Keywords

Comments

Euler transform of length 3 sequence [8, -5, 1]. - Michael Somos, Oct 03 2018

Links

Crossrefs

Cf. A299255.
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 (2 n + 1) (n^2 + n + 1) - Mod[n - 1, 3, -1]) / 9; (* Michael Somos, Oct 03 2018 *)
LinearRecurrence[{3,-3,2,-3,3,-1},{1,8,31,81,168,303},50] (* Harvey P. Dale, May 15 2025 *)
  • PARI
    Vec((1 + x)^5 / ((1 - x)^4*(1 + x + x^2)) + O(x^60)) \\ Colin Barker, Feb 09 2018
  • PARI
    {a(n) =  (8 * (2*n + 1) * (n^2 + n + 1) + (n%3==0) - (n%3==2)) / 9}; /* Michael Somos, Oct 03 2018 */

Formula

From Colin Barker, Feb 09 2018: (Start)
G.f.: (1 + x)^5 / ((1 - x)^4*(1 + x + x^2)).
a(n) = 3*a(n-1) - 3*a(n-2) + 2*a(n-3) - 3*a(n-4) + 3*a(n-5) - a(n-6) for n>5.
(End)
a(n) = -a(-1-n) for all n in Z. - Michael Somos, Oct 03 2018

A299262 Partial sums of A299256.

Original entry on oeis.org

1, 7, 25, 65, 137, 249, 411, 631, 919, 1283, 1733, 2277, 2925, 3685, 4567, 5579, 6731, 8031, 9489, 11113, 12913, 14897, 17075, 19455, 22047, 24859, 27901, 31181, 34709, 38493, 42543, 46867, 51475, 56375, 61577, 67089, 72921, 79081, 85579, 92423, 99623, 107187, 115125, 123445, 132157, 141269, 150791, 160731, 171099
Offset: 0

Views

Author

N. J. A. Sloane, Feb 07 2018

Keywords

Links

Crossrefs

Cf. A299256.
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[{3,-2,-2,3,-1},{1,7,25,65,137,249},50] (* Harvey P. Dale, Jul 22 2024 *)
  • PARI
    Vec((1 + 2*x)*(1 + 2*x + 2*x^2 + 2*x^3 - x^4) / ((1 - x)^4*(1 + x)) + O(x^60)) \\ Colin Barker, Feb 09 2018

Formula

From Colin Barker, Feb 09 2018: (Start)
G.f.: (1 + 2*x)*(1 + 2*x + 2*x^2 + 2*x^3 - x^4) / ((1 - x)^4*(1 + x)).
a(n) = (6*n^3 + 9*n^2 + 2*n + 12) / 4 for n>0 and even.
a(n) = (6*n^3 + 9*n^2 + 2*n + 11) / 4 for n odd.
a(n) = 3*a(n-1) - 2*a(n-2) - 2*a(n-3) + 3*a(n-4) - a(n-5) for n>5. (End)
E.g.f.: ((12 + 17*x + 27*x^2 + 6*x^3)*cosh(x) + (11 + 17*x + 27*x^2 + 6*x^3)*sinh(x) - 8)/4. - Stefano Spezia, Mar 14 2024
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