Displaying 31-40 of 579 results found.
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1, 6, 15, 37, 74, 131, 213, 330, 475, 653, 882, 1163, 1485, 1862, 2307, 2821, 3398, 4043, 4773, 5598, 6499, 7481, 8574, 9779, 11073, 12470, 13995, 15649, 17414, 19295, 21321, 23502, 25807, 28241, 30846, 33623, 36537, 39602, 42855, 46297, 49898, 53663, 57633, 61818, 66175, 70709, 75474, 80471, 85653, 91034, 96663
LINKS
Index entries for linear recurrences with constant coefficients, signature (2,-2,3,-2,0,0,-2,3,-2,2,-1).
FORMULA
G.f.: (1 +4*x +5*x^2 +16*x^3 +14*x^4 +24*x^5 +18*x^6 +20*x^7 +5*x^8 + x^10 -4*x^11 +4*x^12)/((1 -x)^4*(1 +x)*(1 +x^2)^2*(1 +x +x^2)).
a(n) = 2*a(n-1) - 2*a(n-2) + 3*a(n-3) - 2*a(n-4) - 2*a(n-7) + 3*a(n-8) - 2*a(n-9) + 2*a(n-10) - a(n-11) for n>12.
(End)
MATHEMATICA
CoefficientList[Series[(1 +4*x +5*x^2 +16*x^3 +14*x^4 +24*x^5 +18*x^6 +20*x^7 +5*x^8 + x^10 -4*x^11 +4*x^12)/((1 -x)^4*(1 +x)*(1 +x^2)^2*(1 +x +x^2)), {x, 0, 50}], x] (* G. C. Greubel, Feb 20 2018 *) LinearRecurrence[{2, -2, 3, -2, 0, 0, -2, 3, -2, 2, -1}, {1, 6, 15, 37, 74, 131, 213, 330, 475, 653, 882, 1163, 1485}, 60] (* Harvey P. Dale, Sep 03 2018 *)
PROG
(PARI) Vec((1 + 4*x + 5*x^2 + 16*x^3 + 14*x^4 + 24*x^5 + 18*x^6 + 20*x^7 + 5*x^8 + x^10 - 4*x^11 + 4*x^12) / ((1 - x)^4*(1 + x)*(1 + x^2)^2*(1 + x + x^2)) + O(x^60)) \\ Colin Barker, Feb 15 2018 (Magma) I:=[15, 37, 74, 131, 213, 330, 475, 653, 882, 1163, 1485]; [1, 6] cat [n le 11 select I[n] else 2*Self(n-1) -2*Self(n-2) +3*Self(n-3)-2*Self(n-4)-2*Self(n-7) +3*Self(n-8) -2*Self(n-9)+2*Self(n-10)-Self(n-11): n in [1..30]]; // G. C. Greubel, Feb 20 2018
CROSSREFS
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.
Coordination sequence for "crs" 3D uniform tiling formed from tetrahedra and truncated tetrahedra.
+10
51
1, 6, 18, 48, 78, 126, 182, 240, 330, 390, 522, 576, 758, 798, 1038, 1056, 1362, 1350, 1730, 1680, 2142, 2046, 2598, 2448, 3098, 2886, 3642, 3360, 4230, 3870, 4862, 4416, 5538, 4998, 6258, 5616, 7022, 6270, 7830, 6960, 8682, 7686, 9578, 8448, 10518, 9246
REFERENCES
B. Grünbaum, Uniform tilings of 3-space, Geombinatorics, 4 (1994), 49-56. See tiling #6.
FORMULA
G.f.: (x^6 + 27*x^4 + 30*x^3 + 15*x^2 + 6*x + 1) / (1 - x^2)^3.
a(n) = 3*a(n-2) - 3*a(n-4) + a(n-6) for n>6.
a(n) = (11*n^2 - 6*n + 4) / 2 for n>0 and even.
a(n) = 3 * (3*n^2 + 2*n - 1) / 2 for n odd. (End)
E.g.f.: ((11*x^2 + 15*x + 4)*cosh(x) + (9*x^2 + 5*x - 3)*sinh(x) - 2)/2. - Stefano Spezia, Mar 14 2024
MATHEMATICA
CoefficientList[Series[(x^6+27*x^4+30*x^3+15*x^2+6*x+1)/(1-x^2)^3, {x, 0, 50}], x] (* G. C. Greubel, Feb 20 2018 *)
PROG
(PARI) Vec((1 + 6*x + 15*x^2 + 30*x^3 + 27*x^4 + x^6) / ((1 - x)^3*(1 + x)^3) + O(x^60)) \\ Colin Barker, Feb 09 2018 (Magma) I:=[18, 48, 78, 126, 182, 240, 330]; [1, 6] cat [n le 6 select I[n] else 3*Self(n-2) -3*Self(n-4) + Self(n-6): n in [1..30]]; // G. C. Greubel, Feb 20 2018
CROSSREFS
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.
1, 7, 25, 73, 151, 277, 459, 699, 1029, 1419, 1941, 2517, 3275, 4073, 5111, 6167, 7529, 8879, 10609, 12289, 14431, 16477, 19075, 21523, 24621, 27507, 31149, 34509, 38739, 42609, 47471, 51887, 57425, 62423, 68681, 74297, 81319, 87589, 95419, 102379, 111061
FORMULA
G.f.: (1 + 6*x + 15*x^2 + 30*x^3 + 27*x^4 + x^6) / ((1 - x)^4*(1 + x)^3).
a(n) = (20*n^3 + 33*n^2 - 2*n + 12) / 12 for n even.
a(n) = (20*n^3 + 27*n^2 + 28*n + 9) / 12 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>6. (End)
E.g.f.: ((12 + 75*x + 93*x^2 + 20*x^3)*cosh(x) + (9 + 51*x + 87*x^2 + 20*x^3)*sinh(x))/12. - Stefano Spezia, Mar 14 2024
MATHEMATICA
CoefficientList[Series[(1+6*x+15*x^2+30*x^3+27*x^4+x^6)/((1-x)^4*(1+ x)^3), {x, 0, 50}], x] (* G. C. Greubel, Feb 20 2018 *)
PROG
(PARI) Vec((1 + 6*x + 15*x^2 + 30*x^3 + 27*x^4 + x^6) / ((1 - x)^4*(1 + x)^3) + O(x^60)) \\ Colin Barker, Feb 09 2018 (Magma) I:=[25, 73, 151, 277, 459, 699, 1029]; [1, 7] cat [n le 7 select I[n] else Self(n-1) + 3*Self(n-2) - 3*Self(n-3) - 3*Self(n-4) + 3*Self(n-5) + Self(n-6) - Self(n-7): n in [1..30]]; // G. C. Greubel, Feb 20 2018
CROSSREFS
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.
Coordination sequence for "flu" 3D uniform tiling formed from tetrahedra, rhombicuboctahedra, and cubes.
+10
51
1, 6, 18, 37, 63, 99, 142, 189, 249, 317, 384, 468, 562, 648, 756, 877, 981, 1113, 1262, 1383, 1539, 1717, 1854, 2034, 2242, 2394, 2598, 2837, 3003, 3231, 3502, 3681, 3933, 4237, 4428, 4704, 5042, 5244, 5544, 5917, 6129, 6453, 6862, 7083, 7431, 7877, 8106, 8478, 8962, 9198
COMMENTS
The tiling is called "3-RCO-trille" in Conway, Burgiel, Goodman-Strauss, 2008, p. 297. - Felix Fröhlich, Feb 11 2018
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 #5.
FORMULA
G.f.: (1 + x)^3*(1 + x^2)*(1 + 3*x + 5*x^2 + 3*x^3 + x^4) / ((1 - x)^3*(1 + x + x^2)^3).
a(n) = 3*a(n-3) - 3*a(n-6) + a(n-9) for n>9.
(End)
G.f.: (x^2+1)*(x^4+3*x^3+5*x^2+3*x+1)*(x+1)^3 / (1-x^3)^3. - N. J. A. Sloane, Feb 12 2018 (This confirms my conjecture from Feb 10 2018 and the above conjecture from Colin Barker.) a(n) = (60 + 104*n^2 + (n^2 - 6)*cos(2*n*Pi/3) - 3*sqrt(3)*n*sin(2*n*Pi/3))/27 for n > 0. - Stefano Spezia, Jan 23 2022
MATHEMATICA
CoefficientList[Series[(x^2+1)*(x^4+3*x^3+5*x^2+3*x+1)*(x+1)^3/(1-x^3)^3, {x, 0, 50}], x] (* G. C. Greubel, Feb 20 2018 *)
PROG
(PARI) x='x+O('x^30); Vec((x^2+1)*(x^4+3*x^3+5*x^2+3*x+1)*(x+1)^3/(1-x^3)^3) \\ G. C. Greubel, Feb 20 2018 (Magma) Q:=Rationals(); R<x>:=PowerSeriesRing(Q, 40); Coefficients(R!((x^2+1)*(x^4+3*x^3+5*x^2+3*x+1)*(x+1)^3/(1-x^3)^3)); // G. C. Greubel, Feb 20 2018
CROSSREFS
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.
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
LINKS
Index entries for linear recurrences with constant coefficients, signature (1,0,3,-3,0,-3,3,0,1,-1).
FORMULA
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)
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
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 *)
PROG
(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 (Magma) Q:=Rationals(); R<x>:=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
CROSSREFS
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.
Coordination sequence for "hal" 3D uniform tiling.
+10
51
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
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.
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
FORMULA
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)
CROSSREFS
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.
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
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
CROSSREFS
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.
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
COMMENTS
Euler transform of length 6 sequence [5, -1, 1, -1, 1, -1]. - Michael Somos, Oct 03 2018
FORMULA
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.
EXAMPLE
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
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 *)
PROG
(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 */
CROSSREFS
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.
Coordination sequence for "pcu-i" 3D uniform tiling.
+10
51
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
REFERENCES
B. Grünbaum, Uniform tilings of 3-space, Geombinatorics, 4 (1994), 49-56. See tiling #20.
LINKS
Index entries for linear recurrences with constant coefficients, signature (0,-1,1,0,1,2,0,2,-2,0,-2,-1,0,-1,1,0,1).
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
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 *)
PROG
(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
CROSSREFS
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.
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
LINKS
Index entries for linear recurrences with constant coefficients, signature (1,-1,2,-1,1,1,-2,2,-4,2,-2,1,1,-1,2,-1,1,-1).
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
PROG
(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
CROSSREFS
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.
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