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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A355053 Number of unoriented multidimensional n-ominoes with cell centers determining n-3 space.

Original entry on oeis.org

1, 11, 77, 412, 2009, 8869, 36988, 146578, 560498, 2078927, 7530385, 26734692, 93360884, 321454484, 1093599885, 3681897625, 12284317088, 40660245162, 133638662066, 436488290069, 1417680926923, 4581355626106
Offset: 4

Views

Author

Robert A. Russell, Jun 16 2022

Keywords

Comments

Multidimensional polyominoes are connected sets of cells of regular tilings with Schläfli symbols {oo}, {4,4}, {4,3,4}, {4,3,3,4}, etc. Each tile is a regular orthotope (hypercube). For unoriented polyominoes, chiral pairs are counted as one.

Examples

			a(4)=1 as there is only one tetromino in one-space. a(5)=11 because there are 5 achiral and 6 chiral pairs of pentominoes in 2-space, excluding the 1-D straight pentomino.

Links

Crossrefs

Cf. A355052 (oriented), A355054 (chiral), A355055 (achiral) A355056 (asymmetric), A191092 (fixed), A000081 (rooted trees), A049430 (Lunnon's DE).
Other dimensions: A036364 (n-2), A000055 (n-1), A355048 (orthoplex).

Programs

  • Mathematica
    sb[n_,k_]:= sb[n,k] = b[n+1-k,1] + If[n<2k, 0, sb[n-k,k]];
b[1,1] := 1; b[n_,1] := b[n,1] = Sum[b[i,1]sb[n-1,i]i,{i,1,n-1}]/(n-1);
b[n_,k_] := b[n,k] = Sum[b[i,1]b[n-i,k-1],{i,1,n-1}];
nmax = 30; B[x_] := Sum[b[i,1]x^i,{i,0,nmax}]
Drop[CoefficientList[Series[(50B[x]^6+3B[x]^7+30B[x]^2B[x^2]^2+3B[x]^3B[x^2](6+B[x^2])+3B[x]^5(37+2B[x^2])+12B[x]^4(1+3B[x^2])+B[x](57B[x^2]^2+6B[x^2]^3+6B[x^4]+6B[x^2]B[x^4])+4(3B[x^2]^2+11B[x^2]^3+B[x^3]^2+B[x^6]))/24+B[x]^2(112B[x]^5+9B[x]^6+3B[x^2]^2+4B[x]B[x^2]^2+B[x]^2B[x^2](14+B[x^2])+8B[x]^3(1+4B[x^2])+B[x]^4(167+10B[x^2]))/(8(1-B[x]))+B[x]^5(46B[x]^3+6B[x]^4+3B[x^2]+B[x]^2(67+2B[x^2])+B[x](2+6B[x^2]))/(2(1-B[x])^2)+B[x]^6(153B[x]^2+75B[x]^3+12B[x]^4+3B[x^2]+B[x](4+3B[x^2]))/(6(1-B[x])^3)+B[x]^9(21+4B[x])/(2(1-B[x])^4)+3B[x]^10/(2(1-B[x])^5)+B[x^2](6B[x]^3B[x^2]+2B[x]^4B[x^2]+13B[x^2]^2+19B[x^2]^3+2B[x]^2B[x^2](1+3B[x^2])+B[x^4]+B[x^2]B[x^4]+B[x](35B[x^2]^2+5B[x^2]^3+B[x^4]+B[x^2]B[x^4]))/(4(1-B[x^2]))+B[x^2]^4(5+3B[x^2]+B[x](8+B[x^2]))/(1-B[x^2])^2+2B[x^2]^5(1+B[x])/(1-B[x^2])^3+2B[x]B[x^3]^2/(6(1-B[x^3]))+B[x]B[x^4]^2/(2(1-B[x^4]))+B[x]^2B[x^2]^2(7B[x]^2+5B[x]^3+3B[x^2]+B[x](2+B[x^2]))/(2(1-B[x])(1-B[x^2]))+B[x]^5B[x^2]^2(3+2B[x])/((1-B[x])^2(1-B[x^2]))+B[x]^6B[x^2]^2/((1-B[x])^3(1-B[x^2]))+B[x]^2B[x^2]^4/((1-B[x])(1-B[x^2])^2)+B[x^2]B[x^4]^2(1+B[x])/(2(1-B[x^2])(1-B[x^4])),{x,0,nmax}],x],4]

Formula

a(n) = A355052(n) - A355054(n) = (A355052(n) + A355055(n)) / 2 = A355054(n) + A355055(n).
a(n) = A049430(n,n-3), the third diagonal of Lunnon's DE array.
G.f.: (50B(x)^6+3B(x)^7+30B(x)^2B(x^2)^2+3B(x)^3B(x^2)(6+B(x^2))+3B(x)^5(37+2B(x^2))+12B(x)^4(1+3B(x^2))+B(x)(57B(x^2)^2+6B(x^2)^3+6B(x^4)+6B(x^2)B(x^4))+4(3B(x^2)^2+11B(x^2)^3+B(x^3)^2+B(x^6)))/24 + B(x)^2(112B(x)^5+9B(x)^6+3B(x^2)^2+4B(x)B(x^2)^2+B(x)^2B(x^2)(14+B(x^2))+8B(x)^3(1+4B(x^2))+B(x)^4(167+10B(x^2)))/(8(1-B(x))) + B(x)^5(46B(x)^3+6B(x)^4+3B(x^2)+B(x)^2(67+2B(x^2))+B(x)(2+6B(x^2)))/(2(1-B(x))^2) + B(x)^6(153B(x)^2+75B(x)^3+12B(x)^4+3B(x^2)+B(x)(4+3B(x^2)))/(6(1-B(x))^3) + B(x)^9(21+4B(x))/(2(1-B(x))^4) + 3B(x)^10/(2(1-B(x))^5) + B(x^2)(6B(x)^3B(x^2)+2B(x)^4B(x^2)+13B(x^2)^2+19B(x^2)^3+2B(x)^2B(x^2)(1+3B(x^2))+B(x^4)+B(x^2)B(x^4)+B(x)(35B(x^2)^2+5B(x^2)^3+B(x^4)+B(x^2)B(x^4)))/(4(1-B(x^2))) + B(x^2)^4(5+3B(x^2)+B(x)(8+B(x^2)))/(1-B(x^2))^2 + 2B(x^2)^5(1+B(x))/(1-B(x^2))^3 + 2B(x)B(x^3)^2/(6(1-B(x^3))) + B(x)B(x^4)^2/(2(1-B(x^4))) + B(x)^2B(x^2)^2(7B(x)^2+5B(x)^3+3B(x^2)+B(x)(2+B(x^2)))/(2(1-B(x))(1-B(x^2))) + B(x)^5B(x^2)^2(3+2B(x))/((1-B(x))^2(1-B(x^2))) + B(x)^6B(x^2)^2/((1-B(x))^3(1-B(x^2))) + B(x)^2B(x^2)^4/((1-B(x))(1-B(x^2))^2) + B(x^2)B(x^4)^2(1+B(x))/(2(1-B(x^2))(1-B(x^4))), where B(x) is the generating function for rooted trees with n nodes in A000081.

A355051 Number of asymmetric orthoplex n-ominoes with cell centers determining n-3 space.

Original entry on oeis.org

6, 67, 412, 1926, 7856, 29057, 101105, 335081, 1072653, 3337131, 10154700, 30330869, 89226443, 259092076, 744095757, 2116643127, 5971171140, 16722250081, 46529076097, 128722040503, 354276958783, 970546150818
Offset: 7

Views

Author

Robert A. Russell, Jun 16 2022

Keywords

Comments

Orthoplex polyominoes are connected sets of cells of regular tilings with Schläfli symbols {}, {4}, {3,4}, {3,3,4}, {3,3,3,4}, etc. These are tilings of regular orthoplexes projected on their circumspheres. Orthoplex polyominoes are equivalent to multidimensional polyominoes that do not extend more than two units along any axis, i.e., fit within a 2^d cube. An asymmetric polyomino has a symmetry group of order 1.

Examples

			a(7)=6 because there are 6 asymmetric heptominoes in 2^4 space. See trunks 1, 6, 8, 12, 27, and 28 in the linked Trunk Generating Functions.

Links

Crossrefs

Cf. A355047 (oriented), A355048 (unoriented), A355049 (chiral) A355050 (achiral), A004111 (rooted asymmetric).
Other dimensions: A036369 (n-2), A000220 (n-1), A355056 (multidimensional).

Programs

  • Mathematica
    sa[n_, k_] := sa[n, k] = a[n+1-k, 1] + If[n < 2 k, 0, -sa[n-k, k]];
a[1, 1] := 1; a[n_, 1] := a[n, 1] = Sum[a[i, 1] sa[n-1, i] i, {i, 1, n-1}]/(n-1);
a[n_, k_] := a[n, k] = Sum[a[i, 1] a[n-i, k-1], {i, 1, n-1}];
nmax = 30; A[x_] := Sum[a[i, 1] x^i, {i, 0, nmax}]
Drop[CoefficientList[Series[(14 A[x]^6 + 103 A[x]^7 + 24 A[x]^8 - 6 A[x]^4 A[x^2] - 12 A[x]^5 A[x^2] - 24 A[x]^6 A[x^2] - 18 A[x]^2 A[x^2]^2 + 15 A[x]^3 A[x^2]^2 - 14 A[x^2]^3 + 8 A[x] A[x^2]^3 + 6 A[x]^2 A[x^2]^3 + 4 A[x^3]^2 - 4 A[x] A[x^3]^2 + 24 A[x^2] A[x^4] - 18 A[x] A[x^2] A[x^4] - 6 A[x]^2 A[x^2] A[x^4] - 4 A[x^6] + 4 A[x] A[x^6])/(24 (1-A[x])) + A[x]^6 (5 A[x] + 16 A[x]^2 + 6 A[x]^3 - A[x^2] - 2 A[x] A[x^2])/(2 (1-A[x])^2) - A[x^2] (A[x]^4 A[x^2] + 8 A[x] A[x^2]^2 + 2 A[x]^2 A[x^2]^2 + 10 A[x^2]^3 + 5 A[x] A[x^2]^3 - 2 A[x] A[x^4] - 3 A[x^2] A[x^4] - A[x] A[x^2] A[x^4])/(4 (1-A[x^2])) + A[x]^7 (2 + 42 A[x] + 51 A[x]^2 + 24 A[x]^3 - 3 A[x^2])/(12 (1-A[x])^3) - A[x]^2 A[x^2]^2 (2 A[x] + 5 A[x]^3 + 2 A[x^2] - A[x] A[x^2])/(4 (1-A[x]) (1-A[x^2])) + A[x] A[x^3]^2/(1-A[x^3])/3 + A[x]^9 (17 + 8 A[x])/(8 (1-A[x])^4) - A[x]^5 (1 + 4 A[x]) A[x^2]^2/(4 (1-A[x])^2 (1-A[x^2])) - A[x^2]^4 (8 + 17 A[x] + 16 A[x^2] + 8 A[x] A[x^2])/(8 (1-A[x^2])^2) + A[x] (A[x^4]^2/(1-A[x^4]))/4 + 3 A[x]^10/(8 (1-A[x])^5) - A[x]^6 A[x^2]^2/(4 (1-A[x])^3 (1-A[x^2])) - A[x]^2 A[x^2]^4/(8 (1-A[x]) (1-A[x^2])^2) - 3 (1 + A[x]) A[x^2]^5/(4 (1-A[x^2])^3) +3 (1 + A[x]) A[x^2] A[x^4]^2/(4 (1-A[x^2]) (1-A[x^4])), {x,0,nmax}], x], 7]

Formula

G.f.: (14 A(x)^6 + 103 A(x)^7 + 24 A(x)^8 - 6 A(x)^4 A(x^2) - 12 A(x)^5 A(x^2) - 24 A(x)^6 A(x^2) - 18 A(x)^2 A(x^2)^2 + 15 A(x)^3 A(x^2)^2 - 14 A(x^2)^3 + 8 A(x) A(x^2)^3 + 6 A(x)^2 A(x^2)^3 + 4 A(x^3)^2 - 4 A(x) A(x^3)^2 + 24 A(x^2) A(x^4) - 18 A(x) A(x^2) A(x^4) - 6 A(x)^2 A(x^2) A(x^4) - 4 A(x^6) + 4 A(x) A(x^6))/(24 (1 - A(x))) +A(x)^6 (5 A(x) + 16 A(x)^2 + 6 A(x)^3 - A(x^2) - 2 A(x) A(x^2))/(2 (1 - A(x))^2) - A(x^2) (A(x)^4 A(x^2) + 8 A(x) A(x^2)^2 + 2 A(x)^2 A(x^2)^2 + 10 A(x^2)^3 + 5 A(x) A(x^2)^3 - 2 A(x) A(x^4) - 3 A(x^2) A(x^4) - A(x) A(x^2) A(x^4))/(4 (1 - A(x^2))) + A(x)^7 (2 + 42 A(x) + 51 A(x)^2 + 24 A(x)^3 - 3 A(x^2))/(12 (1 - A(x))^3) - A(x)^2 A(x^2)^2 (2 A(x) + 5 A(x)^3 + 2 A(x^2) - A(x) A(x^2))/(4 (1 - A(x)) (1 - A(x^2))) + A(x) A(x^3)^2/(1 - A(x^3))/3 + A(x)^9 (17 + 8 A(x))/(8 (1 - A(x))^4) - A(x)^5 (1 + 4 A(x)) A(x^2)^2/(4 (1 - A(x))^2 (1 - A(x^2))) - A(x^2)^4 (8 + 17 A(x) + 16 A(x^2) + 8 A(x) A(x^2))/(8 (1 - A(x^2))^2) + A(x) (A(x^4)^2/(1 - A(x^4)))/4 + 3 A(x)^10/(8 (1 - A(x))^5) - A(x)^6 A(x^2)^2/(4 (1 - A(x))^3 (1 - A(x^2))) - A(x)^2 A(x^2)^4/(8 (1 - A(x)) (1 - A(x^2))^2) - 3 (1 + A(x)) A(x^2)^5/(4 (1 - A(x^2))^3) + 3 (1 + A(x)) A(x^2) A(x^4)^2/(4 (1 - A(x^2)) (1 - A(x^4))) where A(x) is the generating function for rooted identity trees with n nodes in A004111.

A355054 Number of chiral pairs of multidimensional n-ominoes with cell centers determining n-3 space.

Original entry on oeis.org

6, 54, 297, 1341, 5468, 20519, 72168, 242886, 791780, 2514453, 7814225, 23863941, 71835845, 213601046, 628450974, 1832227629, 5299559865, 15221688836, 43450246045, 123345029035, 348417524877, 979803281560
Offset: 5

Views

Author

Robert A. Russell, Jun 16 2022

Keywords

Comments

Multidimensional polyominoes are connected sets of cells of regular tilings with Schläfli symbols {oo}, {4,4}, {4,3,4}, {4,3,3,4}, etc. Each tile is a regular orthotope (hypercube). Each member of a chiral pair is a reflection but not a rotation of the other.

Examples

			a(5)=6 because there are 6 chiral pairs of pentominoes in 2-space.

Links

Crossrefs

Cf. A355052 (oriented), A355053 (unoriented), A355055 (achiral) A355056 (asymmetric), A191092 (fixed), A045648 (rooted chiral), A195738 (Lunnon's DR), A049430 (Lunnon's DE).
Other dimensions: A036365 (n-2), A045649 (n-1), A355049 (orthoplex).

Programs

  • Mathematica
    sc[n_,k_] := sc[n,k] = c[n+1-k,1] + If[n<2k, 0, sc[n-k,k](-1)^k];
c[1,1] := 1; c[n_,1] := c[n,1] = Sum[c[i,1] sc[n-1,i]i, {i,1,n-1}]/(n-1);
c[n_,k_] := c[n, k] = Sum[c[i, 1] c[n-i, k-1], {i,1,n-1}];
nmax = 30; K[x_] := Sum[c[i,1] x^i, {i,0,nmax}]
Drop[CoefficientList[Series[(12 K[x]^4 + 87 K[x]^5 + 50 K[x]^6 + 3 K[x]^7 + 18 K[x]^3 K[-x^2] + 36 K[x]^4 K[-x^2] + 6 K[x]^5 K[-x^2] - 12 K[-x^2]^2 - 27 K[x] K[-x^2]^2 - 6 K[x]^2 K[-x^2]^2 + 3 K[x]^3 K[-x^2]^2 - 16 K[-x^2]^3 - 6 K[x] K[-x^2]^3 + 4 K[x^3]^2 - 6 K[x] K[-x^4] - 6 K[x] K[-x^2] K[-x^4] + 4 K[-x^6]) / 24 + K[x]^2 (16 K[x]^3 + 159 K[x]^4 + 112 K[x]^5 + 9 K[x]^6 + 14 K[x]^2 K[-x^2] + 32 K[x]^3 K[-x^2] + 10 K[x]^4 K[-x^2] - K[-x^2]^2 + K[x]^2 K[-x^2]^2) / (8 (1-K[x])) + K[x]^5 (2 K[x] + 67 K[x]^2 + 46 K[x]^3 + 6 K[x]^4 + 3 K[-x^2] + 6 K[x] K[-x^2] + 2 K[x]^2 K[-x^2]) / (2 (1-K[x])^2) - K[-x^2] (2 K[x]^2 K[-x^2] + 7 K[-x^2]^2 + 17 K[x] K[-x^2]^2 + 2 K[x]^2 K[-x^2]^2 + 7 K[-x^2]^3 + 5 K[x] K[-x^2]^3 + K[-x^4] + K[x] K[-x^4] + K[-x^2] K[-x^4] + K[x] K[-x^2] K[-x^4]) / (4 (1-K[-x^2])) + K[x]^6 (4 K[x] + 153 K[x]^2 + 75 K[x]^3 + 12 K[x]^4 + 3 K[-x^2] + 3 K[x] K[-x^2]) / (6 (1-K[x])^3) - K[x]^2 K[-x^2]^2 (K[x] + K[-x^2]) / ((1-K[x]) (1-K[-x^2])) + (K[x] K[x^3]^2) / (3 (1-K[x^3])) + K[x]^9 (21 + 4 K[x]) / (2 (1-K[x])^4) - K[-x^2]^4 (6 + 7 K[x] + 2 K[-x^2] + 2 K[x] K[-x^2]) / (2 (1-K[-x^2])^2) + 3 K[x]^10 / (2 (1-K[x])^5) - K[x]^2 K[-x^2]^4 / (2 (1-K[x]) (1-K[-x^2])^2) - (1 + K[x]) K[-x^2]^5 / (1-K[-x^2])^3, {x,0,nmax}], x], 5]

Formula

a(n) = A355052(n) - A355053(n) = (A355052(n) - A355055(n)) / 2 = A355053(n) - A355055(n).
a(n) = A195738(n,n-3) - A049430(n,n-3), diagonals of Lunnon's DR and DE arrays.
G.f.: (12 C(x)^4 + 87 C(x)^5 + 50 C(x)^6 + 3 C(x)^7 + 18 C(x)^3 C(-x^2) + 36 C(x)^4 C(-x^2) + 6 C(x)^5 C(-x^2) - 12 C(-x^2)^2 - 27 C(x) C(-x^2)^2 - 6 C(x)^2 C(-x^2)^2 + 3 C(x)^3 C(-x^2)^2 - 16 C(-x^2)^3 - 6 C(x) C(-x^2)^3 + 4 C(x^3)^2 - 6 C(x) C(-x^4) - 6 C(x) C(-x^2) C(-x^4) + 4 C(-x^6)) / 24 + C(x)^2 (16 C(x)^3 + 159 C(x)^4 + 112 C(x)^5 + 9 C(x)^6 + 14 C(x)^2 C(-x^2) + 32 C(x)^3 C(-x^2) + 10 C(x)^4 C(-x^2) - C(-x^2)^2 + C(x)^2 C(-x^2)^2) / (8 (1-C(x))) + C(x)^5 (2 C(x) + 67 C(x)^2 + 46 C(x)^3 + 6 C(x)^4 + 3 C(-x^2) + 6 C(x) C(-x^2) + 2 C(x)^2 C(-x^2)) / (2 (1-C(x))^2) - C(-x^2) (2 C(x)^2 C(-x^2) + 7 C(-x^2)^2 + 17 C(x) C(-x^2)^2 + 2 C(x)^2 C(-x^2)^2 + 7 C(-x^2)^3 + 5 C(x) C(-x^2)^3 + C(-x^4) + C(x) C(-x^4) + C(-x^2) C(-x^4) + C(x) C(-x^2) C(-x^4)) / (4 (1-C(-x^2))) + C(x)^6 (4 C(x) + 153 C(x)^2 + 75 C(x)^3 + 12 C(x)^4 + 3 C(-x^2) + 3 C(x) C(-x^2)) / (6 (1-C(x))^3) - C(x)^2 C(-x^2)^2 (C(x) + C(-x^2)) / ((1-C(x)) (1-C(-x^2))) + (C(x) C(x^3)^2) / (3 (1-C(x^3))) + C(x)^9 (21 + 4 C(x)) / (2 (1-C(x))^4) - C(-x^2)^4 (6 + 7 C(x) + 2 C(-x^2) + 2 C(x) C(-x^2)) / (2 (1-C(-x^2))^2) + 3 C(x)^10 / (2 (1-C(x))^5) - C(x)^2 C(-x^2)^4 / (2 (1-C(x)) (1-C(-x^2))^2) - (1 + C(x)) C(-x^2)^5 / (1-C(-x^2))^3 where C(x) is the generating function for chiral n-ominoes in n-1 space, one cell labeled in A045648.

A355052 Number of oriented multidimensional n-ominoes with cell centers determining n-3 space.

Original entry on oeis.org

1, 17, 131, 709, 3350, 14337, 57507, 218746, 803384, 2870707, 10044838, 34548917, 117224825, 393290329, 1307200931, 4310348599, 14116544717, 45959805027, 148860350902, 479938536114, 1541025955958, 4929773150983
Offset: 4

Views

Author

Robert A. Russell, Jun 16 2022

Keywords

Comments

Multidimensional polyominoes are connected sets of cells of regular tilings with Schläfli symbols {oo}, {4,4}, {4,3,4}, {4,3,3,4}, etc. Each tile is a regular orthotope (hypercube). This sequence is obtained using the first formula below. For oriented polyominoes, chiral pairs are counted as two.

Examples

			a(4)=1 because there is just one tetromino (with four cells aligned) in 1-space. a(5)=17 because there are 5 achiral and 6 chiral pairs of pentominoes in 2-space, excluding the 1-D straight pentomino.

Links

Crossrefs

Cf. A355053 (unoriented), A355054 (chiral), A355055 (achiral) A355056 (asymmetric), A191092 (fixed), A355047 (orthoplex), A195738 (Lunnon's DR).

Formula

a(n) = A355053(n) + A355054(n) = 2*A355053(n) - A355055(n) = 2*A355054(n) + A355055(n).
a(n) = A195738(n,n-3), the third diagonal of Lunnon's DR array.

A355055 Number of achiral multidimensional n-ominoes with cell centers determining n-3 space.

Original entry on oeis.org

1, 5, 23, 115, 668, 3401, 16469, 74410, 317612, 1287147, 5015932, 18920467, 69496943, 249618639, 879998839, 3053446651, 10452089459, 35360685297, 118416973230, 393038044024, 1294335897888, 4232938101229, 13757913332396
Offset: 4

Views

Author

Robert A. Russell, Jun 16 2022

Keywords

Comments

Multidimensional polyominoes are connected sets of cells of regular tilings with Schläfli symbols {oo}, {4,4}, {4,3,4}, {4,3,3,4}, etc. Each tile is a regular orthotope (hypercube). This sequence is obtained using the first formula below. An achiral polyomino is identical to its reflection.

Examples

			a(4)=1 as there is only one tetromino in one-space. a(5)=5 because there are 5 achiral pentominoes in 2-space, excluding the 1-D straight pentomino.

Links

Crossrefs

Cf. A355052 (oriented), A355053 (unoriented), A355054 (chiral), A355056 (asymmetric), A191092 (fixed), A355050 (orthoplex), A195738 (Lunnon's DR), A049430 (Lunnon's DE).

Formula

a(n) = A355053(n) - A355054(n) = 2*A355053(n) - A355052(n) = A355052(n) - 2*A355054(n).
a(n) = 2*A049430(n,n-3) - A195738(n,n-3), Lunnon's DE and DR arrays.
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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