Original entry on oeis.org
1, 2, 8, 32, 144, 672, 3200, 15360, 73984, 356864, 1722368, 8314880, 40144896, 193830912, 935886848, 4518838272, 21818834944, 105350561792, 508677324800, 2456111022080, 11859152338944, 57261051346944, 276480810549248
Offset: 0
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A002203:= func< n | Round((1+Sqrt(2))^n + (1-Sqrt(2))^n) >;
[2^(n-2)*(2+A002203(n)): n in [0..40]]; // G. C. Greubel, Oct 13 2022
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Table[2^(n-2)*(2+LucasL[n,2]), {n,0,20}] (* Vladimir Reshetnikov, Oct 07 2016 *)
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a(n)=if(n<0,0,polsym(4+4*x-x^2,n)[n+1]/4+2^(n-1))
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[2^(n-2)*(2+lucas_number2(n, 2, -1)) for n in range(41)] # G. C. Greubel, Oct 13 2022
A267652
a(n) = 4*a(n - 1) + 4*a(n - 2) for n>1, a(0)=2, a(1)=3.
Original entry on oeis.org
2, 3, 20, 92, 448, 2160, 10432, 50368, 243200, 1174272, 5669888, 27376640, 132186112, 638251008, 3081748480, 14879997952, 71846985728, 346907934720, 1675019681792, 8087710466048, 39050920591360, 188554524229632, 910421779283968, 4395905214054400, 21225307973353472, 102484852749631488
Offset: 0
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Table[2^(n - 5/2) ((1 + 4 Sqrt[2]) (1 - Sqrt[2])^n - (1 - 4 Sqrt[2]) (1 + Sqrt[2])^n), {n, 0, 25}]
RecurrenceTable[{a[0] == 2, a[1] == 3, a[n] == 4 a[n - 1] + 4 a[n - 2]}, a, {n, 25}] (* Bruno Berselli, Jan 19 2016 *)
LinearRecurrence[{4, 4}, {2, 3}, 20] (* Vincenzo Librandi, Jan 19 2016 *)
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Vec((2-5*x)/(1-4*x-4*x^2) + O(x^100)) \\ Altug Alkan, Jan 19 2016
A320660
Number of business cards required to build an origami level n Jerusalem cube.
Original entry on oeis.org
12, 72, 672, 6048, 55488, 511872, 4738560, 43943424, 407890944, 3787941888, 35186122752, 326885842944, 3037038034944, 28217571901440, 262178452930560, 2436006721486848, 22634041833160704, 210303674768424960, 1954034324430913536, 18155901427591938048
Offset: 0
a(2) = 672 because 456 business cards are needed for the squeleton and 216 more for the panels.
- Eric Baird, L'art fractal, Tangente 150 (2013), 45.
- Thomas Hull, Project Origami: Activities for Exploring Mathematics, A K Peters/CRC Press, 2006.
- Eric Baird, The Jerusalem Cube
- Malachi B-J. Brown, Business Card Origami
- Robert Dickau, Cross Menger (Jerusalem) Cube Fractal
- Origami Resource Center, Jerusalem Cube Fractal (Level 1)
- Franck Ramaharo, An approximate Jerusalem square whose side equals a Pell number, arXiv:1801.00466 [math.CO], 2018.
- Wikipedia, Cube de Jérusalem [In French]
- Index entries for linear recurrences with constant coefficients, signature (12, -16, -80, -48)
At the n-th level, the cube side length is
A000129(n+1), the squeleton requires 6*
A239549(n+1) business cards, and each face requires
A057087(n) units for the panels.
Cf.
A212596 (Origami Menger sponge),
A304960 (Origami Mosely snowflake sponge).
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LinearRecurrence[{12, -16, -80, -48}, {12, 72, 672, 6048}, 20]
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makelist((3/14)*(7*(2 - 2*sqrt(2))^n + 7*(2 + 2*sqrt(2))^n + (21 - 5*sqrt(7))*(4 - 2*sqrt(7))^n + (21 + 5*sqrt(7))*(4 + 2*sqrt(7))^n), n, 0, 20), ratsimp;
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