A005906 -id:A005906 - cp's OEIS frontend

A100156 Structured truncated tetrahedral numbers.

1, 12, 44, 108, 215, 376, 602, 904, 1293, 1780, 2376, 3092, 3939, 4928, 6070, 7376, 8857, 10524, 12388, 14460, 16751, 19272, 22034, 25048, 28325, 31876, 35712, 39844, 44283, 49040, 54126, 59552, 65329, 71468, 77980, 84876, 92167, 99864, 107978, 116520, 125501, 134932

Offset: 1

James A. Record (james.record(AT)gmail.com), Nov 07 2004

Vincenzo Librandi, Table of n, a(n) for n = 1..5000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A100155, A100157 for adjacent structured Archimedean solids; A100145 for more on structured polyhedral numbers. Similar to truncated tetrahedral numbers A005906.

[(1/6)*(11*n^3-3*n^2-2*n): n in [1..40]]; // Vincenzo Librandi, Jul 19 2011

Table[(11n^3-3n^2-2n)/6,{n,40}] (* or *) LinearRecurrence[{4,-6,4,-1},{1,12,44,108},40] (* Harvey P. Dale, Sep 28 2011 *)

vector(50, n, (11*n^3 - 3*n^2 - 2*n)/6) \\ G. C. Greubel, Oct 18 2018

a(n) = (1/6)*(11*n^3 - 3*n^2 - 2*n).
From Harvey P. Dale, Sep 28 2011: (Start)
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4); a(0)=1, a(1)=12, a(2)=44, a(3)=108.
G.f.: x*(2*x*(x+4)+1)/(x-1)^4. (End)
E.g.f.: x*(6 + 30*x + 11*x^2)*exp(x)/6. - G. C. Greubel, Oct 18 2018

A160248 Table read by antidiagonals of "less regular" truncated tetrahedron numbers built of face-centered-cubic sphere packing.

Original entry on oeis.org

1, 6, 4, 19, 16, 10, 44, 40, 31, 20, 85, 80, 68, 52, 35, 146, 140, 125, 104, 80, 56, 231, 224, 206, 180, 149, 116, 84, 344, 336, 315, 284, 246, 204, 161, 120, 489, 480, 456, 420, 375, 324, 270, 216, 165

Offset: 1

Chris G. Spies-Rusk (chaosorder4(AT)gmail.com), May 05 2009, May 11 2009

These also contain 3 existing sequences:
Regular octahedra (A005900) on the x-axis, which represents the increasing edge at truncation.
Regular tetrahedra (essentially A000292) on the y-axis, which represents the increasing remaining original edge.
Regular truncated tetrahedra (A005906) on the diagonal, which represents values where the newly formed edge and the remaining portion of the original tetrahedron edge are of equal length.

Main Title: Polyhedra primer / Peter Pearce and Susan Pearce. Published/Created: New York : Van Nostrand Reinhold, c1978. Description: viii, 134 p. : ill. ; 24 cm. ISBN: 0442264968
Main Title: The book of numbers / John H. Conway, Richard K. Guy. Published/Created: New York, NY : Copernicus c1996. Description: ix, 310 p. : ill. (some col.) ; 24 cm. ISBN: 038797993X

Paste the following formula into cell C3, and fill down and right to desired table size. All volumes 10,000 and under are covered by column AA and row 41.
=((ROW()-2)^3+4*(COLUMN()-2)^3+6*(ROW()-2)^2*(COLUMN()-2)+12*(ROW()-2)*(COLUMN()-2)^2-3*(ROW()-2)^2-12*(COLUMN()-2)^2-12*(ROW()-2)*(COLUMN()-2)+2*(ROW()-2)+8*(COLUMN()-2))/6

v=(y^3+4*x^3+6*y^2*x+12*y*x^2-3*y^2-12*x^2-12*y*x+2*y+8*x)/6

Improvement of the definition's precision by Chris G. Spies-Rusk (chaosorder4(AT)gmail.com), May 19 2009

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A100156 Structured truncated tetrahedral numbers.

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A160248 Table read by antidiagonals of "less regular" truncated tetrahedron numbers built of face-centered-cubic sphere packing.

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