A047837 -id:A047837 - cp's OEIS frontend

A047866 a(n) = ceiling(n(n+1)(n+2)/8).

0, 1, 3, 8, 15, 27, 42, 63, 90, 124, 165, 215, 273, 342, 420, 510, 612, 727, 855, 998, 1155, 1329, 1518, 1725, 1950, 2194, 2457, 2741, 3045, 3372, 3720, 4092, 4488, 4909, 5355, 5828, 6327, 6855, 7410, 7995, 8610, 9256, 9933, 10643, 11385

Offset: 0

Mike Keith

A lower bound to Honaker's triangle problem A047837.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1,0,0,0,0,1,-3,3,-1).

Cf. A047837.

[Ceiling( n*(n+1)*(n+2)/8 ) : n in [0..50]]; // Vincenzo Librandi, Sep 28 2011

Table[Ceiling[n*(n+1)*(n+2)/8], {n, 0, 100}] (* Paolo Xausa, Aug 28 2024 *)

a(n)=(n*(n+1)*(n+2)+6)\8 \\ Charles R Greathouse IV, Oct 07 2015

G.f.: x*(1 + 2*x^2 - x^3 + 3*x^4 - 2*x^5 + 3*x^6)/((1-x)^4*(1+x)*(1+x^2)*(1+x^4)). - R. J. Mathar, Mar 11 2012

A098451 One of three ordered sets of positive integers that solves the minimal magic die puzzle.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 19, 20, 23, 43

Offset: 1

G. L. Honaker, Jr., Sep 07 2004

			The subsets are 43; 20 23; 12 15 16; 3 4 17 19; 1 5 10 13 14; 2 6 7 8 9 11.

Pickover, C. A., The Zen of Magic Squares, Circles and Stars: An Exhibition Of Surprising Structures Across Dimensions, Princeton University Press, 2002 (p.289).

Cf. A047837, A098452, A110548.

A098452 One of three ordered sets of positive integers that solves the minimal magic die puzzle.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 19, 21, 22, 43

Offset: 1

G. L. Honaker, Jr., Sep 07 2004

			The subsets are 43; 21 22; 8 16 19; 5 9 12 17; 3 4 7 14 15; 1 2 6 10 11 13.

Pickover, C. A., The Zen of Magic Squares, Circles and Stars: An Exhibition Of Surprising Structures Across Dimensions, Princeton University Press, 2002 (p.289).

Cf. A047837, A098451, A110548.

A110548 One of the three ordered sets of positive integers that solves the minimal magic die puzzle.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 25, 43

Offset: 1

G. L. Honaker, Jr., Sep 11 2005

Found by Justin Greer & Jonathan Graham (students of Honaker).

			The subsets are 43; 18 25; 13 14 16; 7 10 11 15; 1 5 8 12 17; 2 3 4 6 9 19.

Pickover, C. A., The Zen of Magic Squares, Circles and Stars: An Exhibition Of Surprising Structures Across Dimensions, Princeton University Press, 2002 (p. 289).

Cf. A047837, A098451, A098452.

A047873 a(n) = max_{r=1..n-1} ceiling(t(t(n)-t(r-1))/(n-r+1)), where t() = triangular numbers A000217.

Original entry on oeis.org

1, 3, 8, 15, 27, 43, 65, 94, 130, 175, 229, 294, 369, 456, 557, 671, 800, 944, 1105, 1283, 1479, 1695, 1930, 2187, 2465, 2765, 3090, 3439, 3813, 4213, 4641, 5096, 5580, 6095, 6639, 7216, 7825, 8466, 9143, 9855

Offset: 1

Mike Keith

Another lower bound for Honaker triangle problem (A047837); conjectured to be exact value.

Cf. A047837, A047866.

Empirical g.f.: -x*(x^15 - 3*x^14 + 3*x^13 - 5*x^12 + 5*x^11 - 9*x^10 + 7*x^9 - 10*x^8 + 7*x^7 - 9*x^6 + 5*x^5 - 6*x^4 + 2*x^3 - 3*x^2 - 1) / ((x-1)^4*(x^2-x+1)*(x^2+1)*(x^2+x+1)^2*(x^4-x^2+1)). [Colin Barker, Jan 18 2013]

A340558 a(n) is the smallest prime that can be the apex of a triangle with n rows, all entries being distinct primes, and all row sums equal.

Original entry on oeis.org

2, 19, 53, 131, 269, 503, 853, 1361, 1999, 2879, 3989

Offset: 1

Michel Marcus, Jan 11 2021

A109724(n) is a lower bound for the sum of terms of triangle.

			            19
For n=2: 3  5  11 , so a(2) = 19.

Carlos Rivera, Puzzle 1026. Another pine-like-prime puzzle, Prime Puzzles and Problems Connection.

Cf. A047837, A109724.

cp's OEIS Frontend

A047866 a(n) = ceiling(n*(n+1)*(n+2)/8).

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A098451 One of three ordered sets of positive integers that solves the minimal magic die puzzle.

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A098452 One of three ordered sets of positive integers that solves the minimal magic die puzzle.

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Examples

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A110548 One of the three ordered sets of positive integers that solves the minimal magic die puzzle.

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A047873 a(n) = max_{r=1..n-1} ceiling(t(t(n)-t(r-1))/(n-r+1)), where t() = triangular numbers A000217.

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A340558 a(n) is the smallest prime that can be the apex of a triangle with n rows, all entries being distinct primes, and all row sums equal.

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A047866 a(n) = ceiling(n(n+1)(n+2)/8).