|
%I #199 Jun 29 2024 03:29:05
%S 1,1,1,1,2,1,1,3,3,1,1,4,5,4,1,1,5,7,7,5,1,1,6,9,10,9,6,1,1,7,11,13,
%T 13,11,7,1,1,8,13,16,17,16,13,8,1,1,9,15,19,21,21,19,15,9,1,1,10,17,
%U 22,25,26,25,22,17,10,1,1,11,19,25,29,31,31,29,25,19,11,1,1,12,21,28,33,36,37,36,33,28,21,12,1
%N The rascal triangle, read by rows: T(n,k) (n >= 0, 0 <= k <= n) = k(n-k) + 1.
%C Pascal's triangle is formed using the rule South = West + East, whereas the rascal triangle uses the rule South = (West*East+1)/North. [Anggoro et al.]
%C The n-th diagonal is congruent to 1 mod n-1.
%C Row sums are the cake numbers, A000125. Alternating sum of row n is 0 if n even and (3-n)/2 if n odd. Rows are symmetric, beginning and ending with 1. The number of occurrences of k in this triangle is the number of divisors of k-1, given by A000005.
%C The triangle can be generated by numbers of the form k*(n-k) + 1 for k = 0 to n. Conjecture: except for n = 0,1 and 6 every row contains a prime. - _Amarnath Murthy_, Jul 15 2005
%C Above conjecture needs more exceptions, rows 30 and 54 do not contain primes. - _Alois P. Heinz_, Aug 31 2017
%C From _Moshe Shmuel Newman_, Apr 06 2008: (Start)
%C Consider the semigroup of words in x,y,q subject to the relationships: yx = xyq, qx = xq, qy = yq.
%C Now take words of length n in x and y, with exactly k y's. If there had been no relationships, the number of different words of this type would be n choose k, sequence A007318. Thanks to the relationships, the number of words of this type is the k-th entry in the n-th row of this sequence (read as a triangle, with the first row indexed by zero and likewise the first entry in each row.)
%C For example: with three letters and one y, we have three possibilities: xxy, xyx = xxyq, yxx = xxyqq. No two of them are equal, so this entry is still 3, as in Pascal's triangle.
%C With four letters, two y's, we have the first reduction: xyyx = yxxy = xxyyqq and this is the only reduction for 4 letters. So the middle entry of the fourth row is 5 instead of 6, as in the Pascal triangle. (End)
%C Main diagonals of this triangle sum to polygonal numbers. See A057145. - _Raphie Frank_, Oct 30 2012
%C T(n,k) gives the number of distinct sums of k elements in {1,2,...,n}, e.g., T(5,4) = the number of distinct sums of 4 elements in {1,2,3,4,5}, which is (5+4+3+2) - (4+3+2+1) + 1 = 5. - _Derek Orr_, Nov 26 2014
%C Conjecture: excluding the starting and ending 1's in each row, those that contain only prime numbers are n = 2, 3, 5, 7, 13, and 17. Tested up to row 10^9. - _Rogério Serôdio_, Sep 20 2017
%C The rascal triangle also uses the rule South = (West+East+1)-North. [Abstracts of AMS, Winter 2019, p. 526, 1145-VS-280, refers to Julian Fleron] - _Michael Somos_, Jan 12 2019
%C As a square array read by antidiagonals, selecting terms that give a remainder of 1 when divided by a prime gives evenly sized squares. For example, when each term is divided by 2, showing the remainder looks like:
%C 1 1 1 1 1
%C 1 0 1 0 1
%C 1 1 1 1 1
%C 1 0 1 0 1
%C 1 1 1 1 1 - _Nathaniel J. Strout_, Jan 01 2020
%C T(n,k) is the number of binary words of length n which contain exactly k 1s and have at most 1 ascent. T(n,k) is also the number of ascent sequences avoiding 001 and 210 with length n+1 and exactly k ascents. - _Amelia Gibbs_, May 21 2024
%H Michael De Vlieger, <a href="/A077028/b077028.txt">Table of n, a(n) for n = 0..11475</a> (rows 0 <= n <= 150, flattened).
%H A. Anggoro, E. Liu and A. Tulloch, <a href="http://www.maa.org/publications/periodicals/college-mathematics-journal/rascal-triangle">The Rascal Triangle</a>, College Math. J., Vol. 41, No. 5, Nov. 2010, pp. 393-395.
%H D. C. Fielder & C. O. Alford, <a href="/A000108/a000108_19.pdf">An investigation of sequences derived from Hoggatt Sums and Hoggatt Triangles</a>, Application of Fibonacci Numbers, 3 (1990) 77-88. Proceedings of 'The Third Annual Conference on Fibonacci Numbers and Their Applications,' Pisa, Italy, July 25-29, 1988. (Annotated scanned copy)
%H Julian Fleron, <a href="http://www.artofmathematics.org/blogs/jfleron/tackling-rascals-triangle-how-inquiry-challenges-what-we-know-and-how-we-know-it">Tackling Rascals’ Triangle - How Inquiry Challenges What We Know and How We Know It</a>, Discovering the Art of Mathematics, December 15 2015.
%H Amelia Gibbs and Brian K. Miceli, <a href="https://arxiv.org/abs/2405.11045">Two Combinatorial Interpretations of Rascal Numbers</a>, arXiv:2405.11045 [math.CO], 2024.
%H Jena Gregory, Brandt Kronholm & Jacob White, <a href="https://doi.org/10.1007/s00010-023-00987-6">Iterated rascal triangles</a>, Aequationes mathematicae, 2023.
%H Zachary Hoelscher and Eyvindur Ari Palsson, <a href="https://arxiv.org/abs/2011.14502">Counting Restricted Partitions of Integers into Fractions: Symmetry and Modes of the Generating Function and a Connection to omega(t)</a>, arXiv:2011.14502 [math.NT], 2020.
%H Brian Hopkins, <a href="http://www.jstor.org/stable/10.4169/college.math.j.45.2.082">Editorial: Anonymity and Youth</a>, The College Mathematics Journal, 45 (Number 2, 2014), 82. - From _N. J. A. Sloane_, Apr 05 2014
%H Philip K. Hotchkiss, <a href="https://arxiv.org/abs/1907.11159">Generalized Rascal Triangles</a>, arXiv:1907.11159 [math.HO], 2019.
%H Philip K. Hotchkiss, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL23/Hotchkiss/hotchkiss4.html">Generalized Rascal Triangles</a>, Journal of Integer Sequences, Vol. 23, 2020.
%H Philip K. Hotchkiss, <a href="https://arxiv.org/abs/1907.07749">Student Inquiry and the Rascal Triangle</a>, arXiv:1907.07749 [math.HO], 2019.
%H Iva Kodrnja and Helena Koncul, <a href="https://arxiv.org/abs/2405.10747">Number of Polynomials Vanishing on a Basis of S_m(Gamma_0(N))</a>, arXiv:2405.10747 [math.NT], 2024. See p. 10.
%H L. McHugh, <a href="http://digital.ipcprintservices.com/display_article.php?id=647987">CMJ Article Shows Collaboration Is Not Limited by Geography ... or Age</a>, MAA Focus (Magazine), Vol. 31, No. 1, 2011, p. 13.
%H Franck Ramaharo, <a href="https://arxiv.org/abs/1902.08989">A generating polynomial for the two-bridge knot with Conway's notation C(n,r)</a>, arXiv:1902.08989 [math.CO], 2019. See p. 8.
%F As a square array read by antidiagonals, a(n, k) = 1 + n*k. a(n, k) = a(n-1, k) + k. Row n has g.f. (1+(n-1)x)/(1-x)^2, n >= 0. - _Paul Barry_, Feb 22 2003
%F Still thinking of square arrays. Let f:N->Z and g:N->Z be given and I an integer, then define a(n, k) = I + f(n)*g(k). Then a(n, k)*a(n-1, k-1) = a(n-1, k)*a(n, k-1) + I*(f(n) - f(n-1))*(g(k) - g(k-1)) for suitable n and k. S= (E*W + 1)/N. arises with I = 1, and f = g = id. - _Terry Lindgren_, Apr 10 2011
%F Using the above: Having just read J. Fleron's nice article in Discovering the Art of Mathematics on the rascal triangle, it is neat to note and straightforward to show that when I = 1, a(n, k) + a(n-1, k-1) = a(n-1, k) + a(n, k-1) + (f(n) - f(n-1))*(g(k) - g(k-1)), so with I = 1, and f = g = id, we have S+N = E+W + 1, as his students discovered. - _Terry Lindgren_, Nov 28 2016
%F T(n, k) = A128139(n-1, k-1). - _Gary W. Adamson_, Jul 02 2012
%F O.g.f. (1 - x*(1 + t) + 2*t*x^2)/((1 - x)^2*(1 - t*x)^2) = 1 + (1 + t)*x + (1 + 2*t + t^2)*x^2 + .... Cf. A105851. - _Peter Bala_, Jul 26 2015
%F T(n, k) = 0 if n < k, T(n, 0) = 1, T(n,n) = 1, for n >= 0, and T(n, k) = (T(n-1, k-1)*T(n-1, k) + 1)/(T(n-2, k-1)) for 0 < k < n. See the first comment referring to the triangle with its apex in the middle. - _Wolfdieter Lang_, Dec 19 2017
%e Third diagonal (1,3,5,7,...) consists of the positive integers congruent to 1 mod 2.
%e The triangle T(n, k) begins:
%e n\k 0 1 2 3 4 5 6 7 8 9 10 ...
%e 0: 1
%e 1: 1 1
%e 2: 1 2 1
%e 3: 1 3 3 1
%e 4: 1 4 5 4 1
%e 5: 1 5 7 7 5 1
%e 6: 1 6 9 10 9 6 1
%e 7: 1 7 11 13 13 11 7 1
%e 8: 1 8 13 16 17 16 13 8 1
%e 9: 1 9 15 19 21 21 19 15 9 1
%e 10: 1 10 17 22 25 26 25 22 17 10 1
%e ... reformatted. - _Wolfdieter Lang_, Dec 19 2017
%e As a square array read by antidiagonals, the first rows are:
%e 1, 1, 1, 1, 1, 1, ...
%e 1, 2, 3, 4, 5, 6, ...
%e 1, 3, 5, 7, 9, 11, ...
%e 1, 4, 7, 10, 13, 16, ...
%e 1, 5, 9, 13, 17, 21, ...
%p A077028 := proc(n,k)
%p if n <0 or k<0 or k > n then
%p 0;
%p else
%p k*(n-k)+1 ;
%p end if;
%p end proc: # _R. J. Mathar_, Jul 28 2016
%t t[n_, k_] := k (n - k) + 1; t[0, 0] = 1; Table[ t[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* _Robert G. Wilson v_, Jul 06 2012 *)
%o (PARI) {T(n, k) = if( k<0 || k>n, 0, k * (n - k) + 1)}; /* _Michael Somos_, Mar 20 2011 */
%Y Cf. A000125, A003991, A077029, A105851.
%Y The maximum value for each antidiagonal is given by sequence A033638.
%Y Equals A004247(n) + 1.
%K nonn,tabl
%O 0,5
%A _Clark Kimberling_, Oct 19 2002
%E Better definition based on Murthy's comment of Jul 15 2005 and the Anggoro et al. paper. - _N. J. A. Sloane_, Mar 05 2011
|
