

A077028


The rascal triangle, read by rows: T(n,k) (n >= 0, 0 <= k <= n) = k(nk) + 1.


38



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, 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, 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
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OFFSET

0,5


COMMENTS

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.]
The nth diagonal is congruent to 1 mod n1.
Row sums are the cake numbers, A000125. Alternating sum of row n is 0 if n even and (3n)/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 k1, given by A000005.
The triangle can be generated by numbers of the form k*(nk) + 1 for k = 0 to n. Conjecture: except for n = 0,1 and 6 every row contains a prime.  Amarnath Murthy, Jul 15 2005
Above conjecture needs more exceptions, rows 30 and 54 do not contain primes.  Alois P. Heinz, Aug 31 2017
From Moshe Shmuel Newman, Apr 06 2008: (Start)
Consider the semigroup of words in x,y,q subject to the relationships: yx = xyq, qx = xq, qy = yq.
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 kth entry in the nth row of this sequence (read as a triangle, with the first row indexed by zero and likewise the first entry in each row.)
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.
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)
Main diagonals of this triangle sum to polygonal numbers. See A057145.  Raphie Frank, Oct 30 2012
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
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
The rascal triangle also uses the rule South = (West+East+1)North. [Abstracts of AMS, Winter 2019, p. 526, 1145VS280, refers to Julian Fleron]  Michael Somos, Jan 12 2019
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:
1 1 1 1 1
1 0 1 0 1
1 1 1 1 1
1 0 1 0 1
1 1 1 1 1  Nathaniel J. Strout, Jan 01 2020


LINKS

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened).
A. Anggoro, E. Liu and A. Tulloch, The Rascal Triangle, College Math. J., Vol. 41, No. 5, Nov. 2010, pp. 393395.
D. C. Fielder & C. O. Alford, An investigation of sequences derived from Hoggatt Sums and Hoggatt Triangles, Application of Fibonacci Numbers, 3 (1990) 7788. Proceedings of 'The Third Annual Conference on Fibonacci Numbers and Their Applications,' Pisa, Italy, July 2529, 1988. (Annotated scanned copy)
Julian Fleron, Tackling Rascals’ Triangle  How Inquiry Challenges What We Know and How We Know It, Discovering the Art of Mathematics, December 15 2015.
Zachary Hoelscher and Eyvindur Ari Palsson, Counting Restricted Partitions of Integers into Fractions: Symmetry and Modes of the Generating Function and a Connection to omega(t), arXiv:2011.14502 [math.NT], 2020.
Brian Hopkins, Editorial: Anonymity and Youth, The College Mathematics Journal, 45 (Number 2, 2014), 82.  From N. J. A. Sloane, Apr 05 2014
Philip K. Hotchkiss, Generalized Rascal Triangles, arXiv:1907.11159 [math.HO], 2019.
Philip K. Hotchkiss, Student Inquiry and the Rascal Triangle, arXiv:1907.07749 [math.HO], 2019.
L. McHugh, CMJ Article Shows Collaboration Is Not Limited by Geography ... or Age, MAA Focus (Magazine), Vol. 31, No. 1, 2011, p. 13.
Franck Ramaharo, A generating polynomial for the twobridge knot with Conway's notation C(n,r), arXiv:1902.08989 [math.CO], 2019.


FORMULA

As a square array read by antidiagonals, a(n, k) = 1 + n*k. a(n, k) = a(n1, k) + k. Row n has g.f. (1+(n1)x)/(1x)^2, n >= 0.  Paul Barry, Feb 22 2003
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(n1, k1) = a(n1, k)*a(n, k1) + I*(f(n)  f(n1))*(g(k)  g(k1)) for suitable n and k. S= (E*W + 1)/N. arises with I = 1, and f = g = id.  Terry Lindgren, Apr 10 2011
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(n1, k1) = a(n1, k) + a(n, k1) + (f(n)  f(n1))*(g(k)  g(k1)), so with I = 1, and f = g = id, we have S+N = E+W + 1, as his students discovered.  Terry Lindgren, Nov 28 2016
T(n, k) = A128139(n1, k1).  Gary W. Adamson, Jul 02 2012
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
T(n, k) = 0 if n < k, T(n, 0) = 1, T(n,n) = 1, for n >= 0, and T(n, k) = (T(n1, k1)*T(n1, k) + 1)/(T(n2, k1)) for 0 < k < n. See the first comment referring to the triangle with its apex in the middle.  Wolfdieter Lang, Dec 19 2017


EXAMPLE

Third diagonal (1,3,5,7,...) consists of the positive integers congruent to 1 mod 2.
The triangle T(n, k) begins:
n\k 0 1 2 3 4 5 6 7 8 9 10 ...
0: 1
1: 1 1
2: 1 2 1
3: 1 3 3 1
4: 1 4 5 4 1
5: 1 5 7 7 5 1
6: 1 6 9 10 9 6 1
7: 1 7 11 13 13 11 7 1
8: 1 8 13 16 17 16 13 8 1
9: 1 9 15 19 21 21 19 15 9 1
10: 1 10 17 22 25 26 25 22 17 10 1
... reformatted.  Wolfdieter Lang, Dec 19 2017
As a square array read by antidiagonals, the first rows are:
1, 1, 1, 1, 1, 1, ...
1, 2, 3, 4, 5, 6, ...
1, 3, 5, 7, 9, 11, ...
1, 4, 7, 10, 13, 16, ...
1, 5, 9, 13, 17, 21, ...


MAPLE

A077028 := proc(n, k)
if n <0 or k<0 or k > n then
0;
else
k*(nk)+1 ;
end if;
end proc: # R. J. Mathar, Jul 28 2016


MATHEMATICA

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 *)


PROG

(PARI) {T(n, k) = if( k<0  k>n, 0, k * (n  k) + 1)}; /* Michael Somos, Mar 20 2011 */


CROSSREFS

Cf. A000125, A003991, A077029, A105851.
The maximum value for each antidiagonal is given by sequence A033638.
Equals A004247(n) + 1.
Sequence in context: A255741 A132892 A174448 * A114225 A193515 A259874
Adjacent sequences: A077025 A077026 A077027 * A077029 A077030 A077031


KEYWORD

nonn,tabl


AUTHOR

Clark Kimberling, Oct 19 2002


EXTENSIONS

Better definition based on Murthy's comment of Jul 15 2005 and the Anggoro et al. paper.  N. J. A. Sloane, Mar 05 2011


STATUS

approved



