login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A077028 The rascal triangle, read by rows: T(n,k) (n >= 0, 0 <= k <= n) = k(n-k) + 1. 36
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 (list; table; graph; refs; listen; history; text; internal format)
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 n-th diagonal is congruent to 1 mod n-1.

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.

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

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 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.)

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, 1145-VS-280, 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. 393-395.

D. C. Fielder & C. O. Alford, An investigation of sequences derived from Hoggatt Sums and Hoggatt Triangles, 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)

Julian Fleron, Tackling Rascals’ Triangle - How Inquiry Challenges What We Know and How We Know It, Discovering the Art of Mathematics, December 15 2015.

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 two-bridge 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(n-1, k) + k. Row n has g.f. (1+(n-1)x)/(1-x)^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(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

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

T(n, k) = A128139(n-1, k-1). - 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(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

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*(n-k)+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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified July 12 12:18 EDT 2020. Contains 335661 sequences. (Running on oeis4.)