HP Prime: Sums and Differences of Triangular Numbers, Triangular Numbers functions

HP Prime:  Sums and Differences of Triangular Numbers, Triangular Numbers functions

  

Benjamin Vitale, author of the Fun With Numb3rs blog (https://benvitalenum3ers.wordpress.com) wrote:

Consider the following system of equations:

T_a + T_b = T_c

T_a – T_b = T_d

Where T represents triangular numbers, which is defined by:

T_n = n * (n+1)/2 where n is a positive integer

Source Link: https://benvitalenum3ers.wordpress.com/2015/04/13/triangular-number-system-of-equations/

(Retrieved April 16, 2015)

Then:

T_a + T_b = T_c

T_a – T_b = T_d

Let:

a*(a+1)/2 + b*(b+1)/2 = C

a*(a+1)/2 – b*(b+1)/2 = D

C and D need to be tested to determine if they are both triangular numbers, which is fairly easy to check.  Without loss of generality, let:

C = n*(n+1)/2

2*C = n*(n+1)

2*C = n^2 + n

0 = n^2 + n – 2*C

Solving for n:

n = (-1 ± √(1^2 – 4*-1*2*C))/2

Since we are looking integers where n>0, we can state that:

n = (-1 + √(1^2 – 4*-1*2*C))/2

n = (-1 + √(1 – 8*C))/2

n = -.5 + √((1 – 8*C)/4)

n = -.5 + √(.25 – 2*C)

If frac(n) = 0, we have a solution.   (frac stands the fraction function)

The program TRIINTEST will list pairs of triangular numbers.  The argument of TRIINTEST(n), where n is upper limit of T_n to be tested.

HP Prime Program:  TRIINTEST

EXPORT TRIINTTEST(u)

BEGIN

// 2014-04-18

LOCAL a,b,c,d;

LOCAL A,B,C,D;

// matrix starter

LOCAL m:=[[0,0]];

LOCAL s:=2;

FOR a FROM 2 TO u DO

FOR b FROM 1 TO a DO

A:=(a^2+a)/2;

B:=(b^2+b)/2;

C:=A+B;

D:=A-B;

c:=−.5+√(.25+2*C);

d:=−.5+√(.25+2*D);

IF FP(c)==0 AND FP(d)==0 THEN

ADDROW(m,[A,B],s);

s:=s+1;

END;

END;

END;

// clean up

DELROW(m,1);

RETURN m;

END;

Finding pairs from T_1 to T_50:

[[ 3, 3 ]

[ 21, 15 ]

[ 105, 105 ]

[ 171, 105 ]

[ 703, 378 ]

[ 990, 780 ]]

Here are the functions TRINUM and ITRINUM:

TRINUM(n):  returns the nth triangular number. 

ITRINUM(t):  test whether t is a triangular number.  If t is not a triangular number, -1 is returned.

Program TRINUM:

EXPORT TRINUM(N)

BEGIN

// triangular number

RETURN N*(N+1)/2;

END;

Program ITRINUM:

EXPORT ITRINUM(T)

BEGIN

// inverse triangular number

// returns −1 if not a triangular

// number

LOCAL t:=(−.5+√(.25+2*T));

IF FP(t)==0 THEN

RETURN t;

ELSE

RETURN −1;

END;

END;

Examples:

TRINUM(15) = 120  (15^th triangular number)

ITRINUM(276) = 23   (276 is the 23^rd triangular number)

ITRINUM(572) returns -1   (572 is not a triangular number)

This blog is property of Edward Shore.  2015