The Museum of HP Calculators

3 Batrachions for the HP-41

This program is supplied without representation or warranty of any kind. Jean-Marc Baillard and The Museum of HP Calculators therefore assume no responsibility and shall have no liability, consequential or otherwise, of any kind arising from the use of this program material or any part thereof.

Overview

1°) Hofstadter's Batrachion
2°) Conway's Batrachion
3°) Mallows's Batrachion

-These programs compute the successive terms of 3 recursive sequences

1°) Hofstadter's Batrachion

-The sequence is defined by H(1) = H(2) = 1 ; H(n) = H(n-H(n-1)) + H(n-H(n-2))

Data Registers: R00 = 0 ; R01 = H(1) ; R02 = H(2) ; .......... ; Rnn = H(n)
Flags: /
Subroutines: /

01 LBL "HOF"
02 .1
03 %
04 1
05 STO 01
06 ST+ Y
07 0
08 STO 00
09 STO 02
10 LBL 01
11 RCL Z
12 X<>Y
13 -
14 RDN
15 RCL IND T
16 X<>Y
17 ST- T
18 X<> L
19 X<>Y
20 RCL IND T
21 +
22 STO IND Z
23 ISG Z
24 GTO 01
25 END

( 43 bytes / SIZE nnn+1 )

      STACK        INPUTS      OUTPUTS

           Z             /        1+n.nnn

           Y             /         H(n-1)

           X             n          H(n)

Example: 100 XEQ "HOF" >>>> H(100) = 56 ( in 39 seconds ) and H(n) is in register Rnn for n < 101

-The firs few terms are:

n 1 2 3 4 5 6 7 8 9 10

H(n) 1 1 2 3 3 4 5 5 6 6

2°) Conway's Batrachion

-The sequence is defined by C(1) = C(2) = 1 ; C(n) = C(C(n-1)) + C(n-C(n-1)) ( n > 2 )

Data Registers: R00 unused ; R01 = C(1) ; R02 = C(2) ; .......... ; Rnn = C(n)
Flags: /
Subroutines: /

01 LBL "CNW"
02 3
03 X<=Y?
04 GTO 00
05 SIGN
06 RTN
07 LBL 00
08 X<>Y
09 E3
10 /
11 +
12 1
13 STO 01
14 STO 02
15 LBL 01
16 RCL Y
17 X<>Y
18 -
19 X<>Y
20 RCL IND Y
21 RCL IND L
22 +
23 STO IND Y
24 ISG Y
25 GTO 01
26 END

( 42 bytes / SIZE nnn+1 )

      STACK        INPUTS      OUTPUTS

           Y             /        1+n.nnn*

           X             n          C(n)

* if n > 2

Example: 100 XEQ "CNW" >>>> C(100) = 57 ( in 30 seconds ) and C(n) is in register Rnn for n < 101

-The first few terms are

n 1 2 3 4 5 6 7 8 9 10

C(n) 1 1 2 2 3 4 4 4 5 6

-We have C(2^k) = 2^k-1
-The graph of the ratio C(n)/n looks like a hopping frog and C(n)/n tends to 1/2 as n tends to infinity.

C(n)/n
   |       *
   |     * *          *
   |   *     *      *    *              *
   | *       *   *        *       *       *
   | *         **            * *             *
--|*-------*----------*------------*------------------------- n

3°) Mallows's Batrachion

-The sequence is defined by M(1) = M(2) = 1 ; M(n) = M(M(n-2)) + M(n-M(n-2))

Data Registers: R00 = 0 ; R01 = M(1) ; R02 = M(2) ; .......... ; Rnn = M(n)
Flags: /
Subroutines: /

01 LBL "MLW"
02 .1
03 %
04 1
05 STO 01
06 STO 02
07 ST+ Y
08 0
09 STO 00
10 LBL 01
11 X<>Y
12 RCL Z
13 X<>Y
14 -
15 RDN
16 RCL IND T
17 RCL IND L
18 +
19 STO IND Z
20 ISG Z
21 GTO 01
22 END

( 38 bytes / SIZE nnn+1 )

      STACK        INPUTS      OUTPUTS

           Z             /        1+n.nnn

           Y             /         M(n-1)

           X             n          M(n)

Example: 260 XEQ "MLW" >>>> M(260) = 180 ( in 85 seconds ) and M(n) is in register Rnn for n < 261

-The first few terms are

n 1 2 3 4 5 6 7 8 9 10

M(n) 1 1 2 3 3 4 5 6 6 7

Reference: Clifford A. Pickover - "Keys to Infinity" - John Wiley & Sons - ISBN 0-471-11857-5

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