This task is a variation of the short story by Arthur C. Clarke.

(Solvers should be aware of the consequences of completing this task.)

In detail, to specify what is meant by a   “name”:

The integer 1 has 1 name     “1”.
The integer 2 has 2 names   “1+1”,   and   “2”.
The integer 3 has 3 names   “1+1+1”,   “2+1”,   and   “3”.
The integer 4 has 5 names   “1+1+1+1”,   “2+1+1”,   “2+2”,   “3+1”,   “4”.
The integer 5 has 7 names   “1+1+1+1+1”,   “2+1+1+1”,   “2+2+1”,   “3+1+1”,   “3+2”,   “4+1”,   “5”.


Task
Display the first 25 rows of a number triangle which begins:

                                      1
                                    1   1
                                  1   1   1 
                                1   2   1   1
                              1   2   2   1   1
                            1   3   3   2   1   1

Where row   



n


{\displaystyle n}

   corresponds to integer   



n


{\displaystyle n}

,   and each column   



C


{\displaystyle C}

   in row   



m


{\displaystyle m}

   from left to right corresponds to the number of names beginning with   



C


{\displaystyle C}

.

A function   



G
(
n
)


{\displaystyle G(n)}

   should return the sum of the   



n


{\displaystyle n}

-th   row.

Demonstrate this function by displaying:   



G
(
23
)


{\displaystyle G(23)}

,   



G
(
123
)


{\displaystyle G(123)}

,   



G
(
1234
)


{\displaystyle G(1234)}

,   and   



G
(
12345
)


{\displaystyle G(12345)}

.

Optionally note that the sum of the   



n


{\displaystyle n}

-th   row   



P
(
n
)


{\displaystyle P(n)}

   is the     integer partition function.

Demonstrate this is equivalent to   



G
(
n
)


{\displaystyle G(n)}

   by displaying:   



P
(
23
)


{\displaystyle P(23)}

,   



P
(
123
)


{\displaystyle P(123)}

,   



P
(
1234
)


{\displaystyle P(1234)}

,   and   



P
(
12345
)


{\displaystyle P(12345)}

.



Extra credit
If your environment is able, plot   



P
(
n
)


{\displaystyle P(n)}

   against   



n


{\displaystyle n}

   for   



n
=
1
…
999


{\displaystyle n=1\ldots 999}

.

Related tasks
 Partition function P


 
/* ARM assembly AARCH64 Raspberry PI 3B */
/*  program integerName64.s   */ 
 
/*******************************************/
/* Constantes file                         */
/*******************************************/
/* for this file see task include a file in language AArch64 assembly*/
.include "../includeConstantesARM64.inc"
 
.equ MAXI,   524
 
/*********************************/
/* Initialized data              */
/*********************************/
.data
sMessResult:        .asciz "Total  : @  pour @ \n"
szMessError:        .asciz "Number too large !!.\n"
szCarriageReturn:   .asciz "\n"
/*********************************/
/* UnInitialized data            */
/*********************************/
.bss
sZoneConv:        .skip 24
tbNames:          .skip 8 * MAXI
/*********************************/
/*  code section                 */
/*********************************/
.text
.global main 
main:                                 // entry of program 
 
    mov x0,#5
    bl functionG
 
    mov x0,#23
    bl functionG
 
    mov x0,#123
    bl functionG
 
    mov x0,#524
    bl functionG
 
    mov x0,#1234
    bl functionG
100:                                  // standard end of the program 
    mov x0, #0                        // return code
    mov x8, #EXIT                     // request to exit program
    svc #0                            // perform the system call
 
qAdrszCarriageReturn:     .quad szCarriageReturn
qAdrsMessResult:          .quad sMessResult
qAdrtbNames:              .quad tbNames
qAdrsZoneConv:            .quad sZoneConv
/******************************************************************/
/*            compute function G                                 */ 
/******************************************************************/
/* x0 contains N */
functionG:
    stp x1,lr,[sp,-16]!          // save  registres
    stp x2,x3,[sp,-16]!          // save  registres
    stp x4,x5,[sp,-16]!          // save  registres
    cmp x0,#MAXI + 1
    bge 2f
    mov x3,x0
    mov x2,#1
1:                               // loop compute every item
    mov x0,x2
    bl computeNumber
    add x2,x2,#1
    cmp x2,x3
    ble 1b
 
    ldr x1,qAdrsZoneConv         // result display
    bl conversion10              // call decimal conversion
    ldr x0,qAdrsMessResult
    ldr x1,qAdrsZoneConv         // insert conversion in message
    bl strInsertAtCharInc
    mov x4,x0
    mov x0,x3
    ldr x1,qAdrsZoneConv         // result display
    bl conversion10              // call decimal conversion
    mov x0,x4
    ldr x1,qAdrsZoneConv         // insert conversion in message
    bl strInsertAtCharInc
    bl affichageMess
    mov x0,#0
    b 100f
2:
    ldr x0,qAdrszMessError
    bl affichageMess
    mov x0,#-1
100:
    ldp x4,x5,[sp],16           // restaur des  2 registres
    ldp x2,x3,[sp],16           // restaur des  2 registres
    ldp x1,lr,[sp],16           // restaur des  2 registres
    ret
qAdrszMessError:         .quad szMessError
/******************************************************************/
/*            random door test strategy                           */ 
/******************************************************************/
/* x0 contains N */
computeNumber:
    stp x1,lr,[sp,-16]!          // save  registres
    stp x2,x3,[sp,-16]!          // save  registres
    stp x4,x5,[sp,-16]!          // save  registres
    stp x6,x7,[sp,-16]!          // save  registres
    ldr x6,qAdrtbNames           // table address
    mov x1,#1
    str x1,[x6]                  // init item 0
    mov x1,#0
    str x1,[x6,x0,lsl #3]        // init item N
    mov x2,#1                    // indice
1:
    add x3,x2,x2, lsl #1
    sub x4,x3,#1
    mul x4,x2,x4
    lsr x4,x4,#1
    subs x3,x0,x4                // compute new indice
    blt 90f
    tst x2,#1                    // indice owen ?
    beq 2f
    ldr x4,[x6,x3,lsl #3]
    ldr x5,[x6,x0,lsl #3]
    add x5,x5,x4                 // addition
    str x5,[x6,x0,lsl #3]
    b 3f
2:                               // else substrac
    ldr x4,[x6,x3,lsl #3]
    ldr x5,[x6,x0,lsl #3]
    sub x5,x5,x4
    str x5,[x6,x0,lsl #3]
3:
    subs x3,x3,x2                // compute new indice
    blt 90f
 
    tst x2,#1                    // owen ?
    beq 4f
    ldr x4,[x6,x3,lsl #3]
    ldr x5,[x6,x0,lsl #3]
    add x5,x5,x4
    str x5,[x6,x0,lsl #3]
    b 5f
4:
    ldr x4,[x6,x3,lsl #3]
    ldr x5,[x6,x0,lsl #3]
    sub x5,x5,x4
    str x5,[x6,x0,lsl #3]
5:
    add x2,x2,#1
    cmp x2,x0
    ble 1b
90:
   ldr x0,[x6,x0,lsl #3]         // return last item of table
100:
    ldp x6,x7,[sp],16           // restaur des  2 registres
    ldp x4,x5,[sp],16           // restaur des  2 registres
    ldp x2,x3,[sp],16           // restaur des  2 registres
    ldp x1,lr,[sp],16           // restaur des  2 registres
    ret
/********************************************************/
/*        File Include fonctions                        */
/********************************************************/
/* for this file see task include a file in language AArch64 assembly */
.include "../includeARM64.inc"
 