Numbers

Machine Related Operations

• Numbers
  • Fibonacci numbers
  • Finding the n^th Fibonacci number
  • Harmonic numbers
  • n^th harmonic number
  • Bernoulli numbers
  • Euler numbers

Source: HP-45 Applications Book (HP 00045-90320 Rev. B Reorder 00045-66001, Dec 1974)

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Numbers

Fibonacci numbers

Formula:

In a Fibonacci sequence, each term is the sum of the two preceding terms.

f_i = f_i-1 + f_i-2

f_i represents the i^th term in the sequence.

Example:

Develop the Fibonacci sequence with f₁ = 1, f₂ = 1.

Answer:

1, 1, 2, 3, 5, 8, 13, 21, …

LINE	DATA	OPERATIONS	DISPLAY	REMARKS
1	f1	ENTER
2	f2
3		ENTER ENTER R↓ R↓ +	fi	Perform 3 for i=3,4,…

; XEQ F ENTER
        LBL     F
        CLx
        ENTER
        !
@005:   PSE
        eqn     'REGX+REGY'
        GTO     @005

Source: Morten Nygaard Åsnes (208623)

Finding the n^th Fibonacci number

Formula:

Example:

Find the 12^th Fibonacci number in the sequence 1, 1, 2, 3, 5, 8 …

Answer:

144

LINE	DATA	OPERATIONS	DISPLAY	REMARKS
1	n	ENTER 5 √x STO
2		A ENTER 1 + 2
3		÷ x<>y y˟ RCL A
4		÷ DISPLAY 1'FIX 0

Harmonic numbers

Formula:

The Harmonic numbers H_i (i = 1, 2, …) are

1, 1+¹⁄₂, 1+¹⁄₂+¹⁄₃, 1+¹⁄₂+¹⁄₃+¹⁄₄, …

or

1, ³⁄₂, ¹¹⁄₆, ²⁵⁄₁₂, …

Example:

Display the sequence in decimal form.

Answer:

1.00, 1.50, 1.83, 2.08, …

LINE	DATA	OPERATIONS	DISPLAY	REMARKS
1		ENTER STO H
2				Perform 2-4 for i=1,2,…
3		1 + ENTER 1/x RCL
4		H + STO H	Hi

n^th harmonic number

Formula:

Example:

Find the 7^th Harmonic number.

Answer:

2.59

Note:

E = 0.5772156649 is Euler's constant (usually denoted by the lowercase Greek letter γ).

LINE	DATA	OPERATIONS	DISPLAY	REMARKS
1		STO A 1/x ENTER
2		ENTER ENTER 1 2 0
3		1/x × × 1 2
4		1/x - × 2 1/x
5		+ ×
6	E	+ RCL A ln
7		+

Bernoulli Numbers

The Bernoulli numbers B_l, B₂, B₃ … are defined by

specifically

¹⁄₆, ¹⁄₃₀, ¹⁄₄₂, ¹⁄₃₀, ⁵⁄₆₆, ⁶⁹¹⁄₂₇₃₀, ⁷⁄₆, …

Example:

The 16^th Bernoulli number = 7.09 (i takes the values 1, 2).

LINE	DATA	OPERATIONS	DISPLAY	REMARKS
1	n	ENTER 2 × ENTER ENTER
2		ENTER ENTER
3		STO B		Perform 3-6 for i=1,2,…
4	i	ENTER 2 × 1 +		until change Ai does not change
5		x<>y y˟ 1/x RCL B
6		+	Ai
7		R↓ ! 2 ×
8		RCL B × R↓ R↓
9		2 x<>y y˟ 1 -
10		x<>y π x<>y y˟
11		× ÷

; XEQ B ENTER
        LBL     B
        STO     C
        1
        x>y?
        RTN
        STO     X
        RCL+    X
        x<=y?
        GTO     @12
        1/x
        +/-
        RTN
@12:
        x!=y?
        GTO     @18
        6
        1/x
        RTN
@18:
        RMDR
        0
        x!=y?
        RTN
        4
        RCL     C
        x!=y?
        GTO     @31
        30
        1/x
        +/-
        RTN
@31:
        6
        x!=y?
        GTO     @38
        42
        1/x
        RTN
@38:
        x<>y
        FIX     9
        XEQ     ZETA
        STO     X
        RCL+    X
        1
        +/-
        RCL     C
        2
        /
        y^x
        *
        +/-
        pi
        STO     X
        RCL+    X
        1e-9
        -
        RCL     C
        y^x
        /
@58:
        RCL     C
        *
        DSE     C
        GTO     @58
        RTN
ZETA:
        +/-
        STO     B
        1
        STO     A
        ENTER
@07:
        CLx
        !               ; REGX = 1
        RCL     A
        +
        STO     A
        RCL     B
        y^x
        -
        +/-
        LASTx
        RND             ; the accuracy is determined by the display format.
        x!=0?
        GTO     @07
        !               ; REGX = 1
        2
        RCL     B
        y^x
        STO     X
        RCL+    X
        -
        /
        ABS
        RTN

Source: Jean-Marc Baillard (41bernu)

Euler Numbers

Compute the n_th Euler number

Note:

The Euler numbers are 1, 5, 61, 1385, 50521, …

Formula:

The Euler numbers E¹, E², E³, … are defined by

Example:

The 5^th Euler number = 50521.

LINE	DATA	OPERATIONS	DISPLAY	REMARKS
1	n	ENTER 2 × ENTER ENTER
2		ENTER ! x<>y 2
3		+ 2 x<>y y˟ ×
4		x<>y 1 ÷ π
5		x<>y y˟ ÷ DISPLAY
6		1'FIX 0