SU1474627A2 - Generator of sequential generalized fibonacci p-numbers under arbitrary conditions - Google Patents

Abstract

The invention relates to computing and can be used to construct devices for controlling numbers represented in Fibonacci codes. The aim of the invention is to expand the scope of application by separating a sequence of even p = Fibonacci numbers. The generator contains counting cells 1.1-1.3, each of which contains trigger 2, first 3 and second 4 AND elements, AND-NE 5 element and reversing counter 6, registers 7.1 and 7.3, adder 8, block 9 controls, switch 10, informational 11, additional informational 12 and installation 13 inputs of the generator, informational 14, and additional informational 15 outputs of the generator, 1 dw., 2 tab.

Description

The invention relates to computing, can be used in the construction of devices for controlling numbers represented in Fibonacci codes, and is an improvement to the auth.t. No. 662926,

The aim of the invention is to expand the scope of application by identifying a sequence of even p-Fibonaccio numbers.

The drawing shows the functional diagram of the generator for the case.

The generator contains counting cells 1.1-1.3, each of which contains trigger 2, first and second elements 3 and 4, element 5 and reversing counter 6, registers 7.1–7.3, adder 8, control block 9, switch 10 and information 11, additional informational 12 and installation 13 inputs of the generator, the inet number of the first even number n0, 3, and the number by which the number of the first even number is recalculated is three. Thus, we have the following even numbers p and 1 Fibonacci numbers: 2,8,34,144 etc corresponding numbers 3,6,9,12 etc. (3,,, and

0 etc.) (Table 1).

,, 2ri-1 7, In this case, the numbers of the first three even terms are as follows: §,, n 7, and the number for which the recalculation of the numbers of the first three even numbers is seven. Thus, we have the following even p-numbers of 2-Fibonacci numbers: 2,4,6,28,, 406,872,1278, etc. corresponding numbers 4,6,7,11,13,

20 14,18,20,2) etc. (4,6,7;,,,,,, etc.) (Table 1)

,, 2p + 1-1 15; Moreover, the numbers of the first seven even-numbered members are 30

formational 14 and additional introductory ones: n 5,, poe 10,, formational 15 generator outputs. ,, Thus, the generator operates as follows.

In the proposed generator, the regularity of alternation of even and odd numbers of a sequence is used. The numbers of even numbers of a sequence are determined on the basis of the following expression

n n0j + (2P + 1 - 1) хК where n is the number of the first 2. even numbers of the sequence

(, 2.0.o, 2p-1);

K is a natural number.

35

P05

So, we have the following even p-numbers of 3-Fibonacci numbers: 2,4,10,14,26, 36,50,250,476,1252, etc., corresponding to the numbers 5,7,10,11,13,14,15, 20, 22,25,26,28,29,30 etc. (5,7,10,11, 3,14,15;,, 10 + 15 25, 11 + 15 26, 13 + 15 28, 14 + 15 29. 15 + 15 30, etc.).

Sequences for - 3 are presented in table.

Suppose that, a. On the basis of the recurrence relation (1), the sequence of 2 Fibonacci numbers given in Table 2.

We take the selection of even numbers of a sequence of generalized Fibonacci numbers, with,,, when, based on the recurrence relation

Oh when

(i) NO at i 0; (1) N04 (i-l) + No4 (i-l-p)

when i О, when Nft is the initial condition;

p - arbitrary natural

number;

Cp (i) is the i-e Fibonacci number, we have three numerical sequences given in Table 1. For simplicity, sequences are shown with an initial condition equal to one, which does not change the essence of the stated.

, 2P -1 1, 2 - 3. With

This is the number of the first even number n0, 3, and the number to which the numbers of the first even number are recalculated is three. Thus, we have the following even numbers of the row 1 Fibonacci numbers: 2,8,34,144, etc. corresponding to the numbers 3, 6,9,12 etc. (3,,, and

0 etc.) (Table 1).

,, 2ri-1 7; In this case, the numbers of the first three even members are as follows: §,, n 7, and the number to which the numbers of the first three even numbers are recalculated is seven. Thus, we have the following even p-numbers of 2-Fibonacci numbers: 2,4,6,28,, 406,872,1278, etc. corresponding numbers 4,6,7,11,13,

0 14,18,20,2) etc. (4,6,7;,,,,,, etc.) (Table 1)

,, 2р + 1-1 15, In this case, the numbers of the first seven even-numbered members after

5 blowing: n 5,, poe 10,,,,. Thus

five

0

five

0

five

P05

So, we have the following even p-numbers of 3-Fibonacci numbers: 2,4,10,14,26, 36,50,250,476,1252, etc., corresponding to the numbers 5,7,10,11,13,14,15, 20, 22,25,26,28,29,30 etc. (5,7,10,11, 3,14,15;,, 10 + 15 25, 11 + 15 26, 13 + 15 28, 14 + 15 29. 15 + 15 30, etc.).

Sequences for - 3 are presented in table.

Suppose that, a. On the basis of the recurrence relation (1), the sequence of 2 Fibonacci numbers given in Table 2.

At the initial time, at the input of 1 G, the initial condition, a number, is entered in the first register 7.1. At the same time, the remaining registers 7.2 and 7.3 contain zeros. The values corresponding to the numbers of the first three (2 - 1 3) even-numbered terms of the sequence are entered with the signal from input 13 at each counter 6.1 - 6.3 at inputs 12.1-12.3 of the device. So, the number 4 is entered into the counter 6., the number 6 is entered into the counter 6.2, the number 7 is counted at the counter 6.3. At the same time, the signal at input 13 triggers 2.1-2.3 triggers to the initial zero state via their R inputs. The clock signals from the output of the control unit 9 are received simultaneously at the first inputs of all elements AND 3 and 4. Since the flip-flops 2.1-2.3 are in the zero state, the signal from Log.1 from their inverse outputs arriving at the second input of the corresponding elements And 4, allows the passage of clock signals to the input of the reverse counting of the corresponding reversible counters 6. Thus, the reversible counters 1 are initially in the subtraction mode,

When the clock signals arrive, the contents of the reversible counters 1 are reduced by one. In this case, in each clock cycle the contents of registers 7.1 and 7.3 are summed up, after which the result of the summation is received by register 7.1, and the contents of register 7.1 are rewritten into register 7.2, the previous contents of which are received by register 7.3. Thus, in each clock cycle at the output of register 7.1 ( output 14), the results of the summation in the previous cycle, corresponding to the next number in the sequence of generalized Fibonacci numbers, appear according to the recurrence relation (1) (Table 2). The process of reducing the content of the reversible counters 6 occurs before they overflow. Since the number 4 was written to the reversing counter 6.1, it overflows first. The overflow signal of this counter (Log.O) from its first overflow output goes to the first input of the AND-KE 5.1 element, and therefore the output of the IE 5.1 element comes in the signal of Log.1, which is fed to the control input of the switch 10, and the number from the register output 7.1, corresponding to number 4, is fed to the output 15 of the generator. As can be seen from Table 2, this number q (4) 10 is an even number. Thus, the first even member of the 2-Fibonacci sequence is selected. At the same time, the overflow signal of the 6/1 rover counter enters the S-input of the 2 „1 trigger and sets it to the single state. Log.1 signal comes from its direct output to the second input of the element 3.1 and after this, the clock signals from the output of the control block 9 begin to arrive at the input of the direct count of the reversing counter 6.1, which goes into the summation mode with,

After the overflow counter 6c 1 overflows, after each 1 7 clock signals, the signal from the second overflow output goes to the second input of the AND-NOT 5.1 element, and the output of the last signal is the Log.1 signal that allows the output 15 of the device to pass This register output register 7,1 The sixth clock signal overflows the reversible counter 6.2 "Overflow signal (Log.O)" from its first overflow output enters the first input of the AND-HE element 5.2. The Log signal, 1 from its output, arriving at the control input of the switch 10, allows the output of the device 15 to pass a number developed by the register 7.1 and corresponding to the number 6 "

As can be seen from Table 2, this number is an even cp (6) 20 ". Thus, the second even number of the Fibonacci 2-sequence sequence is singled out. At the same time, the overflow signal of the reversible counter 6.2 goes to the S-input of the trigger 2.2 and sets it to in a single state. The signal Log.1 from its direct output enters the second input of the element I Zc2, and after that the clock signals begin to arrive at the input of the direct account of the reversible counter 602, which goes into the summation mode. The reversible counter 6.2 in this mode also starts summing the clock signals of the control unit 9

By overflowing the reversible counter 6.2, which occurs every 2 Pt1 - 1 7 clock signals, the signal from the second overflow output goes to the second input of the NAND element, and from its output to the control input of the switch 10, and allows the next even number from the sequence produced by the register 7.1 in this clock. Thus, the selection of even members of the sequence of generalized Fibonacci numbers with numbers 6,13,20, etc. is provided. As can be seen from Table 2, these numbers correspond to the following even numbers cf (6) 20, C (13) 300, etc.

In the seventh cycle, the reversible counter 6.3 overflows and the overflow signal from its first overflow output goes to the first input.

element AND NOT 5.3. Signal Log. its output enters the control input of the switch 10, thereby allowing the passage of the next even number of a sequence of generalized 2 Fibonacci numbers to the output 15 of the generator, whose number corresponds to this seventh cycle. As can be seen from Table 2, this number q (7) 30 is an even number. Thus, the third is highlighted, i.e. even (2 -1) number of the sequence of generalized 2 Fibonacci numbers. Simultaneously, the redirection signal can be entered into a new initial condition on input 12 to form and extract even numbers of the following sequence of generalized 2 Fibonacci numbers.

Claims (1)

Invention Formula Generator of a sequence of generalized p-Fibonacci numbers with arbitrary initial conditions. auth, sv. №662926, about tl and h and y y and i- so that, in order to expand about from the first output of the overflow - 15 fields of application due to the release sequences of even p-Fibonacci numbers, a switch and 2 -3 counting cells are entered into it, each of which contains a reversible counter, trigger-20 GER elements AND elements and NAND, the output of the first element AND of the counting cell is connected to the summing input of the reversible counter, the first output of which1 is connected to the first input by the summation element and further, the IS-NOT item and the S-input of the trigger, the output of the reversible counter of connections connects the clock with the clock signal. By overflowing the reversing yen with the second input of the NAND element, counter 6.3, every 2 pn forward and inverse outputs of the clock trigger signal, the signal from the second is connected to the first inputs of the corresponding overflow output to the first and second elements equal to the switch input 10 and A-And, the output of the second element And is connected with the subtracting input of the reversible counter, the second inputs of the elements AND of all the counting cells are combined and connected to the first output of the control unit, the C inputs of the reversible counters and the R inputs of the triggers of the counting cells are information input 40 reversible inputs counters; cells, which are the corresponding additional information on the reversing counter 6.3, are fed to the S-input of the trigger 2.3 and sets it to a single state. The signal Log.1 from the direct output of the trigger 2.3 arrives at the second input of the And 3.3 element, and further the clock signals arrive at the input of the direct account of the reversible counter 6.3. This counter also goes to decides the passage of the next number to exit 15 in this tact. Thus, the selection of the even terms of the sequence of generalized 2 Fibonacci numbers with numbers 7,14,21, etc. is provided. As can be seen from table 2, the following even numbers correspond to these numbers: cf (7) 30, cf (14) 440, etc. Thus, at the output of the generator (Table 2), all the even numbers of the sequence of the generalized 2 Fibonacci numbers are allocated: 10, dami generator, the outputs of the elements 20,30,140,300,440,2030, etc., respectively AND-NOT counting cells are combined and matching numbers 4,6,7; 11,13,14; 45 are connected to the control input of the switch 18,20,21, etc. It is noteworthy that by the signal connected to the corresponding information overflow from the second output by the output outputs of the first register, the execution of the reversible counter 6.3 rods are additional information generator, if there are the necessary outputs of the generator . , 1- -t1 -., -.-, inLiLl l Mlbl ll Ill 11.1 2 3 5 8 13 2 34 55 89 144 233 377 6Ю I I I I I 2 3 4 6 9 Э 19 28 I I 2 3 4 5 7 10 14 Table 1 13 four 15 (6 П J18 19 20 J2I J22 J23 | 24J25 41 60 88 179 1v9 271 406 595 872 1278 19 26 36 50 69 95 131 181 250 345 476 657 907 ГI 2 of 4 5 I 6 1 7 8 9 lO ll l2 l3 JI4 l5 J l 6 17 Ј 18 2 5 5 5 10 15 20 30 45 65 95 140 205 300 440 645 945 1385 2030 table 2