General-Recurrence Sequence are defined below and exemplified (found

in State Archives, Rm, Valcea, Romania).

A. 1, 2, 5, 26, 29, 677, 680, 701, 842, 845, 866, 1517, 458330, 458333, 458354, ...

(ss2(n) is the smallest number, strictly greater than the previous one, which is the squares sum of two previous distinct terms of the sequence;

in our particular case the first two terms are 1 and 2.

Recurrence definition:

(1) The numbers a <= b belong to SS2;

(2) If b, c belong to SS2, then b

(3) Only numbers, obtained by rules [(1) and/or (2)] applied a finite

number of times, belong to SS2.

The sequence (set) SS2 is increasingly ordered.

[ Rule (1) may be changed by: the given numbers a

B. 1, 1, 2, 4, 5, 6, 16, 17, 18, 20, 21, 22, 25, 26, 27, 29, 30, 31, 36, 37, 38, 40, 41, 42, 43, 45, 46, ...

(SS1(n) is the smallest number, strictly greater than the previous one, (for n>=3), which is the squares sum of one or more previous distinct terms of the sequence;

in our particular case the first term is 1.)

Recurrence definition:

(1) The number a belongs to SS1;

(2) If b

(3) Only numbers, obtained by rules [(1) and/or (2)] applied a finite number of times, belong to SS1.

The sequence (set) SS1 is increasingly ordered.

[ Rule (1) may be changed by: the given numbers a

C. 1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 14, 15, 16, 18, 19, 21, ...

(NSS2(n) is the smallest number, strictly greater than the previous one, which is NOT the squares sum of two previous distinct terms of the sequence;

In our particular case the first two terms are 1 and 2.)

Recurrence definition:

(1) The numbers a <= b belong to NSS2;

(2) If b, c belong to NSS2, then b

(3) Only numbers, obtained by rules [(1) and/or (2)] applied a finite number of times, belong to NSS2.

The sequence (set) NSS2 is increasingly ordered.

[Rule (1) may be changed by; the given numbers a

D. 1, 2, 3, 6, 7, 8, 11, 12, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 38, 39, 42, 43, 44, 47, ...

(NSS1(n) is the smallest number, strictly greater than the previous one, which is NOT the squares sum of one or more of the previous distinct terms of the sequence; in our particular case the first term is 1.)

Recurrence definition:

(1) The number a belongs to NSS1;

(2) If b

any other numbers belong to NSS1;

(3) Only numbers, obtained by rules [(1) and/or (2)] applied a finite number of times, belong to NSS1.

[ Rule (1) may be changed by: the given numbers a

E. 1, 2, 9, 730, 737, 389017001, 389017008,389017729, ...

(CS2(n) is the smallest number, strictly greater than the previous one, which is the cubes sum of two previous distinct terms of the sequence;

in our particular case the first two terms are 1 and 2.)

Recurrence definition:

(1) The numbers a <= b belong to CS2;

(2) If c,d belong to CS2, then c

(3) Only numbers, obtained by rules [(1) and/or (2)] applied a finite number of times, belong to CS2.

The sequence (set) CS2 is increasingly ordered.

[ Rule (1) may be changed by: the given numbers a

F. 1, 1, 2, 8, 9, 10, 512, 513, 514, 520, 521, 522, 729, 730, 731, 737, 738, 739, 1241, ...

(CS1(n) is the smallest number, strictly greater than the previous one (for n >= 3), which is the cubes sum of one or more previous distinct terms of the sequence;

in our particular case the first term is 1;

Recurrence definition:

(1) The number a belongs to CS1;

(2) If b

(3) Only numbers, obtained by rules [(1) and/or (2)] applied a finite number of times, belong to CS1.

The sequence (set) CS1 is increasingly ordered.

[ Rule (1) may be changed by: the given numbers a

G. 1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 29, 30, 31, 32, 33, 34, 36, 37, 38, ...

(NCS2(n) is the smallest number, strictly greater than the previous one, which is NOT the cubes sum of two previous distinct terms of the sequence;

in our particular case the first two terms are 1 and 2.)

Recurrence definition:

(1) The numbers a <= b belong to NCS2.

(2) If c,d belong to NCS2, then c

(3) Only numbers, obtained by rules [(1) and/or (2)] applied a finite number of times, belong to NCS2.

The sequence (set) NCS2 is increasingly ordered.

[ Rule (1) may be changed by: the given numbers a

H. 1, 2, 3, 4, 5, 6, 7, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 29, 30, 31, 32, 33, 34, 37, 38, 39, ...

(NCS1(n) is the smallest number, strictly greater than the previous one, which is NOT the cubes sum of one or more previous distinct terms of the sequence;

in our particular case the first term is 1.)

Recurrence definition:

(1) The number a belongs to NCS1.

(2) If b

(3) Only numbers, obtained by rules [(1) and/or (2)] applied a finite number of times, belong to NCS1.

The sequence (set) NCS1 is increasingly ordered.

[ Rule (1) may be changed by: the given numbers a

I. General recurrence type sequence:

General recurrence definition:

Let k >= j be natural numbers, and a

Then:

(1) The elements a

(2) If m

(3) Only numbers, obtained by rules [(1) and/or (2)] applied a finite number of times, belong to SGR.

The sequence (set) SGR is increasingly ordered.

Method of construction of the general recurrence sequence:

-level 1: the given elements a

-level 2: apply the relationship R for all combinations of j elements among a

the results belong to SGR too;

order all elements of levels 1 and 2 together,

-level i+1:

if b

the results belong to SGR too;

order all elements of levels i and i+1 together;

and so on . . .

*Originally appeared in