NEW FAMILY OF SEQUENCES

THE GENERATION OF NEW SEQUENCES (Part D)

Picture of a square

Introduction

Recently a new method for the generation of squares of squares were produced in Part IA through Part IVE. In addition, a new interleaved sequence of numbers was developed from this work and the sequence awarded the Sloane number A178218. OEIS has also published four other numbers based on this particular type of sequence. These numbers are A214345, A214493, A214393 and A214405.

This site will show that these four sequences as well as an infinitude of others are generated from the square sequence method developed in Part IA and ending with Part IVE .

Some Background

The previous parts showed that a tuple (a1,b1,c1) can be converted into a different tuple (a,b,c) basically a transformation of the type (a1,b1,c1) ⇒ (a,b,c). In addition, the initial tuples start out with the tuples (1,b1,1) or (1,1,c1) in which the b1s and the c1s have the following values:

b1 =  k
b1 = −k
c1 =  k
c1 = −k

where k is any natural integer from 1 to ∞ used in calculating f and the denominator d = 2(2b1 − c1 − 1 ) in the equation:

f = [2e2n2 + (4c1 − 4b1) en +(1 − 2b12 + c12)] / {2(2b1 − c1 − 1)}

This equation is critical is that it is the initial starting point for generating the interleaved sequences.

Table and General Sequence

The tables listed below although produced according to the methods of Parts IA through IVE, were actually computed and outputted by a computer program. Four sequences are apparent as highlighted in color. The one in green, the one in teal, the one in pink and the one in white. For example each green tuple ends with a number which is repeated in the next green tuple. Likewise for the teal, the pink and white tuples.

  
n
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Table I
a1 b1 c1
191
1139
11717
12125
12533
12941
13349
13757
14165
14573
14981
15389
15797
161105
165113
169121
  
f = S/d
-5
-8
-9
-8
-5
0
7
16
27
40
55
72
91
112
135
161
Table II
a b c
-44-4
-751
-888
-71317
-42028
12941
84056
175373
286892
4185113
56104136
73125161
92148188
133173217
136200248
161229281
  
Δ
0
-24
0
120
384
840
1536
2520
3840
5544
7680
10296
13440
17160
21504
26520
For the tuple (1,9,1)   f is calculated to be:
f = [2e2n2 + (4 − 36)en − 160]/2×16 = [2e2n2 − 32en − 160]/32
Setting e = 4 and g = 8 affords f = n2 − 4n − 5
Substituting this value of f in
(f + 1, f + 4n + 9, f + 8n + 1) as was shown in Part IA
gives the general equations for the complete sequence

a = (n2 − 4n − 5 + 1 ) = (n2 − 4n − 4 )
b = (n2 − 4n − 5 + 4n + 9 ) = (n2 + 4)
c = (n2 − 4n − 5 + 8n + 1 ) = (n2 + 4n − 4)

Separation of Sequences

To separate out the four sequences and generate the equations for each, the following method was found. All we need are two equations to generate the interleaved sequences in this case we wil take a and b. Since two sequences may be produced from table II above, a factor fs = 4 will be used to convert a ⇒ new a and b ⇒ new b. This is accomplished by substituting the value of 4n for n in the equations for a and b, to give the equations for the first sequence:

a = (16n2 − 16n − 4 )
b = (16n2 + 4)

Using these equations (or copying directly from Table II) we obtain for the first sequence whose Sloane number is listed in oeis as A216871:

-4, 4, -4, 20, 28, 68, 92, 148, 188, 260, 316, 404, 476, 580, 668, 788, 892, 1028, 1148, 1300, 1436, 1604, 1756, 1940, 2108, 2308, 2492

Since this sequence is obtained from the (1,9,1) tuple, it occurs twice, on both this page (Part D) and at Part C.

In addition, when each entry is divided by 4 the sequence is identical to the Sloane number A214345:
5, 7, 17, 23, 37, 47, 65, 79, 101, 119, 145, 167, 197, 223, 257, 287, 325, 359, 401, 439, 485, 527, 577, 623, 677, 727, 785, 839

To obtain the equations for the second sequence we use the second line of table II (1,13,9) and solve for f.

For the tuple (1,13,9)   f is calculated to be:
f = [2e2n2 + (36 − 52)en − 256]/2×16 = [2e2n2 − 16en − 256]/32
and by setting e = 4 affords the general equation f = n2 − 2n − 8
Substituting this value of f in
(f + 1, f + 4n + 13, f + 8n + 9) as was shown in Part IA
gives the general equations for the complete sequence starting at point (1,13,9)

a = (n2 − 2n − 8 + 1 ) = (n2 − 2n − 7 )
b = (n2 − 2n − 8 + 4n + 13 ) = (n2 + 2n + 5)
c = (n2 − 2n − 8 + 8n + 9) = (n2 + 6n + 1)

Substituting the value of 4n for n in the equations for a and b as was done above, affords the equations for the second sequence:

a = (16n2 − 8n − 7 )
b = (16n2 + 8n + 5)

Using these equations (or copying directly from Table II) we obtain for the second sequence identical to the Sloane No. A214405 except for the first three numbers which are also part of the sequence:

-7, 5, 1, 29, 41, 85, 113, 173, 217, 293, 353, 445, 521, 629, 721, 845, 953, 1093, 1217, 1373, 1513, 1685, 1841, 2029, 2201, 2405, 2593, 2813, 3017

To obtain the equations for the third sequence we use the third line of table II (1,17,17) and solve for f.

For the tuple (1,17,17)   f is calculated to be:
f = [2e2n2 + (0 − 0)en − 288]/2×16 = [2e2n2 − 288]/32
and by setting e = 4 affords the general equation f = n2 − 9
Substituting this value of f in
(f + 1, f + 4n + 17, f + 8n + 17) as was shown in Part IA
gives the general equations for the complete sequence starting at point (1,17,17)

a = (n2 − 9 + 1 ) = (n2 − 8 )
b = (n2 − 9 + 4n + 17 ) = (n2 + 4n + 8)
c = (n2 − 9 + 8n + 17) = (n2 + 8n + 8)

Substituting the value of 4n for n in the equations for a and b as was done above, affords the equations for the third sequence:

a = (16n2 − 8 )
b = (16n2 + 16n + 8)

Using these equations (or copying directly from Table II) we obtain for the third sequence whose Sloane number is listed in oeis as A216865:

-8, 8, 8, 40, 56, 104, 136, 200, 248, 328, 392, 488, 568, 680, 776, 904, 1016, 1160, 1288, 1448, 1592, 1768, 1928, 2120, 2296, 2504, 2696

This sequence is identical to the first sequence in Part C having the equations:

a = (16n2 − 32n + 8 )
b = (16n2 − 16n + 8)

which starts at 8,8,-8,8,8... and is, thefore, present in both families.

In addition, dividing each entry by 8 the sequence produces the identical sequence with the Sloane number A178218:
1, 1, -1, 1, 1, 5, 7, 13, 17, 25, 31, 41, 49, 61, 71, 85, 97, 113, 127, 145, 161, 181, 199, 221, 241, 265, 287, 313, 337, 365, 391, 421, 449, 481, 511, 545, 577, 613, 647, 685, 721, 761, 799, 841, 881, 925, 967, 1013

To obtain the equations for the fourth sequence we use the fourth line of table II (1,21,25) and solve for f.

For the tuple (1,21,25)   f is calculated to be:
f = [2e2n2 + (100 − 84)en − 256]/2×16 = [2e2n2 + 16en − 256]/32
and by setting e = 4 affords the general equation f = n2 + 2n − 8
Substituting this value of f in
(f + 1, f + 4n + 21, f + 8n + 25) as was shown in Part IA
gives the general equations for the complete sequence starting at point (1,21,25)

a = (n2 + 2n − 8 + 1 ) = (n2 + 2n − 7 )
b = (n2 + 2n − 8 + 4n + 21 ) = (n2 + 6n + 13)
c = (n2 + 2n − 8 + 8n + 25) = (n2 + 10n + 17)

Substituting the value of 4n for n in the equations for a and b as was done above, affords the equations for the fourth sequence:

a = (16n2 + 8n − 7 )
b = (16n2 + 24n + 13)

Using these equations (or copying directly from Table II) we obtain for the fourth sequence identical tothe Sloane No. A214393:

-7, 13, 17, 53, 73, 125, 161, 229, 281, 365, 433, 533, 617, 733, 833, 965, 1081, 1229, 1361, 1525, 1673, 1853, 2017, 2213, 2393, 2605, 2801

So in effect what we have done is to produce four new sequences via a completely new route (the Sloane numbers A214405 and A214393 are already known). Previous methods would have generated one sequence without the knowledge that others existed. In addition, all four sequences are related via the general sequence formula (before factoring in fs) and thus constitute a family of interleaved sequences.

This concludes Part D. Go back to homepage.

Copyright © 2012 by Eddie N Gutierrez. E-Mail: Fiboguti89@Yahoo.com