A000217 and related

[Antreas P. Hatzipolakis]

D Number: A000217 (Formerly M2535 and N1002)
Sequence:  0,1,3,6,10,15,21,28,36,45,55,66,78,91,105,120,136,153,171,
           190,210,231,253,276,300,325,351,378,406,435,465,496,528,561,
           595,630,666,703,741,780,820,861,903,946,990,1035,1081,1128,
           1176,1225,1275
Name:      Triangular numbers: C(n+1,2) = n(n+1)/2.

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1. Triangular numbers Tx which are the sum of two triangular numbers Ty, Tz
(with y,z <x): 6,21,36,55,66,91,120,136,....

 x   Tx                     Ty   Tz
_____________________________________
 3   T3  =   6 =  3 +   3 = T2 + T2
 6   T6  =  21 =  6 +  15 = T3 + T5
 8   T8  =  36 = 15 +  21 = T5 + T6
10  T10  =  55 = 10 +  45 = T4 + T9
11  T11  =  66 = 21 +  45 = T6 + T9
13  T13  =  91 = 36 +  55 = T8 + T10
15  T15  = 120 = 15 + 105 = T5 + T14
16  T16  = 136 = 45 +  91 = T9 + T13
............

Sequence of x's: 3,6,8,10,11,13,15,16,....  =  A012132

Theorem (Sierpinski): x^2 + (x+1)^2 is composite.

x^2   (x+1)^2
-----------------------------
 3^2 +  4^2 =  25 =  5 X   5
 6^2 +  7^2 =  85 =  5 X  17
 8^2 +  9^2 = 145 =  5 X  29
10^2 + 11^2 = 221 = 13 X  17
11^2 + 12^2 = 265 =  5 X  53
13^2 + 14^2 = 365 =  5 X  73
15^2 + 16^2 = 481 = 13 X  37
16^2 + 17^2 = 545 =  5 X 109
.......

Sierpinski's Sequence of Composites: 25,85,145,221,265,365,481,545,....
[Numbers of the form x^2 + (x+1)^2, where x's are: A012132]

2. Conjecture (Schinzel): There are infinitely many triangular numbers
which are not the sum of two triangular numbers.

Schinzel's Sequence: 1,3,10,15,28,45,78,105,153,190,.....


Antreas