Sum of a nonzero pentagonal number and a nonzero square in at least one way

[Jonathan Post]

Happy holidays!  Are there any corrections or extensions before I submit this?

Sum of a nonzero pentagonal number and a nonzero square in at least one way.

{A000326(i) + A000290(j) for i, j > 0}.
{i(3*i-1)/2 + j^2 for i, j > 0}.

2, 5, 6, 9, 10, 13, 14, 16, 17, 21, 23, 26, 28, 30, 31, 36, 37, 38,
39, 41, 44, 47, 48, 50, 51, 52, 54, 55, 58, 60, 61, 65, 67, 69, 71,
74, 76, 79, 82, 84, 86, 87, 93, 95, 96, 99, 100, 101, 103, 105, 106,
108, 112, 115, 116, 117, 118, 119, 121, 122, 126, 128, 132, 133, 134,
135, 141, 142, 143, 145, 146, 149, 151, 153, 154, 156, 161, 166, 170,
172, 173, 174, 177, 179, 180, 181, 185, 191, 192, 194, 195, 197, 198,
201, 204, 208, 209, 211, 212, 213, 214, 217, 218, 219, 220, 225, 226,
230, 231, 235, 236, 237, 238, 239, 240, 245, 246, 247, 248, 251, 256,
257, 259, 260, 261, 263, 266, 272, 274, 276, 283, 286, 288, 289, 291,
295, 296, 297, 303, 310, 311, 312, 313, 314, 317, 320, 323, 328, 331,
334, 336, 339, 341, 342, 345, 346, 347, 351, 354, 355, 366, 368, 370,
372, 379, 387, 391, 394, 401, 406, 408, 411, 416, 430, 431, 443, 451,
456, 472, 474, 483, 499, 512, 526, 555

This email is incomplete and not double-checked, but includes sums
through the 15th square and 15th pentagonal number.

examples: where P(n) = n-th pentagonal number:
a(1) = P(1) + 1^2 = 1 + 1 = 2.
a(2) = P(1) + 2^2 = 1 + 4 = 5 = P(2).
a(3) = P(2) + 1^2 = 5 + 1 = 6.
a(4) = P(2) + 2^2 = 5 + 4 = 9 = 3^2.
a(5) = P(1) + 3^2 = 1 + 9 = 10 = a(P(2)).
a(8) = P(3) + 2^2 = 12 + 4 = 16 = 4^2.
a(10) = P(2) + 4^2 = 5 + 16 = P(3) + 3^2 = 12 + 9 = 21.
a(12) = P(1) + 5^2 = 1 + 25 = P(4) + 2^2 = 22 + 4 = 26 = a(P(3)).
a(16) = P(5) + 1^2 = 35 + 1 = 36 = 6^2.
a(17) = P(1) + 6^2 = 1 + 36 = P(3) + 5^2 = 12 + 25 = 37.
a(25) = P(5) + 4^2 = 35 + 16 = 51 = P(6).
a(30) = P(6) + 3^2 = 51 + 9 = P(5) + 5^2 = 35 + 25 = 60.
a(35) = P(7) + 1^2 = 70 + 1 = P(5) + 6^2 = 35 + 36 = P(4) + 7^2 = 22 +
49 = 71 = a(P(5)).
a(37) = P(6) + 5^2 = 51 + 25 = P(3) + 8^2 = 12 + 64 = 76.
a(41) = P(7) + 4^2 = 70 + 16 = P(4) + 7^2 = 22 + 64 = 86.
a(43) = P(8) + 1^2 = 92 + 1^2 = P(3) + 9^2 = 12 + 81 = 93.
a(47) = P(6) + 7^2 = 51 + 49 = 100 = 10^2.
a(48) = P(8) + 3^2 = 92 + 9 = P(1) + 10^2 = 1 + 100 = = 101
a(59) = P(9) + 2^2 = 117 + 4 = 121 = 11^2.
a(60) = P(4) + 10^2 = 22 + 100 = P(1) + 11^2 = 1 + 121 = 122.
a(61) = P(9) + 3^2 = 117 + 9 = P(2) + 11^2 = 5 + 121 = 126.
a(64) = P(9) + 4^2 = 117 + 16 = P(3) + 11^2 = 12 + 121 = 133 = a(8^2).
a(x) = P(1) + 12^2 = 1 + 144 = 145 = P(10).
a(z) = P(10) + 2^2 = 145 + 4 = P(2) + 12^2 = 5 + 144 = 149.
a(b) = P(7) + 9^2 = 70 + 81 = P(6) + 10^2 = 51 + 100 = 151.
a(c) = P(8) + 8^2 = 92 + 64 = P(5) + 11^2 = 35 + 121 = P(3) + 12^2 =
12 + 144 = 156.
a(n) = P(9) + 7^2 = 117 + 49 = P(4) + 12^2 = 22+ 144 = 166.
a(n) = P(10) + 5^2 = 145 + 25 = P(7) + 10^2 = 70 + 100 = P(1) + 13^2 =
1 + 169 = 170.
a(h) = P(10) + 6^2 = 145 + 36 = P(9) + 8^2 = 117 + 64 = P(3) + 13^2 =
12 + 169 = 181.
a(o) = P(7) + 11^2 = 70 + 121 = P(4) + 13^2 = 22 + 169 = 191.
a(k) = P(11) + 4^2 = 176 + 16 = P(8) + 10^2 = 92 + 100 = 192.
a(a) = P(11) + 5^2 = 176 + 25 = P(2) + 14^2 = 5 + 196 = 201.
a(b) = P(12) + 2^2 = 210 + 4 = P(7) + 12^2 = 70 + 144 = 214.
a(c) = P(11) + 7^2 = 176 + 49 = 225 = 15^2.
a(i) =  P(12) + 4^2 = 210 + 16 = P(10) + 9^2 = 145 + 81 = P(1) + 15^2
= 1 + 225 = 226.
a(m) = P(6) + 14^2 = 51 + 196 = P(4) + 15^2 = 22 + 225 = 247 = P(13).
a(u) = P(13) + 3^2 = 247 + 9 = 256 = 16^2.
a(f) = P(11) + 9^2 = 176 + 81 = P(1) + 16^2 = 1 + 256 = 257.
a(o) = P(9) + 12^2 = 117 + 144 = P(2) + 16^2 = 5 + 256 = 261.
a(y) = P(10) + 11^2 = 145 + 121 = P(7) + 14^2 = 70 + 196 = 266.
a(d) = P(11) + 10^2 = 176 + 100 = P(6) + 15^2 = 51 + 225 = 276.
a(s) = P(10) + 12^2 = 145 + 144 = 289 = 17^2.
a(j) = P(14) + 2^2 = 287 + 4 = P(5) + 16^2 = 35 + 256 = 291.
a(f) = P(14) + 3^2 = 287 + 9 = P(13) + 7^2 = 247 + 49 = 296.
a(b) = P(15) + 1^2 = 330 + 1 = P(12) + 11^2 = 210 + 121 = 331.
a(v) = P(14) + 9^2 = 287 + 81 = P(13) + 11^2 = 247 + 121 = 368.
a(r) = P(15) + 7^2 = 330 + 49 = P(12) + 13^2 = 210 + 169 = 379.
a(t) = P(11) + 15^2 = 176 + 225 = P(1) + 20^2 = 1 + 400 = 401.
a(u) = P(12) + 15^2 = 210 + 225 = P(5) + 20^2 = 35 + 400 = 435.

So the numbers which can be so represented in exactly two ways begin:
21, 26, 37, 60, 76, 86, 93, 101, 122, 126, 133, 149, 151, 166, 170,
181, 191, 192, 201, 214, 247, 257, 261, 266, 276, 291, 296, 331, 368,
379, 401, 435.

The numbers which can be so represented in exactly three ways begin:
71, 156, 170, 181, 226.

The squares which can be so represented begin: 9, 16, 36, 100, 121,
225, 256, 289.

The pentagonal numbers which can be so represented begin: 5, 51, 145, 247.

Are almost all positive integers in this sequence, and, if so, what is
the largest value in the complement? The largest square in the
complement?  The largest pentagonal number in the complement?