A292565 Take 0, skip 3 * 1 + 1, take 1, skip 3 * 2 + 1, take 2, skip 3 * 3 + 1, ...
5, 13, 14, 25, 26, 27, 41, 42, 43, 44, 61, 62, 63, 64, 65, 85, 86, 87, 88, 89, 90, 113, 114, 115, 116, 117, 118, 119, 145, 146, 147, 148, 149, 150, 151, 152, 181, 182, 183, 184, 185, 186, 187, 188, 189, 221, 222, 223, 224, 225, 226, 227, 228, 229, 230, 265
Offset: 1
Examples
k| A292564(n)^2 | a(n)^2 | Sum
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0| 0^2 (= 0)
1| 3^2 + 4^2 = 5^2 (= 25)
2| 10^2 + 11^2 + 12^2 = 13^2 + 14^2 (= 365)
3| 21^2 + 22^2 + 23^2 + 24^2 = 25^2 + 26^2 + 27^2 (= 2030)
4| 36^2 + 37^2 + 38^2 + 39^2 + 40^2 = 41^2 + 42^2 + 43^2 + 44^2 (= 7230)
| ...
Row 3 is proved by the following:
(25^2 - 24^2) + (26^2 - 23^2) + (27^2 - 22^2) = 49*1 + 49*3 + 49*5 = 7^2*3^2 = 21^2.
Row k is proved by the same way.
Links
- Michael Boardman, Proof Without Words: Pythagorean Runs, Math. Mag., 73 (2000), 59.
Programs
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Mathematica
Block[{s = Array[{# - 1, 3 # + 1} &, 12], r}, r = Range@ Total@ Flatten@ s; Map[Function[{a, b}, {First@ #, Set[r, Drop[Last@ #, b]]} &@ TakeDrop[r, a]] @@ # &, s][[All, 1]] // Flatten] (* Michael De Vlieger, Sep 25 2017 *)
Formula
Sum_{n = (k-1)*k/2+1 .. k*(k+1)/2} a(n)^2 = Sum_{n = k*(k+1)/2 .. (k+1)*(k+2)/2-1} A292564(n)^2 = A059255(k) for k > 0.
a(n) = n + 4 + (3k^2 + 11k)/2 where k = floor((sqrt(2*n) - 1/2)). - Jon E. Schoenfield, Sep 30 2017