A363284 Numbers that are square or square pyramidal.

0, 1, 4, 5, 9, 14, 16, 25, 30, 36, 49, 55, 64, 81, 91, 100, 121, 140, 144, 169, 196, 204, 225, 256, 285, 289, 324, 361, 385, 400, 441, 484, 506, 529, 576, 625, 650, 676, 729, 784, 819, 841, 900, 961, 1015, 1024, 1089, 1156, 1225, 1240, 1296, 1369, 1444, 1496

Offset: 1

Author

Clark Kimberling, May 25 2023

Keywords

• nonn

Comments

This sequence essentially consists of the numbers in A363269 arranged in increasing order. Although A363269 is a linear recurrence sequence, it appears that this sequence is not.

4900 is the only nontrivial case of a square number that is also square pyramidal (proved by Watson). - Peter Munn, Jul 30 2023

References

• W. Ljunggren, New solution of a problem proposed by E. Lucas, Norsk Mat. Tidsskr. 34 (1952), pp 65-72.
• David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, 1987, entry 24, p 101.

Links

• Michael A. Bennett, Lucas' Square Pyramid Problem Revisited.
• E. Lucas, Problem 1180, Nouvelles Ann. Math. (2) 14 (1875), p 336.
• G. N. Watson, The problem of the square pyramid, Messenger of Mathematics 48 (1918), pp. 1-22.

Crossrefs

Cf. A000290, A000330, A363269.

Programs

Mathematica

c[1] = 1; c[2] = 1;
c[n_] := If[OddQ[n], c[n - 2] + n, c[n - 2] + c[n - 1]]
u = Table[c[n], {n, 1, 120}]  (* A363269 *)
FindLinearRecurrence[u]
Union[u] (* this sequence *)

Extensions

Name simplified and 0 prefixed to data by Peter Munn, Jul 30 2023