A039596 Numbers that are simultaneously triangular and square pyramidal.

0, 1, 55, 91, 208335

Offset: 1

Author

Felice Russo

Keywords

• fini
• nonn
• full

Comments

Equivalent to 0^2 + 1^2 + 2^2 + 3^2 + ... + r^2 = 0 + 1 + 2 + 3 + ... + s = n for some r and s.

Examples

			1^2 + 2^2 + 3^2 + 4^2 + 5^2 = 1 + 2 + 3 + ... + 10 = 55, so 55 is in the sequence.
		

References

• Joe Roberts, Lure of the Integers, page 245 (entry for 645).
• David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, p. 108.

Links

• R. Finkelstein and H. London, On triangular numbers which are sums of consecutive squares, J. Number Theory 4 (1972), 455-462.
• M. Gardner, Letter to N. J. A. Sloane, circa Aug 11 1980, concerning A001110, A027568, A039596, etc.
• H. E. Thomas Jr., Problem 5634, Amer. Math. Monthly, 75 (1968), p. 1018.

Crossrefs

Intersection of A000217 and A000330.

Cf. A053611, A053612.

Programs

Maple

q:= n-> issqr(8*n+1):
select(q, [sum(j^2, j=1..n)$n=0..100])[];  # Alois P. Heinz, Oct 17 2024

Extensions

Additional comments from Jud McCranie, Mar 19 2000

Zero inserted by Daniel Mondot, Sep 07 2023