A378245 Numbers that are both k-gonal and (k+1)-gonal for some k >= 3.

1, 36, 1225, 9801, 40755, 41616, 121771, 297045, 631125, 1212751, 1413721, 2158695, 3617601, 5773825, 8851275, 13117251, 18886285, 26523981, 36450855, 48024900, 49146175, 65151801, 85076025, 94109401, 109597411, 139468635, 175520325, 218664901, 269900415, 330314391

Offset

1,2

Links

Table of n, a(n) for n=1..30.

Example

a(2) = 36 is both the 8th triangular and the 6th square number.
a(3) = 1225 is both the 49th triangular and the 35th square number.
a(5) = 40755 is both the 165th pentagonal number and the 143th hexagonal number.

Prog

(PARI) upto(limit) = my(terms=List(1)); for(k=3, oo, my(found=0); for(n=2, oo, my(a = (2*n - 1)^2, b = (4*n*(3*n - 5) + 6), c = (8*(n-1)^2 + 1), s = (a*k^2 - b*k + c), v = n * (n*k - k - 2*n + 4) / 2); if(issquare(s), my(t = sqrtint(s) + k - 3); if(t % (2*(k-1)) == 0, listput(terms, v); found += 1)); if(v >= limit, break)); if(found == 0, break)); Vec(vecsort(terms)); \\ Daniel Suteu, Dec 08 2024

Crossrefs

Cf. A057145, A342300.

Cf. A001110, A036353, A046180, A048903, A048906, A048924 and A203627 (subsequences).

The subdiagonal of A189216 is also a subsequence.

Sequence in context: A215858 A103278 A004294 * A075760 A113938 A001110

Adjacent sequences: A378242 A378243 A378244 * A378246 A378247 A378248

Keyword

nonn

Author

Kelvin Voskuijl, Nov 20 2024

Extensions

a(12)-a(30) from Pontus von Brömssen, Dec 07 2024

Status

approved