OFFSET
3,1
COMMENTS
From David A. Corneth, Nov 06 2024: (Start)
a(n) <= (n^4 - 2*n^3 + 38*n^2 - 85*n + 72)/6 for n >= 5. Proof:
A polygonal number is of the form P(m, n) = m/2 * ((n - 2) * m - n + 4).
We have P(n - 5, n) + P(n - 4, n) + P(n, n) = P(n - 6, n) + P(n - 2, n) + P(n - 1, n) = (3*n^3 - 18*n^2 + 21*n) / 2.
This lets us find the upper bound on a(n) by making two lists from 1 through n + 3. From one of them we remove n-2, n-1 and n + 3 and from the other we remove n-3, n+1 and n+2. The sum for remaining polygonal numbers is the same giving an upper bound on a(n) which turns out to be (n^4 - 2*n^3 + 38*n^2 - 85*n + 72)/6 (End)
LINKS
Eric Weisstein's World of Mathematics, Polygonal Number
FORMULA
From David A. Corneth, Nov 06 2024: (Start)
a(n) >= A006484(n).
Conjecture: a(n) = (n^4 - 2*n^3 + 38*n^2 - 85*n + 72)/6 for n >= 5. (End)
Conjectured g.f.: x^3*(19 - 5*x - 98*x^2 + 199*x^3 - 171*x^4 + 72*x^5 - 12*x^6) / (1 - x)^5.
EXAMPLE
a(3) = 19 = 1 + 3 + 15 = 3 + 6 + 10.
a(4) = 90 = 1^2 + 2^2 + 6^2 + 7^2 = 1^2 + 3^2 + 4^2 + 8^2.
CROSSREFS
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Nov 05 2024
EXTENSIONS
a(12)-a(36) from Michael S. Branicky, Nov 06 2024
More terms from David A. Corneth, Nov 10 2024
STATUS
approved