A104395 Sums of 5 distinct positive pentatope numbers (A000332).

126, 182, 217, 237, 247, 251, 266, 301, 321, 331, 335, 357, 377, 386, 387, 391, 412, 421, 422, 426, 441, 442, 446, 451, 455, 456, 477, 497, 507, 511, 532, 542, 546, 551, 561, 562, 566, 576, 581, 586, 591, 595, 606, 616, 620, 626, 630, 642, 646, 650

Offset: 1

Author

Jonathan Vos Post, Mar 05 2005

Keywords

• easy
• nonn

Comments

Hyun Kwang Kim asserts that every positive integer can be represented as the sum of no more than 8 pentatope numbers; but in this sequence we are only concerned with sums of nonzero distinct pentatope numbers.

References

• Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 55-57, 1996.

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Hyun Kwang Kim, On Regular Polytope Numbers, Proc. Amer. Math. Soc., 131 (2003), 65-75.
• Eric Weisstein's World of Mathematics, Pentatope Number.

Crossrefs

Cf. A000332, A100009, A102857, A104392, A104393, A104394.

Programs

Maple

N:= 1000: # for terms <= N
ptop:= n -> n*(n+1)*(n+2)*(n+3)/24:
P:= 1:
for i from 1 while ptop(i) < N do
  P:= P * (1 + x*y^ptop(i))
od:
sort(map(degree,convert(convert(series(coeff(P,x,5),y,N+1),polynom),list)));
# Robert Israel, Nov 20 2023

Formula

a(n) = Ptop(g) + Ptop(h) + Ptop(i) + Ptop(j) + Ptop(k) for some positive g=/=h=/=i=/=j=/=k and Ptop(n) = binomial(n+3,4).

Extensions

Extended by Ray Chandler, Mar 05 2005