OFFSET
0,3
COMMENTS
Write n (uniquely) as n = C(n_t,t) + C(n_{t-1},t-1) + ... + C(n_v,v) where n_t > n_{t-1} > ... > n_v >= v >= 1. Then M_t(n) = C(n_t-1,t-1) + C(n_{t-1}-1,t-2) + ... + C(n_v-1,v-1).
From Samuel Harkness, Sep 30 2022: (Start)
a(n) is the smallest number of balls needed on the base layer to stack n balls.
All nonrepeating terms other than a(0) occur at tetrahedral numbers + 1 (n = A000292 + 1).
The value of the nonrepeating terms other than a(0) are the Central Polygonal numbers (A000124). (End)
REFERENCES
D. E. Knuth, The Art of Computer Programming, Vol. 4, Fascicle 3, Section 7.2.1.3, Table 3.
LINKS
Samuel Harkness, Table of n, a(n) for n = 0..10000
B. M. Abrego, S. Fernandez-Merchant, and B. Llano, An Inequality for Macaulay Functions, J. Int. Seq. 14 (2011) # 11.7.4.
MAPLE
lowpol := proc(n, t) local x::integer ; x := floor( (n*factorial(t))^(1/t)) ; while binomial(x, t) <= n do x := x+1 ; od ; RETURN(x-1) ; end:
C := proc(n, t) local nresid, tresid, m, a ; nresid := n ; tresid := t ; a := [] ; while nresid > 0 do m := lowpol(nresid, tresid) ; a := [op(a), m] ; nresid := nresid - binomial(m, tresid) ; tresid := tresid-1 ; od ; RETURN(a) ; end:
M := proc(n, t) local a ; a := C(n, t) ; add( binomial(op(i, a)-1, t-i), i=1..nops(a)) ; end:
A123579 := proc(n) M(n, 3) ; end:
for n from 0 to 120 do printf("%d, ", A123579(n)) ; od ; # R. J. Mathar, Mar 14 2007
MATHEMATICA
c = 0; T = {0}; For[r = 1, r <= 7, r++, For[n = 1, n <= r, n++, c++; For[m = 1, m <= n, m++, AppendTo[T, c]]]]; Take[T, 75] (* Samuel Harkness, Sep 30 2022 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Nov 12 2006
STATUS
approved