A123579 The Kruskal-Macaulay function M_3(n).

0, 1, 2, 3, 3, 4, 5, 5, 6, 6, 6, 7, 8, 8, 9, 9, 9, 10, 10, 10, 10, 11, 12, 12, 13, 13, 13, 14, 14, 14, 14, 15, 15, 15, 15, 15, 16, 17, 17, 18, 18, 18, 19, 19, 19, 19, 20, 20, 20, 20, 20, 21, 21, 21, 21, 21, 21, 22, 23, 23, 24, 24, 24, 25, 25, 25, 25, 26, 26, 26, 26, 26, 27, 27, 27

Offset: 0

Author

N. J. A. Sloane, Nov 12 2006

Keywords

• nonn
• easy

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.

Crossrefs

For M_i(n), i=1, 2, 3, 4, 5 see A000127, A123578, A123579, A123580, A123731.

Cf. A000124, A000292.

Programs

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 *)