A123578 The Kruskal-Macaulay function M_2(n).

0, 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13

Offset: 0

Author

N. J. A. Sloane, Nov 12 2006

Keywords

• nonn
• easy

Comments

Identical to A002024, except for the initial 0.

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).

References

• D. E. Knuth, The Art of Computer Programming, Vol. 4, Fascicle 3, Section 7.2.1.3, Table 3.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• B. M. Abrego, S. Fernandez-Merchant, 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.

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:
A123578 := proc(n) M(n,2) ; end: # R. J. Mathar, Mar 14 2007
a := proc(n) local t, s; t := 1; s := 0;
while t <= n do s := s + 1; t := t + s od; s end:
seq(a(n), n=0..84); # Peter Luschny, Oct 18 2017

Mathematica

lowpol[n_, t_] := Module[{x = Floor[(n*t!)^(1/t)]}, While[Binomial[x, t] <= n, x = x+1]; x-1]; c[n_, t_] := Module[{nresid = n, tresid = t, a = {}, m}, While[nresid > 0, m = lowpol[nresid, tresid]; AppendTo[a, m]; nresid = nresid - Binomial[m, tresid]; tresid = tresid-1]; a]; m[n_, t_] := With[{a = c[n, t]}, Sum[ Binomial[ a[[i]]-1, t-i], {i, 1, Length[a]}]]; a[n_] := m[n, 2]; Table[a[n], {n, 0, 84}] (* Jean-François Alcover, Dec 04 2012, translated from R. J. Mathar's Maple program *)

PARI

A123578(n)=(sqrtint(8*n)+1)\2 \\ M. F. Hasler, Apr 19 2014

Python

from math import isqrt
def A123578(n): return isqrt(n<<3)+1>>1 # Chai Wah Wu, Oct 17 2022