A123580 -id:A123580 - cp's OEIS frontend

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

N. J. A. Sloane, Nov 12 2006

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).

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

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

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

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

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 *)

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

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

A123579 The Kruskal-Macaulay function M_3(n).

Original entry on oeis.org

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

N. J. A. Sloane, Nov 12 2006

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)

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

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.

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

Cf. A000124, A000292.

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

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 *)

A123731 The Kruskal-Macaulay function M_5(n).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 5, 6, 7, 8, 9, 9, 10, 11, 12, 12, 13, 14, 14, 15, 15, 15, 16, 17, 18, 19, 19, 20, 21, 22, 22, 23, 24, 24, 25, 25, 25, 26, 27, 28, 28, 29, 30, 30, 31, 31, 31, 32, 33, 33, 34, 34, 34, 35, 35, 35, 35, 36, 37, 38, 39, 39, 40, 41, 42, 42, 43, 44, 44, 45, 45, 45, 46

Offset: 0

N. J. A. Sloane, Nov 12 2006

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).

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

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

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:
A123731 := proc(n) M(n,5); end:
for n from 0 to 120 do printf("%d, ",A123731(n)); od; # R. J. Mathar, Mar 14 2007

lowpol[n_, t_] := Module[{x = Floor[(n*t!)^(1/t)]}, While[Binomial[x, t] <= n, x = x + 1]; x - 1];
c[n_, t_] := Module[{n0 = n, t0 = t, a = {}, m}, While[n0 > 0, m = lowpol[n0, t0]; AppendTo[a, m]; n0 = n0 - Binomial[m, t0]; t0 = t0 - 1]; a];
M[n_, t_] := With[{a = c[n, t]}, Sum[Binomial[a[[i]] - 1, t - i], {i, 1, Length[a]}]];
A123731[n_] := M[n, 5];
Table[A123731[n], {n, 0, 72}] (* Jean-François Alcover, Mar 30 2023, after R. J. Mathar *)

More terms from R. J. Mathar, Mar 14 2007

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A123578 The Kruskal-Macaulay function M_2(n).

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A123579 The Kruskal-Macaulay function M_3(n).

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A123731 The Kruskal-Macaulay function M_5(n).

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