cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A123575 The Kruskal-Macaulay function L_3(n).

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 1, 2, 2, 3, 5, 5, 5, 6, 6, 7, 9, 9, 10, 12, 15, 15, 15, 16, 16, 17, 19, 19, 20, 22, 25, 25, 26, 28, 31, 35, 35, 35, 36, 36, 37, 39, 39, 40, 42, 45, 45, 46, 48, 51, 55, 55, 56, 58, 61, 65, 70, 70, 70, 71, 71, 72, 74, 74, 75, 77, 80, 80, 81, 83, 86, 90, 90, 91, 93
Offset: 0

Views

Author

N. J. A. Sloane, Nov 12 2006

Keywords

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 L_t(n) = C(n_t,t+1) + C(n_{t-1},t) + ... + C(n_v,v+1).

References

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

Crossrefs

For L_i(n), i=1, 2, 3, 4, 5 see A000217, A111138, A123575, A123576, A123577.
Essentially partial sums of A056558.

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: L := proc(n,t) local a ; a := C(n,t) ; #add( binomial(op(i,a),t+i),i=1..nops(a)) ; add( binomial(op(i,a),t+2-i),i=1..nops(a)) ; end: A123575 := proc(n) L(n,3) ; end: for n from 0 to 80 do printf("%d, ",A123575(n)) ; od ; # R. J. Mathar, May 18 2007
  • Mathematica
    lowpol[n_, t_] := Module[{x = Floor[(n t!)^(1/t)]}, While[Binomial[x, t] <= n, x++] ; x - 1];
c[n_, t_] := Module[{n0 = n, t0 = t, m, a = {}}, While[n0 > 0, m = lowpol[n0, t0]; AppendTo[a, m]; n0 -= Binomial[m, t0]; t0--]; a];
L[n_, t_] := Module[{a = c[n, t]}, Sum[Binomial[a[[i]], t + 2 - i], {i, 1, Length[a]}]];
a[n_] := L[n, 3];
a /@ Range[0, 80] (* Jean-François Alcover, Mar 29 2020, after R. J. Mathar *)

Extensions

More terms from R. J. Mathar, May 18 2007

A123576 The Kruskal-Macaulay function L_4(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 3, 3, 4, 6, 6, 6, 6, 7, 7, 7, 8, 8, 9, 11, 11, 11, 12, 12, 13, 15, 15, 16, 18, 21, 21, 21, 21, 22, 22, 22, 23, 23, 24, 26, 26, 26, 27, 27, 28, 30, 30, 31, 33, 36, 36, 36, 37, 37, 38, 40, 40, 41, 43, 46, 46, 47, 49, 52, 56, 56, 56, 56, 57, 57, 57, 58
Offset: 0

Views

Author

N. J. A. Sloane, Nov 12 2006

Keywords

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 L_t(n) = C(n_t,t+1) + C(n_{t-1},t) + ... + C(n_v,v+1).

References

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

Crossrefs

For L_i(n), i=1, 2, 3, 4, 5 see A000217, A111138, A123575, A123576, A123577.

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: L := proc(n,t) local a ; a := C(n,t) ; add( binomial(op(i,a),t+2-i),i=1..nops(a)) ; end: A123576 := proc(n) L(n,4) ; end: for n from 0 to 80 do printf("%d, ",A123576(n)) ; od ; # R. J. Mathar, May 18 2007
  • Mathematica
    (+ The function L(n,t) is defined in A123575 *)
a[n_] := L[n, 4];
a /@ Range[0, 80] (* Jean-François Alcover, Mar 29 2020 *)

Extensions

More terms from R. J. Mathar, May 18 2007

A123577 The Kruskal-Macaulay function L_5(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 4, 4, 5, 7, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 10, 10, 11, 13, 13, 13, 13, 14, 14, 14, 15, 15, 16, 18, 18, 18, 19, 19, 20, 22, 22, 23, 25, 28, 28, 28, 28, 28, 29, 29, 29, 29, 30, 30, 30, 31, 31, 32, 34, 34, 34, 34, 35, 35, 35, 36, 36
Offset: 0

Views

Author

N. J. A. Sloane, Nov 12 2006

Keywords

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 L_t(n) = C(n_t,t+1) + C(n_{t-1},t) + ... + C(n_v,v+1).

References

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

Crossrefs

For L_i(n), i=1, 2, 3, 4, 5 see A000217, A111138, A123575, A123576, A123577.

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: L := proc(n,t) local a ; a := C(n,t) ; add( binomial(op(i,a),t+2-i),i=1..nops(a)) ; end: A123577 := proc(n) L(n,5) ; end: for n from 0 to 80 do printf("%d, ",A123577(n)) ; od ; # R. J. Mathar, May 18 2007
  • Mathematica
    (* The function L(n,t) is defined in A123575 *)
a[n_] := L[n, 5];
a /@ Range[0, 80] (* Jean-François Alcover, Mar 29 2020 *)

Extensions

More terms from R. J. Mathar, May 18 2007
Showing 1-3 of 3 results.

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