A194847 Write n = C(i,3)+C(j,2)+C(k,1) with i>j>k>=0; sequence gives i values.

2, 3, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10

Offset

0,1

Comments

Each n >= 0 has a unique representation as n = C(i,3)+C(j,2)+C(k,1) with i>j>k>=0. This is the combinatorial number system of degree t = 3, where we get [A194847, A194848, A056558]. For degree t = 2 we get [A002024, A002262] and A138036.

References

D. E. Knuth, The Art of Computer Programming, vol. 4A, Combinatorial Algorithms, Section 7.2.1.3, Eq. (20), p. 360.

Links

Table of n, a(n) for n=0..123.

Formula

Equals A056556(n) + 2.

Example

The i,j,k coordinates for n equal to 0 through 10 are:
0, [2, 1, 0]
1, [3, 1, 0]
2, [3, 2, 0]
3, [3, 2, 1]
4, [4, 1, 0]
5, [4, 2, 0]
6, [4, 2, 1]
7, [4, 3, 0]
8, [4, 3, 1]
9, [4, 3, 2]
10, [5, 1, 0]

Maple

# Given x and a list a, returns smallest i such that x >= a[i].
whereinlist:=proc(x, a)  local i:
if whattype(a) <> list then ERROR(`a not a list`); fi:
for i from 1 to nops(a) do if x < a[i] then break; fi; od:
RETURN(i-1); end:
t3:=[seq(binomial(n, 3), n=0..50)];
t2:=[seq(binomial(n, 2), n=0..50)];
t1:=[seq(binomial(n, 1), n=0..50)];
for n from 0 to 200 do
i3:=whereinlist(n, t3);
i2:=whereinlist(n-t3[i3], t2);
i1:=whereinlist(n-t3[i3]-t2[i2], t1);
L[n]:=[i3-1, i2-1, i1-1];
od:
[seq(L[n][1], n=0..200)];

Prog

(Python)
from math import comb
from sympy import integer_nthroot
def A194847(n): return (m:=integer_nthroot(6*(n+1), 3)[0])+(n>=comb(m+2, 3))+1 # Chai Wah Wu, Nov 05 2024

Crossrefs

The [i,j,k] values are [A194847, A194848, A056558], or equivalently [A056556+2, A056557+1, A056558]. See A194849 for the union list of triples.

Cf. also A002024, A002262, A138036.

Sequence in context: A130256 A335741 A103586 * A262070 A117806 A351115

Adjacent sequences: A194844 A194845 A194846 * A194848 A194849 A194850

Keyword

nonn

Author

N. J. A. Sloane, Sep 03 2011

Status

approved