A009996 Numbers with digits in nonincreasing order.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 20, 21, 22, 30, 31, 32, 33, 40, 41, 42, 43, 44, 50, 51, 52, 53, 54, 55, 60, 61, 62, 63, 64, 65, 66, 70, 71, 72, 73, 74, 75, 76, 77, 80, 81, 82, 83, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 110, 111, 200, 210, 211

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• base
• look

Comments

Base-10 representation Sum_{i=0..m} d(i)*10^i has d(m) >= d(m-1) >= ... >= d(1) >= d(0).

These numbers might be called "Nialpdromes".

A004186(a(n)) = a(n). - Reinhard Zumkeller, Oct 31 2007

Examples

			As 10000 = C(10+6,10) - 6 + C(7+6,1+6) + C(5+5,1+5) + C(4+4,1+4) + C(3+3,1+3) + C(1+2,1+2) + C(0+1,1+1), C(0+0,1+0), a(10000) = 7543100.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
• D. Applegate, M. LeBrun, and N. J. A. Sloane, Dismal Arithmetic, J. Int. Seq. 14 (2011) # 11.9.8.
• David A. Corneth, Table of n, a(n) for n = 1..20000, Jun 03 2014
• Eric Weisstein's World of Mathematics, Digit

Crossrefs

Differs from A032873 and A032907.

Cf. A064222, A152054.

Programs

Mathematica

Select[Range[0,211], GreaterEqual@@IntegerDigits[#]&] (* Ray Chandler, Oct 25 2011 *)

PARI

is(n)=my(d=digits(n)); for(i=2,#d,if(d[i]>d[i-1],return(0))); 1 \\ Charles R Greathouse IV, Jan 02 2014

PARI

\\ This program is optimized for fast calculation of a(n) for large n.
a(n)={my(q,m=10,i,r=0);n--;while(binomial(m+1,10)<=n+m-9,m++);n-=binomial(m,10);n+=m-9;q=m-9;i=q;while(n>0,m=i;while(binomial(m+1,i)<=n,m++);r=10*r+m+1-i;n-=binomial(m,i);i--;);z=q-#digits(r);r*=10^z;r} \\ David A. Corneth, Jun 01 2014

PARI

\\recursive--feed an element a(n)>0 and it gives a(n+1).
nxt(n)={my(r,d=digits(n),y,t); if(d[#d]!=9,y=1; while(y-#d-1&&d[y]==9,y++); t=#d; forstep(i=t,y+1,-1,if(d[i-1]!=d[i],t=i-1;break)); if(t!=#d,d[t+1]++; for(i=t+2,#d,d[i]=0),d[y]++; for(i=y+1,#d,d[i]=0));r=d ,d=vector(#d+1); d[1]=1;for(i=2,#d,d[i]=0); r=d); sum(i=1,#r,10^(#r-i)*r[i])} \\ David A. Corneth, Jun 01 2014

Python

from itertools import count, islice, combinations_with_replacement as mc
def agen(): # generator of terms
    yield 0
    for d in count(1):
        ni = (int("".join(m)) for m in mc("9876543210", d) if m[0]!="0")
        yield from sorted(ni)
print(list(islice(agen(), 70))) # Michael S. Branicky, Jun 24 2022

Formula

Binomial(n+k,k) = (n+k)!/(n!*k!). d(i) is the i-th digit of a(n). q is the number of digits of a(n). Find the highest m such that C(10 + m, 10) - m + 1 <= n. a(n) has m+1 digits. Set n = n - C(10+m,10). Find the highest d(m+1), then d(m), then ..., then d(1) each iteration such that C(d(m+1)+m+1,1+m+1)<=n. Then set n = n-C(d(m+1)+m+1,m+2). If n = 0 then stop. All remaining digits are 0.

Extensions

Corrected by Rick L. Shepherd, Jun 06 2002