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 A009996 Numbers with digits in nonincreasing order. 30
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 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 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 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. EXAMPLE 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. MATHEMATICA Select[Range[0, 211], GreaterEqual@@IntegerDigits[#]&] (* Ray Chandler, Oct 25 2011 *) PROG (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 CROSSREFS Differs from A032873 and A032907. Cf. A064222, A152054. Sequence in context: A084383 A032873 A072543 * A032907 A130576 A334145 Adjacent sequences: A009993 A009994 A009995 * A009997 A009998 A009999 KEYWORD nonn,base,look AUTHOR N. J. A. Sloane EXTENSIONS Corrected by Rick L. Shepherd, Jun 06 2002 STATUS approved

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