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A052217

Numbers whose sum of digits is 3.

3, 12, 21, 30, 102, 111, 120, 201, 210, 300, 1002, 1011, 1020, 1101, 1110, 1200, 2001, 2010, 2100, 3000, 10002, 10011, 10020, 10101, 10110, 10200, 11001, 11010, 11100, 12000, 20001, 20010, 20100, 21000, 30000, 100002, 100011, 100020, 100101

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OFFSET

1,1

COMMENTS

From Joshua S.M. Weiner, Oct 19 2012: (Start)

Sequence is a representation of the "energy states" of "multiplex" notation of 3 quantum of objects in a juggling pattern.

0 = an empty site, or empty hand. 1 = one object resides in the site. 2 = two objects reside in the site. 3 = three objects reside in the site. (See A038447.) (End)

A007953(a(n)) = 3; number of repdigits = #{3,111} = A242627(3) = 2. - Reinhard Zumkeller, Jul 17 2014

Can be seen as a table whose n-th row holds the n-digit terms {10^(n-1) + 10^m + 10^k, 0 <= k <= m < n}, n >= 1. Row lengths are then (1, 3, 6, 10, ...) = n*(n+1)/2 = A000217(n). The first and the n last terms of row n are 10^(n-1) + 2 resp. 2*10^(n-1) + 10^k, 0 <= k < n. - M. F. Hasler, Feb 19 2020

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (terms 1..84 from Vincenzo Librandi, terms 85..1140 from T. D. Noe)

FORMULA

T(n,k) = 10^(n-1) + 10^A003056(k) + 10^A002262(k) when read as a table with row lengths n*(n+1)/2, n >= 1, 0 <= k < n*(n+1)/2. - M. F. Hasler, Feb 19 2020

a(n) = 10^A056556(n-1) + 10^A056557(n-1) + 10^A056558(n-1). - Kevin Ryde, Apr 17 2021

MATHEMATICA

Union[FromDigits/@Select[Flatten[Table[Tuples[Range[0, 3], n], {n, 6}], 1], Total[#]==3&]] (* Harvey P. Dale, Oct 20 2012 *)

Select[Range[10^6], Total[IntegerDigits[#]] == 3 &] (* Vincenzo Librandi, Mar 07 2013 *)

Union[Flatten[Table[FromDigits /@ Permutations[PadRight[s, 18]], {s, IntegerPartitions[3]}]]] (* T. D. Noe, Mar 08 2013 *)

PROG

(Magma) [n: n in [1..100101] | &+Intseq(n) eq 3 ]; // Vincenzo Librandi, Mar 07 2013

(Haskell)

a052217 n = a052217_list !! (n-1)

a052217_list = filter ((== 3) . a007953) [0..]

-- Reinhard Zumkeller, Jul 17 2014

(PARI) isok(n) = sumdigits(n) == 3; \\ Michel Marcus, Dec 28 2015

(PARI) apply( {A052217_row(n, s, t=-1)=vector(n*(n+1)\2, k, t++>s&&t=!s++; 10^(n-1)+10^s+10^t)}, [1..5]) \\ M. F. Hasler, Feb 19 2020

(Python)

from itertools import count, islice

def agen(): yield from (10**i + 10**j + 10**k for i in count(0) for j in range(i+1) for k in range(j+1))

print(list(islice(agen(), 40))) # Michael S. Branicky, May 14 2022

(Python)

from math import comb, isqrt

from sympy import integer_nthroot

def A052217(n): return 10**((m:=integer_nthroot(6*n, 3)[0])-(a:=n<=comb(m+2, 3)))+10**((k:=isqrt(b:=(c:=n-comb(m-a+2, 3))<<1))-((b<<2)<=(k<<2)*(k+1)+1))+10**(c-1-comb(k+(b>k*(k+1)), 2)) # Chai Wah Wu, Dec 11 2024

CROSSREFS

Cf. A069521 to A069530, A069532 to A069537.

Cf. A007953, A218043 (subsequence).

Row n=3 of A245062.