A244217 -id:A244217 - cp's OEIS frontend

A014418 Representation of n in base of Catalan numbers (a classic greedy version).

0, 1, 10, 11, 20, 100, 101, 110, 111, 120, 200, 201, 210, 211, 1000, 1001, 1010, 1011, 1020, 1100, 1101, 1110, 1111, 1120, 1200, 1201, 1210, 1211, 2000, 2001, 2010, 2011, 2020, 2100, 2101, 2110, 2111, 2120, 2200, 2201, 2210, 2211, 10000

Offset: 0

Olivier Gérard

From Antti Karttunen, Jun 22 2014: (Start)
Also called "Greedy Catalan Base" for short.
Note: unlike A239903, this is a true base system, thus A244158(a(n)) = n holds for all n. See also A244159 for another, "less greedy" Catalan Base number system.
No digits larger than 3 will ever appear, because C(n+1)/C(n) approaches 4 from below, but never reaches it. [Where C(n) is the n-th Catalan number, A000108(n)].
3-digits cannot appear earlier than at the fifth digit-position from the right, the first example being a(126) = 30000.
The last digit is always either 0 or 1. (Cf. the sequences A244222 and A244223 which give the corresponding k for "even" and "odd" representations). No term ends as ...21.
No two "odd" terms (ending with 1) may occur consecutively.
A244217 gives the k for which a(k) starts with the digit 1, while A244216 gives the k for which a(k) starts with the digit 2 or 3.
A000108(n+1) gives the position of numeral where 1 is followed by n zeros.
A014138 gives the positions of repunits.
A197433 gives such k that a(k) = A239903(k). [Actually, such k, that the underlying strings of digits/numbers are same].
For the explanations, see the attached notes.
(End)

			A simple weighted sum of Sum_{k} digit(k)*C(k) [where C(k) = A000108(k), and digit(1) is the rightmost digit] recovers the natural number n (which the given numeral a(n) represents) as follows:
a(11) = 201, and indeed 2*C(3) + 0*C(2) + 1*C(1) = 2*5 + 0*2 + 1*1 = 11.
a(126) = 30000, and indeed, 3*C(5) = 3*42 = 126.

Olivier Gérard (first 1001 terms) & Antti Karttunen, Table of n, a(n) for n = 0..16796
Georg Cantor, Über die einfachen Zahlensysteme, Zeitschrift fur Mathematik und Physik, 14 (1869), 121-128.
Aviezri S. Fraenkel, Systems of numeration, The American Mathematical Monthly, Vol. 92, No. 2 (Feb., 1985), pp. 105-114.
Antti Karttunen, A few notes on A014418

Cf. A014420 (gives the sum of digits), A244221 (same sequence reduced modulo 2, or equally, the last digit of a(n)), A244216, A244217, A244222, A244223, A000108, A007623, A197433, A239903, A244155, A244158, A244320, A244318, A244159 (a variant), A244161 (in base-4), A014417 (analogous sequence for Fibonacci numbers).

Cf. also A033552, A176137, A161227, A161228.

CatalanBaseIntDs[n_] := Module[{m, i, len, dList, currDigit}, i = 1; While[n > CatalanNumber[i], i++]; m = n; len = i; dList = Table[0, {len}]; Do[currDigit = 0; While[m >= CatalanNumber[j], m = m - CatalanNumber[j]; currDigit++]; dList[[len - j + 1]] = currDigit, {j, i, 1, -1}]; If[dList[[1]] == 0, dList = Drop[dList, 1]]; FromDigits@ dList]; Array [CatalanBaseIntDs, 50, 0] (* Robert G. Wilson v, Jul 02 2014 *)

from sympy import catalan
def a244160(n):
    if n==0: return 0
    i=1
    while True:
        if catalan(i)>n: break
        else: i+=1
    return i - 1
def a(n):
    if n==0: return 0
    x=a244160(n)
    return 10**(x - 1) + a(n - catalan(x))
print([a(n) for n in range(51)]) # Indranil Ghosh, Jun 08 2017

From Antti Karttunen, Jun 23 2014: (Start)
a(0) = 0, a(n) = 10^(A244160(n)-1) + a(n-A000108(A244160(n))). [Here A244160 gives the index of the largest Catalan number that still fits into the sum].
a(n) = A007090(A244161(n)).
For all n, A000035(a(n)) = A000035(A244161(n)) = A244221(n).
(End)

Description clarified by Antti Karttunen, Jun 22 2014

A244314 Nonnegative integers m such that the Semigreedy Catalan representation A244159(m) contains at least one zero.

Original entry on oeis.org

0, 2, 5, 6, 7, 14, 15, 16, 17, 18, 19, 20, 21, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157

Offset: 0

Antti Karttunen, Jun 25 2014

nonn

Starting offset is zero because A244159(0) = 0 is a borderline case (either one zero, or no zeros if leading zeros are discarded).
From a(1)=2 onward the positions of zeros in A244233.
After zero consists of successive subsequences containing terms from A000108(k) to (A000108(k)+A014138(k-2)-1) computed from k >= 2 onward, as: [2], [5,6,7], [14 .. 21], [42 .. 63], [132 .. 195], [429 .. 624], [1430 .. 2054], [4862 .. 6916], etc.

Antti Karttunen, Table of n, a(n) for n = 0..9891

Subsequence of A244217.

Cf. A000108, A014138, A014143, A244233, A244159, A244217, A244317, A081291, A081293.

(define (A244314 n) (cond ((zero? n) 0) ((= 1 n) 2) (else (+ -1 n (- (A000108 (+ 1 (A244317 n))) (A014143 (- (A244317 n) 2)))))))

a(0) = 0, a(1) = 2, and for n >= 2, a(n) = n + A000108(1+A244317(n)) - A014143(A244317(n)-2) - 1.

A244215 a(0)=0, and for n>=1, if n is the k-th Catalan number (A000108(k)), a(n) = k, otherwise the difference of the indices of the two largest Catalan numbers whose sum is less than or equal to n: a(n) = A244160(n) - A244160(n-A081290(n)).

Original entry on oeis.org

0, 1, 2, 1, 0, 3, 2, 1, 1, 1, 0, 0, 0, 0, 4, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 4, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6

Offset: 0

Antti Karttunen, Jun 23 2014

nonn

After a(0), if the leftmost digit in the Greedy Catalan Base representation of n [= A014418(n)] is larger than 1, then a(n) = 0, otherwise one more than the distance to the next nonzero digit to the right, or to the end of the numeral, if no more nonzero digits are present (i.e., if n is one of the Catalan numbers).

When searching for the two largest Catalan numbers whose sum is less than or equal to n, we first maximize the larger of those two numbers, which is A081290(n) = A000108(A244160(n)), after which we will find the next largest Catalan number that still "fits into" n. - Antti Karttunen, Mar 21 2015

Antti Karttunen, Table of n, a(n) for n = 0..4862

A244216 gives the positions of zeros, A244217 the positions of nonzero terms.

Cf. A000108, A014418, A081290, A244160, A163510.

(define (A244215 n) (- (A244160 n) (A244160 (- n (A081290 n)))))

a(n) = A244160(n) - A244160(n-A081290(n)).

For all n >= 1, a(A000108(n)) = n.

Name improved by Antti Karttunen, Mar 21 2015

A244216 Numbers n that are at least twice the size the largest Catalan number less than or equal to n.

Original entry on oeis.org

4, 10, 11, 12, 13, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 264

Offset: 1

Antti Karttunen, Jun 23 2014

nonn

Equally: Numbers k such that if m is the largest Catalan number <= k [= A081290(k)], then k >= 2*m.

Numbers k such that the Greedy Catalan Base representation of k (cf. A014418) starts with digit 2 or 3.

			4 is present, as the largest Catalan number (A000108(k), here C(k)) less than or equal to 4 is C(2) = 2, and 4 >= 2*2.
5 is not present, as the largest Catalan number <= 5 is C(3) = 5, and 5 < 2*5.
10 is present, as the largest Catalan number <= 10 is C(3) = 5, and 10 >= 2*5.
11 is present, as the largest Catalan number <= 11 is C(3) = 5, and 11 >= 2*5.
27 is not present, as the largest Catalan number <= 27 is C(4) = 14, and 27 < 2*14.
28 is present, as the largest Catalan number <= 28 is C(4) = 14, and 28 >= 2*14.

Antti Karttunen, Table of n, a(n) for n = 1..9878

Positions of zeros in A244215. Complement of A244217.

Cf. A000108, A014418, A081290.

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A014418 Representation of n in base of Catalan numbers (a classic greedy version).

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A244314 Nonnegative integers m such that the Semigreedy Catalan representation A244159(m) contains at least one zero.

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A244215 a(0)=0, and for n>=1, if n is the k-th Catalan number (A000108(k)), a(n) = k, otherwise the difference of the indices of the two largest Catalan numbers whose sum is less than or equal to n: a(n) = A244160(n) - A244160(n-A081290(n)).

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A244216 Numbers n that are at least twice the size the largest Catalan number less than or equal to n.

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