A244155 -id:A244155 - 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

A239903 List of Restricted-Growth Strings a_{k-1}a_{k-2}...a_{2}a_{1}, with k=2 and a_1 in {0,1} or k>2, a_{k-1}=1 and a_{j+1}>=1+a_j, for k-1>j>0.

Original entry on oeis.org

0, 1, 10, 11, 12, 100, 101, 110, 111, 112, 120, 121, 122, 123, 1000, 1001, 1010, 1011, 1012, 1100, 1101, 1110, 1111, 1112, 1120, 1121, 1122, 1123, 1200, 1201, 1210, 1211, 1212, 1220, 1221, 1222, 1223, 1230, 1231, 1232, 1233, 1234, 10000, 10001, 10010, 10011

Offset: 0

N. J. A. Sloane, Apr 06 2014

We write the nonnegative integers as restricted growth strings (so called by J. Arndt in his book fxtbook.pdf, p. 325) in such a way that the Catalan numbers (cf. A000108) are expressed: 1=1, 10=2, 100=5, 1000=14, etc., 10...0 (with k zeros) = the k-th Catalan number. Once the entries of a restricted-growth string grow above 9, one would need commas or parentheses, say, to separate those entries. See Dejter (2017) for the precise definition.
In the paper "A system of numeration for middle-levels", restricted growth strings (RGSs) are defined as sequences that begin with either 0 or 1, with each successive number to the right being at least zero and at most one greater than its immediate left neighbor. Moreover, apart from case a(0), the RGSs are finite integer sequences of restricted growth which always start with 1 as their first element b_1 in position 1, and from then on, each successive element b_{i+1} in the sequence is restricted to be in range [0,(b_i)+1].
This sequence gives all such finite sequences in size-wise and lexicographic order, represented as decimal numbers by concatenating the integers of such finite sequences (e.g., from [1,2,0,1] we get 1201). The 58784th such sequence is [1, 2, 3, 4, 5, 6, 7, 8, 9, 9], thus a(58784) = 1234567899, after which comes the first RGS, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], where an element larger than 9 is present, which means that the decimal system employed here is unambiguous only up to n=58784. Note that 58785 = A000108(11)-1.
Also, if one considers Stanley's interpretation (u) of Catalan numbers, "sequences of a_1, a_2, ..., a_n of integers such that a_1 = 0 and 0 <= a_{i+1} <= a_{i} + 1" (e.g., 000, 001, 010, 011, 012 for C_3), and discards their initial zero, then one has a bijective correspondence with Dejter's RGSs of one element shorter length, which in turn are in bijective correspondence with the first C_n terms of this sequence (by discarding any leading zeros), from a(0) to a(C_n - 1). From this follows that the k-th Catalan number, A000108(k) (k>0), is represented in this system as 1 followed by k-1 zeros: a(1)=1, a(2)=10, a(5)=100, a(14)=1000, etc., and also that there exist exactly A000245(k) RGSs of length k.
Note how this differs from other number representations utilizing Catalan numbers, A014418 and A244159, in that while the latter are base-systems, where a simple weighted Sum_{k} digit(k)*C(k) recovers the natural number n (which the n-th numeral of such system represents), in contrast here it is the sum of appropriate terms in Catalan's Triangle (A009766, A030237), obtained by unranking a unique instance of a certain combinatorial structure (one of the Catalan interpretations), that gives a correspondence with a unique natural number. (Cf. also A014486.)
This sequence differs from "Semigreedy Catalan Representation", A244159, for the first time at n=10, where a(10) = 120, while A244159(10) = 121. That is also the first position where A244158(a(n)) <> n.
Please see Dejter's preprint for a more formal mathematical definition and how this number system is applied in relation to Havel's Conjecture on the existence of Hamiltonian cycles in the middle-levels graphs.
a(n) is given by the concatenation (with leading zeros removed) of the terms of row n + 23714 of A370222. - Paolo Xausa, Feb 17 2024

			Catalan's Triangle T(row,col) = A009766 begins with row n=0 and 0<=col<=n as:
  Row 0: 1
  Row 1: 1, 1
  Row 2: 1, 2,  2
  Row 3: 1, 3,  5,  5
  Row 4: 1, 4,  9, 14, 14
  Row 5: 1, 5, 14, 28, 42,  42
  Row 6: 1, 6, 20, 48, 90, 132, 132
  (the leftmost diagonal of 1s is "column 0").
  ...
For example, for n=38, we find that A081290(38)=14, which occurs on row A081288(n)-1 = 4, in columns A081288(n)-1 and A081288(n)-2, i.e., as T(4,4) and T(4,3). Thus we subtract 38-14 to get 24, and we see that the next term downward on the same diagonal, 28, is too large to accommodate into the same sum, so we go one diagonal up, starting now from T(3,2) = 5. This fits in, so we now have 24 - 5 = 19, and also the next term on the same diagonal, T(4,2) = 9, fits in, so we now have 19-9 = 10. The next term on the same diagonal, T(5,2) = 14, would not fit in anymore, so we rewind ourselves back to penultimate column, but one step up from where we started on this diagonal, so T(2,1) = 2, which fits in, 10 - 2 = 8, also the next one T(3,1) = 3, 8 - 3 = 5, and the next one T(4,1) = 4, 5 - 4 = 1, after which comes T(5,1) = 5 > 1, thus we jump to T(1,0) = 1, 1-1 = 0, and T(2,0)=1 would not fit anymore, thus next time the row would be zero, and the algorithm is ready with 1 (14), 2 (5+9), 3 (2+3+4) and 1 (1) terms collected, whose total sum 14+5+9+2+3+4+1 = 38, thus a(38) = 1231.
For n=20, the same algorithm results in 1 (14), 1 (5), 0 (not even the first tentative term T(2,1) = 2 from the column 1 would fit, so it is skipped), and from one row higher we get the needed 1 (1), so the total sum of these is 14+5+0+1 = 20, thus a(20) = 1101.

D. E. Knuth, The Art of Computer Programming, Vol. 2: Seminumerical Algorithms, third edition, Addison-Wesley, 1977, p. 192.
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999, Exercise 19, interpretation (u).

Antti Karttunen, Table of n, a(n) for n = 0..16796
Joerg Arndt, Matters Computational: Ideas, Algorithm, Source Code, Springer, 2011; freely downloadable here: Fxtbook.
Georg Cantor, Über die einfachen Zahlensysteme, Zeitschrift fur Mathematik und Physik, 14 (1869), 121-128.
Italo J. Dejter, A system of numeration for middle levels, arXiv:1012.0995 [math.CO], (2014).
Italo J. Dejter, The role of restricted growth strings in the two middle levels of the Boolean lattice B_(2k+1), University of Puerto Rico, 2018.
Italo J. Dejter, Reinterpreting Mütze's theorem via natural enumeration of ordered rooted trees, arXiv:1911.02100 [math.CO], (2019).
Italo J. Dejter, A numeral system for the middle-levels graphs, Elec. J. Graph Theory and Applications (2021) Vol. 9, No. 1, 137-156. See p. 138.
Italo J. Dejter, Edge-supplementary arc factorizations of odd graphs, uniform 2-factors and Hamilton cycles, arXiv:2203.05326 [math.CO], 2022.
Italo J. Dejter, Integer sequence of Dyck-path single changes, arXiv:2209.11122 [math.CO], 2022.
Italo J. Dejter, A Dyck-word tree that controls all odd graphs, arXiv:2306.14249 [math.CO], 2023.
Aviezri S. Fraenkel, The Use and Usefulness of Numeration Systems, Inf. Comput., 81(1), (1989), 46-61.
Aviezri S. Fraenkel, Systems of numeration, IEEE Symposium on Computer Arithmetic, (1983), 37-42.
Aviezri S. Fraenkel, Systems of numeration, The American Mathematical Monthly, Vol. 92, No. 2 (Feb., 1985), pp. 105-114.
Antti Karttunen, Catalan ranking and unranking functions, OEIS Wiki.
R. P. Stanley, Exercises on Catalan and Related Numbers, interpretation (u) of Catalan numbers, on page 224 (page 4/27 in the November 1 2000 version of the PDF-file).

Cf. A000108 (Catalan numbers), A000245 (their first differences), A009766 (Catalan's triangle), A236855 (the sum of elements in k-th RGS), A236859 (for n>=1, gives the length of the initial ascent 123... in term a(n)), A244159 (different kinds of Catalan number systems).
Other Catalan combinatorial structures represented as integer sequences: A014486/A063171: Dyck words, parenthesizations, etc., A071156/A071158: Similar restricted words encoded with help of A007623 (Integers written in factorial base), A071153/A079436 (Łukasiewicz words).
Cf. A007001, A007623, A014418, A071154, A071159, A081288, A081290, A081291, A082853, A126307, A244155, A244156, A244157, A244158, A370222.

function CatalanNumerals(z)
    z == 0 && return 0
    f(n) = factorial(n)
    t(j, k) = div(f(k+j)*(k-j+1), f(j)*f(k+1))
    k, i = 2, 0
    while z >= t(i, i + 1) i += 1 end
    dig = fill(0, i); dig[1] = 1
    x = z - t(i - 1, i)
    m = i - 1
    while x > 0
        w, s, p = 0, 0, 0
        while w <= x
            p = w
            w += t(m - 1, m + s)
            s += 1
        end
        dig[k] = s - 1
        m -= 1; k += 1; x -= p
    end
    s = ""; for d in dig s *= string(d) end
    parse(Int, s)
end
[CatalanNumerals(n) for n in 0:42] |> println # Peter Luschny, Nov 10 2019

function [ c ] = catrep(z)
i=0; x=0; y=0; s=0;
while z>=(factorial(2*i+1)*(2))/(factorial(i)*factorial(i+2))
i=i+1;
end
y=(factorial(2*i-1)*(2))/(factorial(i-1)*factorial(i+1));
a=zeros(1,i); a(1,1)=1; k=2; x=z-y; m=1;
while x>0
w=0; s=0; p=0;
while w<=x
p=w;
w=w+(factorial(2*i-2*m+s-1)*(s+2))/(factorial(i-1-m)*factorial(i-m+s+1));
s=s+1;
end
m=m+1; a(1,k)=s-1; k=k+1; x=x-p;
end
a
end

A239903full = With[{r = 2*Range[2, 11]-1}, Reverse[Map[FromDigits[r-#] &, Rest[Select[Subsets[Range[2, 21], {10}, 125477], Min[r-#] >= 0 &]]]]];
A239903full[[;;100]] (* Paolo Xausa, Feb 17 2024 *)

define (t(j,k), (factorial(k+j)*(k-j+1))/(factorial(j)*factorial(k+1)));
i:0;
x:19;
z:0;y:0;s:0;
while x>=t(i,i+1) do (i:i+1);
y:t(i-1,i);a:zeromatrix(1,i);a[1,1]:1;k:2;z:x-y;m:1;
while (z>0) do (
w:0,s:0,p=0,
while (w<=z) do (
p:w,
w:w+t(i-1-m,i-m+s),
s:s+1
),
m:m+1,
a[1,k]:s-1,k:k+1,
z:z-p
);
print(a);

\\ Valid for n<58786 (=A000108(11)).
nxt(w)=if(w[1]==#w, vector(#w+1, i, i>#w), my(k=1); while(w[k]>w[k+1], w[k]=0; k++); w[k]++; w)
seq(n)={my(a=vector(n), w=[1]); a[1]=0; for(i=2, #v, a[i]=fromdigits(Vecrev(w)); w=nxt(w)); a} \\ Andrew Howroyd, Jan 24 2023

(define (A239903_only_upto_16794 n) (if (zero? n) n (A235049 (A071159 (A081291 n))))) ;; Gives correct results only up to 16794.
;; The following gives correct results all the way up to n=58784.
(define (A239903 n) (baselist-as-decimal (A239903raw n)))
(definec (A239903raw n) (if (zero? n) (list) (let loop ((n n) (row (A244160 n)) (col (- (A244160 n) 1)) (srow (- (A244160 n) 1)) (catstring (list 0))) (cond ((or (zero? row) (negative? col)) (reverse! (cdr catstring))) ((> (A009766tr row col) n) (loop n srow (- col 1) (- srow 1) (cons 0 catstring))) (else (loop (- n (A009766tr row col)) (+ row 1) col srow (cons (+ 1 (car catstring)) (cdr catstring))))))))
(define (baselist-as-decimal lista) (baselist->n 10 lista))
(define (baselist->n base bex) (let loop ((bex bex) (n 0)) (cond ((null? bex) n) (else (loop (cdr bex) (+ (* n base) (car bex)))))))
;; From Antti Karttunen, Apr 14-19 2014

To find an RGS corresponding to natural number n, one first finds a maximum row index k such that T(k,k-1) <= n in the Catalan Triangle (A009766) illustrated in the Example section. Note that as the last two columns of this triangle consist of Catalan numbers (that is, T(k,k-1) = T(k,k) = A000108(k)), it means that the first number to be subtracted from n is A081290(n) which occurs as a penultimate element of the row A081288(n)-1, in the column A081288(n)-2. The unranking algorithm then proceeds diagonally downwards, keeping the column index the same, and incrementing the row index, as long as it will encounter terms such that their total sum stays less than or equal to n.
If the total sum of encountered terms on that diagonal would exceed n, the algorithm jumps back to the penultimate column of the triangle, but one row higher from where it started the last time, and again starts summing the terms as long as the total sum stays <= n.
When the algorithm eventually reaches either row zero or column less than zero, the result will be a list of numbers, each element being the number of terms summed from each diagonal, so that the diagonal first traversed appears as the first 1 (as that first diagonal will never allow more than one term), and the number of terms summed from the last traversed diagonal appears the last number in the list. These lists of numbers are then concatenated together as decimal numbers.
These steps can also be played backwards in order to recover the corresponding decimal integer n from such a list of numbers, giving a "ranking function" which will be the inverse to this "unranking function".
For n=1..16794 (where 16794 = A000108(10)-2), a(n) = A235049(A071159(A081291(n))). - Antti Karttunen, Apr 14 2014
Alternative, simpler description of the algorithm from Antti Karttunen, Apr 21 2014: (Start)
Consider the following square array, which is Catalan triangle A009766 without its rightmost, "duplicate" column, appropriately transposed (cf. also tables A030237, A033184 and A054445):
   Row| Terms on that row
   ---+--------------------------
    1 |   1   1   1    1    1 ...
    2 |   2   3   4    5    6 ...
    3 |   5   9  14   20   27 ...
    4 |  14  28  48   75  110 ...
    5 |  42  90 165  275  429 ...
    6 | 132 297 572 1001 1638 ...
To compute the n-th RGS, search first for the greatest Catalan number C_k which is <= n (this is A081290(n), found as the first term of row A081288(n)-1). Then, by a greedy algorithm, select from each successive row (moving towards the top of table) as many terms from the beginning of that row as will still fit into n, subtracting them from n as you go. The number of terms selected from the beginning of each row gives each element of the n-th RGS, so that the number of terms selected from the topmost row (all 1's) appears as its last element.
(End)

Description, formula and examples edited/rewritten by Italo J Dejter, Apr 13 2014 and Antti Karttunen, Apr 18 2014

A197433 Sum of distinct Catalan numbers: a(n) = Sum_{k>=0} A030308(n,k)*C(k+1) where C(n) is the n-th Catalan number (A000108). (C(0) and C(1) not treated as distinct.)

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 7, 8, 14, 15, 16, 17, 19, 20, 21, 22, 42, 43, 44, 45, 47, 48, 49, 50, 56, 57, 58, 59, 61, 62, 63, 64, 132, 133, 134, 135, 137, 138, 139, 140, 146, 147, 148, 149, 151, 152, 153, 154, 174, 175, 176, 177, 179, 180, 181, 182, 188, 189, 190, 191, 193, 194, 195, 196

Offset: 0

Philippe Deléham, Oct 15 2011

Replace 2^k with A000108(k+1) in binary expansion of n.
From Antti Karttunen, Jun 22 2014: (Start)
On the other hand, A244158 is similar, but replaces 10^k with A000108(k+1) in decimal expansion of n.
This sequence gives all k such that A014418(k) = A239903(k), which are precisely all nonnegative integers k whose representations in those two number systems contain no digits larger than 1. From this also follows that this is a subsequence of A244155.
(End)

Antti Karttunen, Table of n, a(n) for n = 0..8191

Characteristic function: A176137.
Subsequence of A244155.
Cf. A000108, A030308, A197432, A014418, A239903, A244158, A244159, A244230, A244231, A244232, A244315, A244316.
Cf. also A060112.
Other sequences that are built by replacing 2^k in binary representation with other numbers: A022290 (Fibonacci), A029931 (natural numbers), A059590 (factorials), A089625 (primes), A197354 (odd numbers).

nmax = 63;
a[n_] := If[n == 0, 0, SeriesCoefficient[(1/(1-x))*Sum[CatalanNumber[k+1]* x^(2^k)/(1 + x^(2^k)), {k, 0, Log[2, n] // Ceiling}], {x, 0, n}]];
Table[a[n], {n, 0, nmax}] (* Jean-François Alcover, Nov 18 2021, after Ilya Gutkovskiy *)

For all n, A244230(a(n)) = n. - Antti Karttunen, Jul 18 2014

G.f.: (1/(1 - x))*Sum_{k>=0} Catalan number(k+1)*x^(2^k)/(1 + x^(2^k)). - Ilya Gutkovskiy, Jul 23 2017

Name clarified by Antti Karttunen, Jul 18 2014

A244158 If n = Sum c_i * 10^i then a(n) = Sum c_i * Cat(i+1), where Cat(k) = A000108(k).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 5

Offset: 0

Antti Karttunen, Jun 22 2014

This sequence converts any number from various "Catalan Base number systems" (when represented as decimal numbers) back to the integer the numeral represents: e.g. we have a(A014418(n)) = n and a(A244159(n)) = n (except for the latter this is eventually broken by the shortcomings of the decimal representation used, while for the former it works for all n, because no digits larger than 3 will ever appear in the terms of A014418).
A197433 is similar, but replaces 2^k with A000108(k+1) in binary expansion of n.
For 1- and 2-digit numbers the same as A156230. - R. J. Mathar, Jun 27 2014

Index entries for sequences related to decimal expansion of n

Cf. A000108, A014418, A244159, A197433, A244155, A244156, A244157.

Differs from A028897 and A081594 for the first time at n=100, which here is a(100) = 5.

A244158 := proc(n)
    local dgs,k ;
    dgs := convert(n,base,10) ;
    add( op(k,dgs)*A000108(k),k=1..nops(dgs)) ;
end proc: # R. J. Mathar, Jan 31 2015

A244156 Numbers n such that when the n-th Catalan restricted growth string [b_k, b_{k-1}, ..., b_2, b_1] (see A239903) is viewed as a simple numeral in Catalan Base: b_kC(k) + b_{k-1}C(k-1) + ... + b_2C(2) +b_1C(1) it differs from n. Here C(m) = A000108(m).

Original entry on oeis.org

10, 11, 12, 13, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 52, 53, 54, 55, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105

Offset: 1

Antti Karttunen, Jun 22 2014

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16156

Complement of A244155. Positions of nonzeros in A244157.

Cf. A000108, A239903, A014418.

A244157 a(n) = difference between n and the n-th Catalan restricted growth string [b_k, b_{k-1}, ..., b_2, b_1] (see A239903) when it is viewed as a simple numeral in Catalan Base: b_kC(k) + b_{k-1}C(k-1) + ... + b_2C(2) +b_1C(1). Here C(m) = A000108(m).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 4, 4, 4, 4, 4, 5, 5, 5, 5, 7, 7, 7, 7, 7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 4, 4, 4, 4, 4, 5, 5, 5, 5, 7, 7, 7, 7, 7, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 18

Offset: 0

Antti Karttunen, Jun 22 2014

nonn

Antti Karttunen, Table of n, a(n) for n = 0..16796

A244155 gives the positions of zeros, A244156 the positions of nonzeros.

Cf. A000108, A239903, A014418, A244158.

(define (A244157 n) (- n (CatBaseSum (A239903raw n)))) ;; A239903raw given in A239903.
(define (CatBaseSum lista) (let loop ((digits (reverse lista)) (i 1) (s 0)) (if (null? digits) s (loop (cdr digits) (+ i 1) (+ s (* (car digits) (A000108 i)))))))

a(n) = n - A244158(A239903(n)) up to 58784, after which the "digits" in Catalan restricted growth strings grow larger than 9 and their decimal representation used in A239903 starts corrupting the results.
At n=58785 (= C(11)-1, where C(k) = the k-th Catalan number, A000108(k)), the correct value for this sequence is a(58785) = 58785 - ((1*C(10)) + (2*C(9)) + (3*C(8)) + (4*C(7)) + (5*C(6)) + (6*C(5)) + (7*C(4)) + (8*C(3)) + (9*C(2)) + (10*C(1))) = 25181.
Use the Scheme-program given in the Program sections of this entry and A239903 (the function A239903raw) to get correct results for all n.

cp's OEIS Frontend

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

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A239903 List of Restricted-Growth Strings a_{k-1}a_{k-2}...a_{2}a_{1}, with k=2 and a_1 in {0,1} or k>2, a_{k-1}=1 and a_{j+1}>=1+a_j, for k-1>j>0.

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A197433 Sum of distinct Catalan numbers: a(n) = Sum_{k>=0} A030308(n,k)*C(k+1) where C(n) is the n-th Catalan number (A000108). (C(0) and C(1) not treated as distinct.)

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A244158 If n = Sum c_i * 10^i then a(n) = Sum c_i * Cat(i+1), where Cat(k) = A000108(k).

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A244156 Numbers n such that when the n-th Catalan restricted growth string [b_k, b_{k-1}, ..., b_2, b_1] (see A239903) is viewed as a simple numeral in Catalan Base: b_k*C(k) + b_{k-1}*C(k-1) + ... + b_2*C(2) +b_1*C(1) it differs from n. Here C(m) = A000108(m).

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A244157 a(n) = difference between n and the n-th Catalan restricted growth string [b_k, b_{k-1}, ..., b_2, b_1] (see A239903) when it is viewed as a simple numeral in Catalan Base: b_k*C(k) + b_{k-1}*C(k-1) + ... + b_2*C(2) +b_1*C(1). Here C(m) = A000108(m).

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A244156 Numbers n such that when the n-th Catalan restricted growth string [b_k, b_{k-1}, ..., b_2, b_1] (see A239903) is viewed as a simple numeral in Catalan Base: b_kC(k) + b_{k-1}C(k-1) + ... + b_2C(2) +b_1C(1) it differs from n. Here C(m) = A000108(m).

A244157 a(n) = difference between n and the n-th Catalan restricted growth string [b_k, b_{k-1}, ..., b_2, b_1] (see A239903) when it is viewed as a simple numeral in Catalan Base: b_kC(k) + b_{k-1}C(k-1) + ... + b_2C(2) +b_1C(1). Here C(m) = A000108(m).