cp's OEIS Frontend

Note that a(33604) = A000217(10) = 55 because the sum is computed from the underlying list (vector) of numbers, and thus is not subject to any corruption by decimal representation as A244159 itself is.

Equivalent description: partition n "greedily" as terms of A197433, i.e. n = A197433(i) + A197433(j) + ... + A197433(k), always using the largest term of A197433 that still "fits in" (i.e. is <= n remaining). Then a(n) = A000120(i) + A000120(j) + ... + A000120(k).

The first value larger than nine occurs at a(33604) = 10. (33604 = A014143(9)). Note that this sequence is not subject to any corruption by decimal representation as A244159 itself is.

A014143 gives records up to that A014143(9) = 33604, but thereafter, the next record occurs at 57317, for which a(57317) = 11, although A014143(10) = 116103. This is explained by that A244159raw(57317) = [2,3,4,5,6,7,8,9,10,11] and A244159raw(116103) = [1,2,3,4,5,6,7,8,9,10,11] (where A244159raw means the underlying representation, before it is maimed by the decimal representation).

This change is explained by the fact that A014143(n-1) + A014138(n) > A000108(n+1) for n = 1..9, but for n >= 10, A014143(n-1) + A014138(n) < A000108(n+1).

For example, although 16808 = 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), its representation in A244159 system is 1000000123, as 16808 = 1*C(10) + 1*C(3) + 2*C(2) + 3*C(1).

Note that a(33604) = 10! = 3628800 because the product is computed from the underlying list (vector) of numbers, and thus is not subject to any corruption by decimal representation as A244159 itself is.

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.

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)

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

These are called "Jacobsthal Representation Numbers" in Horadam's 1996 paper.

Sum_{i=0..} digit(i)*A001045(2+digit(i)) recovers n from such representation a(n), where digit(0) stands for the least significant digit (at the right), and A001045(k) gives the k-th Jacobsthal number.

No larger digits than 2 will occur, which allows representing the same sequence in a more compact form by base-3 coding in A265746.

Sequence A197911 gives the terms with no digit "2" in their representation, while its complement A003158 gives the terms where "2" occurs at least once.

Numbers beginning with digit "2" in this representation are given by A020988(n) [= 2*A002450(n) = 2*A001045(2n)].

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)

Apart from 0, each n occurs A000245(n) times.

For n >= 1, a(n) gives the largest k such that C(k) <= n, where C(k) stands for the k-th Catalan number, A000108(k).

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