This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
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.
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
a(0) = 0, because no numbers are needed to form an empty sum, which is zero. For n=1 we need just A001045(2) = 1, thus a(1) = 1. For n=2 we need A001045(2) + A001045(2) = 1 + 1, thus a(2) = 2. For n=4 we need A001045(3) + A001045(2) = 3 + 1, thus a(4) = 2. For n=6 we form the greedy sum as A001045(4) + A001045(2) = 5 + 1, thus a(6) = 2. Alternatively, we could form the sum as A001045(3) + A001045(3) = 3 + 3, but the number of summands in that case is no less. For n=7 we need A001045(4) + A001045(2) + A001045(2) = 5 + 1 + 1, thus a(7) = 3. For n=8 we need A001045(4) + A001045(3) = 5 + 3, thus a(8) = 2. For n=10 we need A001045(4) + A001045(4) = 5 + 5, thus a(10) = 2.
jacob[n_] := (2^n - (-1)^n)/3; maxInd[n_] := Floor[Log2[3*n + 1]]; A265745[n_] := A265745[n] = 1 + A265745[n - jacob[maxInd[n]]]; A265745[0] = 0; Array[A265745, 100, 0] (* Amiram Eldar, Jul 21 2023 *)
A130249(n) = floor(log(3*n + 1)/log(2)); A001045(n) = (2^n - (-1)^n) / 3; A265745(n) = {if(n == 0, 0, my(d = n - A001045(A130249(n))); if(d == 0, 1, 1 + A265745(d)));} \\ Amiram Eldar, Jul 21 2023
def greedyJ(n): n1 = (3*n+1).bit_length() - 1; return (2**n1 - (-1)**n1)//3 def a(n): return 0 if n == 0 else 1 + a(n - greedyJ(n)) print([a(n) for n in range(107)]) # Michael S. Branicky, Jul 11 2021
pell[1] = 1; pell[2] = 2; pell[n_] := pell[n] = 2*pell[n - 1] + pell[n - 2]; a[n_] := Module[{s = {}, m = n, k}, While[m > 0, k = 1; While[pell[k] <= m, k++]; k--; AppendTo[s, k]; m -= pell[k]; k = 1]; Plus @@ IntegerDigits[Total[3^(s - 1)], 3]]; Array[a, 100, 0] (* Amiram Eldar, Mar 12 2022 *)
4 is a term since its Catalan representation, A014418(4) = 20, has the sum of digits A014420(4) = 2 + 0 = 2 and 4 is divisible by 2. 9 is a term since its Catalan representation, A014418(9) = 120, has the sum of digits A014420(9) = 1 + 2 + 0 = 3 and 9 is divisible by 3.
c[n_] := c[n] = CatalanNumber[n]; q[n_] := Module[{s = {}, m = n, i}, While[m > 0, i = 1; While[c[i] <= m, i++]; i--; m -= c[i]; AppendTo[s, i]]; Divisible[n, Plus @@ IntegerDigits[Total[4^(s - 1)], 4]]]; Select[Range[216], q]
4 is a term since 4 and 5 are both Catalan-Niven numbers: the Catalan representation of 4, A014418(20) = 20, has the sum of digits 2+0 = 2 and 4 is divisible by 2, and the Catalan representation of 5, A014418(5) = 100, has the sum of digits 1+0+0 = 1 and 5 is divisible by 1.
c[n_] := c[n] = CatalanNumber[n]; q[n_] := Module[{s = {}, m = n, i}, While[m > 0, i = 1; While[c[i] <= m, i++]; i--; m -= c[i]; AppendTo[s, i]]; Divisible[n, Plus @@ IntegerDigits[Total[4^(s - 1)], 4]]]; Select[Range[2300], q[#] && q[#+1] &]
4 is a term since 4, 5 and 6 are all Catalan-Niven numbers: the Catalan representation of 4, A014418(20) = 20, has the sum of digits 2+0 = 2 and 4 is divisible by 2, the Catalan representation of 5, A014418(5) = 100, has the sum of digits 1+0+0 = 1 and 5 is divisible by 1, and the Catalan representation of 6, A014418(6) = 101, has the sum of digits 1+0+1 = 2 and 6 is divisible by 2.
c[n_] := c[n] = CatalanNumber[n]; catNivQ[n_] := Module[{s = {}, m = n, i}, While[m > 0, i = 1; While[c[i] <= m, i++]; i--; m -= c[i]; AppendTo[s, i]]; Divisible[n, Plus @@ IntegerDigits[Total[4^(s - 1)], 4]]]; seq[count_, nConsec_] := Module[{cn = catNivQ /@ Range[nConsec], s = {}, c = 0, k = nConsec + 1}, While[c < count, If[And @@ cn, c++; AppendTo[s, k - nConsec]]; cn = Join[Rest[cn], {catNivQ[k]}]; k++]; s]; seq[30, 3]
144 is a term since 144, 145, 146 and 147 are all divisible by the sum of the digits in their Catalan representation:
k A014418(k) A014420(k) k/A014420(k)
--- ---------- ---------- ------------
144 100210 4 36
145 100211 5 29
146 101000 2 73
147 101001 3 49
c[n_] := c[n] = CatalanNumber[n]; catNivQ[n_] := Module[{s = {}, m = n, i}, While[m > 0, i = 1; While[c[i] <= m, i++]; i--; m -= c[i]; AppendTo[s, i]]; Divisible[n, Plus @@ IntegerDigits[Total[4^(s - 1)], 4]]]; seq[count_, nConsec_] := Module[{cn = catNivQ /@ Range[nConsec], s = {}, c = 0, k = nConsec + 1}, While[c < count, If[And @@ cn, c++; AppendTo[s, k - nConsec]]; cn = Join[Rest[cn], {catNivQ[k]}]; k++]; s]; seq[5, 4]
For n=18, using the alternative description, we see that it is partitioned into the terms of A197433 as a greedy sum A197433(11) + A197433(1) = 17 + 1. Thus a(18) = A000120(11) + A000120(1) = 3+1 = 4. For n=128, we see that is likewise represented as A197433(31) + A197433(31) = 64 + 64. Thus a(128) = 2*A000120(31) = 10.
As the 0th Catalan String is empty, indicated by A239903(0)=0, a(0)=0. As the 18th Catalan String is [1,0,1,2] (A239903(18)=1012), a(18) = 1+0+1+2 = 4. Note that although the range of validity of A239903 is inherently limited by the decimal representation employed, it doesn't matter here: We have a(58785) = 55, as the corresponding 58785th Catalan String is [1,2,3,4,5,6,7,8,9,10], even though A239903 cannot represent that unambiguously.
A236855list[m_] := With[{r = 2*Range[2, m]-1}, Reverse[Map[Total[r-#] &, Select[Subsets[Range[2, 2*m-1], {m-1}], Min[r-#] >= 0 &]]]]; A236855list[6] (* Generates C(6) terms *) (* Paolo Xausa, Feb 19 2024 *)
(define (A236855 n) (apply + (A239903raw n))) (define (A239903raw n) (if (zero? n) (list) (let loop ((n n) (row (- (A081288 n) 1)) (col (- (A081288 n) 2)) (srow (- (A081288 n) 2)) (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)))))))) ;; Alternative definition: (define (A236855 n) (let ((x (A071155 (A081291 n)))) (- (A034968 x) (A060130 x))))
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