A297160 -id:A297160 - cp's OEIS frontend

A287156 2-limiting word of the morphism 0->10, 1->21, 2->0.

2, 1, 1, 0, 1, 0, 0, 2, 1, 1, 0, 0, 2, 1, 0, 2, 1, 2, 1, 1, 0, 1, 0, 0, 2, 1, 0, 2, 1, 2, 1, 1, 0, 0, 2, 1, 2, 1, 1, 0, 2, 1, 1, 0, 1, 0, 0, 2, 1, 1, 0, 0, 2, 1, 0, 2, 1, 2, 1, 1, 0, 0, 2, 1, 2, 1, 1, 0, 2, 1, 1, 0, 1, 0, 0, 2, 1, 0, 2, 1, 2, 1, 1, 0, 2, 1

Offset: 1

Clark Kimberling, May 22 2017

Starting with 0, the first 4 iterations of the morphism yield words shown here:
1st:  10
2nd:  2110
3rd:  0212110
4th:  1002010212110
The 2-limiting word is the limit of the words for which the number of iterations is congruent to 2 mod 3.
Let u be the sequence of positions of 0, and likewise, v for 1 and w for 2.  Let U, V, W be the limits of u(n)/n, v(n)/n, w(n)/n, respectively.  Then 1/U + 1/V + 1/W = 1, where
U = 3.079595623491438786010417...,
V = 2.324717957244746025960908...,
W = U + 1 = 4.079595623491438786010417....
If n >=2, then u(n) - u(n-1) is in {1,2,3,4,6}, v(n) - v(n-1) is in {1,2,3,4}, and w(n) - w(n-1) is in {2,3,4,5,7}.

			The 2nd, 5th, and 8th iterates are 2110, 211010021100210212110, 211010021100210212110100210212110021211021101002110021021211002121102110100210212110211010021211010021100210212110.

Clark Kimberling, Table of n, a(n) for n = 1..10000

Cf. A287108, A287157, A287158, A287159, A297160.

s = Nest[Flatten[# /. {0 -> {1, 0}, 1 -> {2, 1}, 2 -> 0}] &, {0}, 11]   (* A287156 *)
Flatten[Position[s, 0]] (* A287157 *)
Flatten[Position[s, 1]] (* A287158 *)
Flatten[Position[s, 2]] (* A287159 *)

A220400 Number of ways to write n as sum of at least 2 consecutive odd positive integers.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 2, 0, 0, 0, 1, 1, 0, 0, 2, 1, 0, 1, 1, 0, 0, 0, 2, 1, 0, 1, 2, 0, 0, 1, 2, 0, 0, 0, 1, 2, 0, 0, 3, 1, 0, 1, 1, 0, 0, 1, 2, 1, 0, 0, 2, 0, 0, 2, 3, 1, 0, 0, 1, 1, 0, 0, 3, 0, 0, 2, 1, 1, 0, 0, 3, 2, 0, 0, 2, 1, 0, 1, 2, 0, 0, 1, 1, 1, 0, 1, 4, 0, 0, 2, 2, 0, 0, 0, 2, 3

Offset: 0

Carl Najafi, Dec 13 2012

nonn

Records occur at 0, 4, 16, 48, 96, 144, 240, 480, 720, 960, 1440, ..., (A297160). - Antti Karttunen, Dec 27 2017

Also number of ways to express n in the form k + (k + 2) + ... + (k + 2*m - 2) = m * (k + m - 1) where k > 0 is odd and m > 0 and m * (m + 1) < n. - David A. Corneth, Dec 27 2017

			For n=16 we can write 1+3+5+7 and 7+9, thus a(16) = 2.
For n = 24, we look for sums of consecutive numbers of m terms of the form m * (k + m - 1) for odd k and m * (m + 1) < 24, i.e., m < 5. We can factorize 24 as such in two positive factors as 1 * 24 = 2 * 12 = 3 * 8 = 4 * 6 giving m = 1, 2, 3 and 4 respectively. Solving for k gives k = 24, k = 11, k = 6 and k = 3 respectively. Of these values, two are odd so a(24) = 2. Superfluously, the corresponding sums are 11 + 13 = 3 + 5 + 7 + 9. - _David A. Corneth_, Dec 28 2017

Antti Karttunen, Table of n, a(n) for n = 0..11111

Cf. A069283 (even numbers lead to this sequence).

Cf. A297160 (positions of records).

nn = 100; t = Table[0, {nn}]; Do[s = odd = 2*n - 1; While[odd = odd + 2; s = s + odd; s <= nn, t[[s]]++], {n, nn/2}]; Join[{0}, t] (* T. D. Noe, Dec 18 2012 *)

a(n) = if(n==0, return(0)); my(d = divisors(n)); (#d + 1) \ 2 - sum(i = 2, (#d + 1) \ 2, (n / d[i] - d[i]) % 2) - 1 \\ David A. Corneth, Dec 27 2017

(define (A220400 n) (let loop ((s 0) (begin 1) (end 1) (sum 1)) (cond ((> begin (/ n 2)) s) ((< sum n) (loop s begin (+ end 2) (+ sum end 2))) ((> sum n) (loop s (+ begin 2) end (- sum begin))) (else (loop (+ 1 s) (+ begin 2) end (- sum begin)))))) ;; Antti Karttunen, Dec 27 2017

More terms from Antti Karttunen, Dec 27 2017

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A287156 2-limiting word of the morphism 0->10, 1->21, 2->0.

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