The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A322051 (Bold, blue-underlined text is an addition; faded, red-underlined text is a ~~deletion~~.)
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A322051

a(n) is the number of initial terms in the row of length 2^n of A322050 that agree with the limiting sequence A322049.
(history; published version)

#28 by N. J. A. Sloane at Sat Dec 29 03:43:37 EST 2018

STATUS

proposed

approved

#27 by Colin Barker at Sat Dec 29 03:24:28 EST 2018

STATUS

editing

proposed

#26 by Colin Barker at Sat Dec 29 03:19:56 EST 2018

FORMULA

Conjectures from Colin Barker, Dec 29 2018: (Start)

G.f.: (1 - x - x^2 + x^3 - 2*x^4 - x^5 + 2*x^6) / ((1 - x)*(1 + x)*(1 - 2*x)).

a(n) = (2^n + 2) / 3 for n even and n>3.

a(n) = (2^n + 1) / 3 for n odd and n>3.

a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) for n>6.

(End)

STATUS

approved

editing

Discussion

Sat Dec 29

03:24

Colin Barker: These conjectures were posted earlier on, but were then deleted.

#25 by M. F. Hasler at Tue Dec 18 22:08:54 EST 2018

STATUS

editing

approved

#24 by M. F. Hasler at Tue Dec 18 22:07:51 EST 2018

NAME

When A322050 is displayed as a triangle, the rows converge to A322049; a(n) is the number of initial terms in the row of length 2^n of A322050 that agree with the limiting sequence A322049.

COMMENTS

Seems to be identical to A005578 with the exception of a(3) = 4. -_ _Omar E. Pol_, Dec 17 2018

EXAMPLE

.n........................

... n i* a(n) first non-matching pair (i* = Index of start in A319018)

............a(n)

...............first non-matching pair

. 0..... 3..... 1..... 5..... 1

. 1..... 5..... 1..... 7..... 5

. 2..... 9..... 2..... 6..... 3

. 3.... 17..... 4..... 8..... 5

. 4.... 33..... 6.... 17.... 15

. 5.... 65.... 11... 145... 141

. 6... 129.... 22.... 73.... 69

. 7... 257.... 43... 734... 726

. 8... 513.... 86... 349... 341

. 9.. 1025... 171.. 3579.. 3563

10.. 2049... 342.. 1696.. 1680

11.. 4097... 683. 17810. 17778

12.. 8193.. 1366.. 8394.. 8362

13. 16385.. 2731. 88553. 88489

14. 32769.. 5462. 41665. 41601

...

EXTENSIONS

Edited by M. F. Hasler, Dec 18 2018

STATUS

approved

editing

Discussion

Tue Dec 18

22:08

M. F. Hasler: See history for comments on this edit made already earlier and unduely reverted.

#23 by N. J. A. Sloane at Tue Dec 18 19:56:34 EST 2018

NAME

When A322050 is displayed as a triangle, the rows converge to A322049; a(n) is the number of initial terms in the row of length 2^n of A322050 that agree with the limiting sequence A322049.

DATA

1, 1, 2, 4, 6, 11, 22, 43, 86, 171, 342, 683, 1366, 2731, 5462, 10923, 21846, 43691, 87382, 174763, 349526, 699051, 1398102

FORMULA

From Paul Curtz, Dec 18 2018: (Start)

a(n) = A001045(n) + (1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, ... = c(n)).

c(n) = A000035(n+1) = period 2: repeat [1, 0] with 1 instead of the second 0 is unknown.

a(n) = A139763(n-1) + (2, 0, 0, 1, 2, 0, 0, 0, 2, 0, 0, 0, 2, ... = d(n)). A139763(-1) = -1.

d(n) is unknown. Period 4: repeat [2, 0, 0, 0] also.

a(2*n) =(4^n + 2)/3. (End)

Conjectures from Colin Barker, Dec 18 2018: (Start)

G.f.: (1 - x - x^2 + x^3 - 2*x^4 - x^5 + 2*x^6) / ((1 - x)*(1 + x)*(1 - 2*x)).

a(n) = (2^n + 2) / 3 for n>3 and even.

a(n) = (2^n + 1) / 3 for n>3 and odd.

a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) for n>6.

(End)

EXAMPLE

n i* a(n) first non-matching pair (i* = Index of start in A319018)

.n........................

...Index of start in A319018

............a(n)

...............first non-matching pair

.0 .....3 .....1 .....5 .....1

.1 .....5 .....1 .....7 .....5

.2 .....9 .....2 .....6 .....3

.3 ....17 .....4 .....8 .....5

.4 ....33 .....6 ....17 ....15

.5 ....65 ....11 ...145 ...141

.6 ...129 ....22 ....73 ....69

.7 ...257 ....43 ...734 ...726

.8 ...513 ....86 ...349 ...341

.9 ..1025 ...171 ..3579 ..3563

10 ..2049 ...342 ..1696 ..1680

11 ..4097 ...683 .17810 .17778

12 ..8193 ..1366 ..8394 ..8362

13 .16385 ..2731 .88553 .88489

14 .32769 ..5462 .41665 .41601

...

From Paul Curtz, Dec 18 2018: (Start)

4th column - 5th one: 4, 2, 3, 3, 2, 4, 4, 8, 8, 16, 16, 32, 32, 64, 64, ... . (End)

...

CROSSREFS

Cf. A000035, A001045, A016116, A047849, A139763.

EXTENSIONS

Edited by M. F. Hasler, Dec 18 2018

STATUS

proposed

approved

#22 by Paul Curtz at Tue Dec 18 16:18:31 EST 2018

STATUS

editing

proposed

Discussion

Tue Dec 18

19:56

N. J. A. Sloane: The 4 terms that Paul Curtz is trying to add:  how were they found?  My guess is that they were based on Hugo's conjecture that "Conjecture: For n >= 5, a(n) = 2*a(n-1)-1 if n is odd, 2*a(n-1) if n is even."  Otherwise they would require a great deal of computation, if one were to compute them honestly.  I am going to revert this.  To those of you who added further changes while this was being reviewed, please remeber what I said: stay out of the operating room while a patient is being operated n.

#21 by Paul Curtz at Tue Dec 18 16:13:25 EST 2018

CROSSREFS

Cf. A000035, A001045, A016116, A047849, A139763.

#20 by Paul Curtz at Tue Dec 18 16:09:01 EST 2018

DATA

1, 1, 2, 4, 6, 11, 22, 43, 86, 171, 342, 683, 1366, 2731, 5462, 10923, 21846, 43691, 87382, 174763, 349526, 699051, 1398102

FORMULA

From Paul Curtz, Dec 18 2018 : (Start)

c(n) = A000035(n+1) = period 2: repeat [1, 0] with 1 instead of the second 0 is unknown.

EXAMPLE

From Paul Curtz, Dec 18 2018: (Start)

4th column - 5th one: 4, 2, 3, 3, 2, 4, 4, 8, 8, 16, 16, 32, 32, 64, 64, ... . (End)

#19 by M. F. Hasler at Tue Dec 18 14:29:58 EST 2018

COMMENTS

From Paul Curtz, Dec 18 2018 (Start)

A005578(n) = A001045(n) + period 2: repeat [1, 0]. Compare to c(n).

A005578(n+1) - A005578(n) = A001045(n). (Consequence: 0 together with A005578(n) is an autosequence of the first kind because A001045(n) is. Its companion is A052950(n)). (End)

Discussion

Tue Dec 18

14:34

M. F. Hasler: In Curtz' formulas, I don't see the point of introducing c(n) without ever referring to ;  the line "d(n) =..." is self-contradictory. What means the second half? This cannot published as is.

14:41

M. F. Hasler: Now I see Curtz' comment about A005578 referred to c(n). But the only link with this sequence is already stated in a clear and simple manner by Pol's comment. The properties of A005578 are not to be studied here.