A118382 -id:A118382 - cp's OEIS frontend

A028356 Simple periodic sequence underlying clock sequence A028354.

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

Offset: 0

N. J. A. Sloane

From Klaus Brockhaus, May 15 2010: (Start)
Continued fraction expansion of (28+sqrt(2730))/56.
Decimal expansion of 1112/9009.
Partial sums of 1 followed by A130151.
First differences of A028357. (End)

Zdeněk Horský, "Pražský orloj" ("The Astronomical Clock of Prague", in Czech), Panorama, Prague, 1988, pp. 76-78.

Michal Křížek, Alena Šolcová and Lawrence Somer, Construction of Šindel sequences, Comment. Math. Univ. Carolin., 48 (2007), 373-388.
N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).
Index entries for linear recurrences with constant coefficients, signature (1,0,-1,1).

Cf. A000034, A028354, A068073, A118382, A118383.

Cf. A177924 (decimal expansion of (28+sqrt(2730))/56), A130151 (repeat 1, 1, 1, -1, -1, -1), A028357 (partial sums of A028356). - Klaus Brockhaus, May 15 2010

&cat [[1, 2, 3, 4, 3, 2]^^20]; // Klaus Brockhaus, May 15 2010

A028356:=n->[1, 2, 3, 4, 3, 2][(n mod 6)+1]: seq(A028356(n), n=0..100); # Wesley Ivan Hurt, Jun 23 2016

CoefficientList[ Series[(1 + 2x + 3x^2 + 4x^3 + 3x^4 + 2x^5)/(1 - x^6), {x, 0, 85}], x]
LinearRecurrence[{1,0,-1,1},{1,2,3,4},120] (* or *) PadRight[{},120,{1,2,3,4,3,2}] (* Harvey P. Dale, Apr 15 2016 *)

def A028356(n): return (1,2,3,4,3,2)[n%6] # Chai Wah Wu, Apr 18 2024

def A():
    a, b, c, d = 1, 2, 3, 4
    while True:
        yield a
        a, b, c, d = b, c, d, a + (d - b)
A028356 = A(); [next(A028356) for n in range(106)] # Peter Luschny, Jul 26 2014

Sum of any six successive terms is 15.
G.f.: (1 + 2*x + 3*x^2 + 4*x^3 + 3*x^4 + 2*x^5)/(1 - x^6).
From Wesley Ivan Hurt, Jun 23 2016: (Start)
a(n) = a(n-1) - a(n-3) + a(n-4) for n>3.
a(n) = (15 - cos(n*Pi) - 8*cos(n*Pi/3))/6. (End)
E.g.f.: (15*exp(x) - exp(-x) - 8*cos(sqrt(3)*x/2)*(sinh(x/2) + cosh(x/2)))/6. - Ilya Gutkovskiy, Jun 23 2016
a(n) = abs(((n+3) mod 6)-3) + 1. - Daniel Jiménez, Jan 14 2023

Additional comments from Robert G. Wilson v, Mar 01 2002

A028355 How the astronomical clock ("Orloj") in Prague would strike 1,2,3,...,24,25,.. (digits follow 12343212343... (A028356), n-th group adds to n).

Original entry on oeis.org

1, 2, 3, 4, 32, 123, 43, 2123, 432, 1234, 32123, 43212, 34321, 23432, 123432, 1234321, 2343212, 3432123, 4321234, 32123432, 123432123, 43212343, 2123432123, 432123432, 1234321234, 32123432123, 43212343212

Offset: 1

N. J. A. Sloane, J. H. Conway

This remarkable sequence is really a sequence of lists rather than numbers.

			1, 2, 3, 4, 3+2=5, 1+2+3=6, 4+3=7, 2+1+2+3=8, 4+3+2=9, 1+2+3+4=10, 3+2+1+2+3=11, 4+3+2+1+2=12, 3+4+3+2+1=13, 2+3+4+3+2=14, 1+2+3+4+3+2=15, ...

Zdenek Horsky, "Prazsky Orloj" ["The Astronomical Clock of Prague", in Czech], Panorama, Prague, 1988, pp. 76-78.

Seiichi Manyama, Table of n, a(n) for n = 1..2500
Michal Krížek, Alena Šolcová and Lawrence Somer, Construction of Šindel sequences, Comment. Math. Univ. Carolin., 48 (2007), 373-388.
N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).

Cf. A028354, A028356, A068962, A118382, A118383.

s[i_] := {1, 2, 3, 4, 3, 2}[[Mod[i, 6, 1]]];
m[k_] := If[k == 1, 0, For[m0 = 1, True, m0++, If[k (k - 1)/2 == Sum[s[i], {i, 1, m0}], Return[m0]]]];
n[k_] := For[n0 = m[k] + 1, True, n0++, If[Sum[s[i], {i, m[k] + 1, n0}] == k, Return[n0]]];
a[k_] := a[k] = Table[s[i], {i, m[k] + 1, n[k]}] // FromDigits; Array[a, 27] (* Jean-François Alcover, Mar 14 2016 *)

Conjectures from Chai Wah Wu, Apr 18 2024: (Start)
a(n) = 1000001*a(n-15) - 1000000*a(n-30) for n > 30.
G.f.: x*(100000*x^28 + 200000*x^27 + 300000*x^26 + 400000*x^25 + 320000*x^24 + 123000*x^23 + 430000*x^22 + 212300*x^21 + 432000*x^20 + 123400*x^19 + 321230*x^18 + 432120*x^17 + 343210*x^16 + 234320*x^15 + 123432*x^14 + 23432*x^13 + 34321*x^12 + 43212*x^11 + 32123*x^10 + 1234*x^9 + 432*x^8 + 2123*x^7 + 43*x^6 + 123*x^5 + 32*x^4 + 4*x^3 + 3*x^2 + 2*x + 1)/(1000000*x^30 - 1000001*x^15 + 1). (End)

A028354 How the astronomical clock ("Orloj") in Prague strikes the hours (digits follow 12343212343... (A028356), n-th group adds to n).

Original entry on oeis.org

1, 2, 3, 4, 32, 123, 43, 2123, 432, 1234, 32123, 43212, 34321, 23432, 123432, 1234321, 2343212, 3432123, 4321234, 32123432, 123432123, 43212343, 2123432123, 432123432, 1, 2, 3, 4, 32, 123, 43, 2123, 432, 1234, 32123, 43212

Offset: 1

J. H. Conway

There is a single bell, which to indicate 5 o'clock, say, strikes thrice then twice.

Zdenek Horsky, "Prazsky Orloj" ["The Astronomical Clock of Prague", in Czech], Panorama, Prague, 1988, pp. 76-78.

Michal Krížek, Alena Šolcová and Lawrence Somer, Construction of Šindel sequences, Comment. Math. Univ. Carolin., 48 (2007), 373-388.
N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).

Cf. A028355, A028356, A068962, A118382, A118383.

s[i_] := {1, 2, 3, 4, 3, 2}[[Mod[i, 6, 1]]]; m[k_] := If[ k == 1, 0, For[m0 = 1, True, m0++, If[k(k-1)/2 == Sum[ s[i], {i, 1, m0}], Return[m0]]]]; n[k_] := For[n0 = m[k]+1, True, n0++, If[Sum[s[i], {i, m[k]+1, n0}] == k, Return[n0]]]; a[k_] := a[k] = If[k>24, a[k-24], Table[ s[i], {i, m[k]+1, n[k]}] // FromDigits]; Array[a, 36] (* Jean-François Alcover, Mar 13 2016 *)

A118383 Unrefined Orloj clock sequences; row n sums to n.

Original entry on oeis.org

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

Offset: 1

Franklin T. Adams-Watters, Apr 26 2006

An Orloj clock sequence is a finite sequence of positive integers that, when iterated, can be grouped so that the groups sum to successive natural numbers. There is one unrefined sequence whose values sum to each n; all other Orloj clock sequences summing to n can be obtained by refining this one. Refining means splitting one or more terms into values summing to that term. (The unrefined sequence for n = 2^k*(2m-1) is the sequence for 2m-1 repeated 2^k times, but any single refinement - possible unless m = 1 - will produce an aperiodic sequence summing to n.) The Orloj clock sequence is the one summing to 15: 1,2,3,4,3,2, with a beautiful up and down pattern.

			For a sum of 5, we have 1,2,2, which groups as 1, 2, 2+1, 2+2, 1+2+2, 1+2+2+1, .... This could be refined by splitting the second 2, to give the sequence 1,2,1,1; note that when this is grouped, the two 1's from the refinement always wind up in the same sum.
The array starts:
1;
1, 1;
1, 2;
1, 1, 1, 1;
1, 2, 2;
1, 2, 1, 2;
1, 2, 3, 1.

Wikipedia, Prague astronomical clock

Cf. A028355, A118382.

Length of row n is A117484(n).

{Orloj(n) = my(found,tri,i,last,r); found = vector(n,i,0); found[n] = 1; tri = 0; for(i = 1, if(n%2==0,n-1,n\2), tri += i; if(tri > n, tri -= n); found[tri] = 1); last = 0; r = []; for(i = 1, n, if(found[i], r = concat(r, [i-last]); last = i)); r}
for (n=1,10,print(Orloj(n)))

Let b(i),0<=i

A117704 Least refined sequence that can be grouped to sum to either natural numbers or odd numbers.

Original entry on oeis.org

1, 2, 1, 2, 3, 1, 5, 1, 5, 4, 3, 8, 9, 4, 6, 9, 2, 12, 3, 10, 9, 5, 15, 1, 15, 8, 9, 16, 2, 19, 6, 14, 15, 6, 22, 3, 20, 13, 11, 24, 1, 26, 10, 17, 22, 6, 29, 6, 24, 19, 12, 32, 1, 32, 15, 19, 30, 5, 36, 10, 27, 26, 12, 39, 4, 36, 21, 20, 39, 3, 43, 15, 29, 34, 11, 46, 8, 39, 28, 20, 49, 50

Offset: 1

Franklin T. Adams-Watters, Apr 27 2006

In other words, least common refinement of the natural numbers and the odd numbers.

			As natural numbers: 1,2,1+2,3+1,5,1+5,4+3,8,...
As odd numbers: 1,2+1,2+3,1+5+1,5+4,3+8,...

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A118382, A005214, A000027, A005408.

a117704 n = a117704_list !! (n-1)
a117704_list = 1 : zipWith (-) (tail a005214_list) a005214_list
-- Reinhard Zumkeller, Aug 03 2011

a(n) = A005214(n) - A005214(n-1).

A117510 Index (sum) of first Orloj clock sequence containing n.

Original entry on oeis.org

1, 3, 7, 11, 19, 21, 35, 63, 45, 95, 99, 93, 119, 105, 165, 429, 273, 285, 581, 385, 345, 495, 749, 651, 665, 1001, 693, 1155, 1197, 1155, 1287, 1463, 693, 1935, 1659, 2331, 2145, 1995, 2915, 2415, 3399, 2115, 4785, 3157, 3519, 5225, 5481, 2871, 5115, 6045

Offset: 1

Franklin T. Adams-Watters, Apr 26 2006

nonn

a(n) >= n(n+1)/2. This is equality for n = 1,2,6,9,14. Are there any others?

Cf. A118382.

A120449 Array by antidiagonals of all primitive Orloj clock striking sequences.

Original entry on oeis.org

1, 11, 1, 111, 2, 1, 1111, 12, 2, 1, 11111, 121, 21, 2, 1, 111111, 212, 22, 3, 2, 1, 1111111, 1212, 122, 112, 3, 2, 1, 11111111, 12121, 1221, 311, 31, 12, 11, 1, 111111111, 21212, 2212, 231, 23, 4, 3, 2, 1, 1111111111, 121212, 21221, 1231, 312

Offset: 1

Franklin T. Adams-Watters, Jul 19 2006

This is the sequences of strikes at each hour, represented by concatenation of the digits. The repeating pattern for each row is in A118382. This table eventually contains non-decimal digits. Row 47 is the first one containing a non-decimal digit.

			The table starts:
1 11 111 1111 11111 111111 ...
1 2 12 121 212 1212 ...
1 2 21 22 122 1221 ...
1 2 3 112 311 231 ...
1 2 3 31 23 312 ...

Cf. A118382, A117510, rows A000042, A028359, A028355.

Showing 1-7 of 7 results.

cp's OEIS Frontend

A028356 Simple periodic sequence underlying clock sequence A028354.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Python

Sage

Formula

Extensions

A028355 How the astronomical clock ("Orloj") in Prague would strike 1,2,3,...,24,25,.. (digits follow 12343212343... (A028356), n-th group adds to n).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Formula

A028354 How the astronomical clock ("Orloj") in Prague strikes the hours (digits follow 12343212343... (A028356), n-th group adds to n).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

A118383 Unrefined Orloj clock sequences; row n sums to n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A117704 Least refined sequence that can be grouped to sum to either natural numbers or odd numbers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Formula

A117510 Index (sum) of first Orloj clock sequence containing n.

Views

Author

Keywords

Comments

Crossrefs

A120449 Array by antidiagonals of all primitive Orloj clock striking sequences.

Views

Author

Keywords

Comments

Examples

Crossrefs