A028355 -id:A028355 - cp's OEIS frontend

A007980 Expansion of (1+x^2)/((1-x)^2*(1-x^3)).

1, 2, 4, 7, 10, 14, 19, 24, 30, 37, 44, 52, 61, 70, 80, 91, 102, 114, 127, 140, 154, 169, 184, 200, 217, 234, 252, 271, 290, 310, 331, 352, 374, 397, 420, 444, 469, 494, 520, 547, 574, 602, 631, 660, 690, 721, 752, 784, 817, 850, 884, 919, 954, 990, 1027, 1064

Offset: 0

N. J. A. Sloane

Molien series for ternary self-dual codes over GF(3) of length 12n containing 11...1.
(1+x)*(1+x^2) / ((1-x)*(1-x^2)*(1-x^3)) is the Poincaré series [or Poincare series] (or Molien series) for H^*(O_3(q); F_2).
a(n) is the position of the n-th triangular number in the running sum of the (pseudo-Orloj) sequence 1,2,1,2,1,2,1...., cf. A028355. - Wouter Meeussen, Mar 10 2002
a(n) = [a(n-1) + (number of even terms so far in the sequence)]. Example: 14 is [10 + 4 even terms so far in the sequence (they are 0,2,4,10)]. See A096777 for the same construction with odd integers. - Eric Angelini, Aug 05 2007
The number of partitions of 2*n into at most 3 parts. - Colin Barker, Mar 31 2015
Also a(n) equals the number of linearly-independent terms at 2n-th order in the power series expansion of a trigonal Rotational Energy Surface. An optimal basis for the expansion follows either decomposition: g1(x) = (1+x)(1+x^2)g2(x) or g1(x) = (1+x^2)x^(-1)g3(x), where g1(x), g2(x), g3(x) are the generating functions for sequences A007980, A001399, A001840. - Bradley Klee, Aug 06 2015
Also a(n) equals the number of linearly-independent terms at 4n-th order in the power series expansion of the symmetrized weight enumerator of a self-dual code of length n over Z4 that contains a vector (+/-)1^n and has all norms divisible by 8. An optimal basis for the expansion follows the decomposition: g1(x) = (1+x)(1+x^2)g2(x) where g1(x), g2(x) are the generating functions for sequences A007980, A001399. (Cf. Calderbank and Sloane, Corollary 5.) - Bradley Klee, Aug 06 2015
Also, a(n) is equal to the number of partitions of 2n+3 of length 3. Letting n=4, there are a(4)=10 partitions of 2n+3=11 of length 3: (9,1,1), (8,2,1), (7,3,1), (7,2,2), (6,4,1), (6,3,2), (5,5,1), (5,4,2), (5,3,3), (4,4,3). - John M. Campbell, Jan 30 2016
a(n) is the number of partitions of n into parts 1 (of two kinds), part 2 (occurring at most once), and parts 3. - Joerg Arndt, Oct 12 2020
Conjecture: a(n) is the maximum number of pieces a triangle can be cut into by n cevians. - Anton Zakharov, Apr 04 2017
Also, a(n) is the number of graphs which are double-triangle descendants of K_5 with n+6 triangles and 3 more vertices than triangles. See Laradji/Mishna/Yeats reference, proposition 3.6 for details. - Karen A. Yeats, Feb 21 2020

			G.f. = 1 + 2*x + 4*x^2 + 7*x^3 + 10*x^4 + 14*x^5 + 19*x^6 + 24*x^7 + ...

A. Adem and R. J. Milgram, Cohomology of Finite Groups, Springer-Verlag, 2nd. ed., 2004; p. 233.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Raghavendra N. Bhat, Cristian Cobeli, and Alexandru Zaharescu, A lozenge triangulation of the plane with integers, arXiv:2403.10500 [math.NT], 2024.
A. R. Calderbank and N. J. A. Sloane, Double circulant codes over Z_4, J. Algeb. Combin., 6 (1997) 119-131 (Abstract, pdf, ps).
Mohamed Laradji, Marni Mishna, and Karen Yeats, Some results on double triangle descendants of K_5, arXiv:1904.06923 [math.CO], 2019.
C. L. Mallows and N. J. A. Sloane, Weight enumerators of self-orthogonal codes over GF(3), SIAM J. Alg. Discrete Methods, 2 (1981), 452-460.
Paul Tabatabai and Dieter P. Gruber, Knights and Liars on Graphs, J. Int. Seq., Vol. 24 (2021), Article 21.5.8.
Anton Zakharov, cevians.
Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).
Index entries for Molien series.
Index entries for two-way infinite sequences.

Cf. A000326, A001399, A001840, A002378, A003215, A004396, A005449, A007980.

Cf. A008796, A028355, A049450, A069905, A096777, A192736, A211540, A238705.

with (combinat):seq(count(Partition((2*n+1)), size=3), n=1..56); # Zerinvary Lajos, Mar 28 2008

Table[Ceiling[n (n+1)/3], {n, 56}]
CoefficientList[Series[(1+x^2)/((1-x)^2*(1-x^3)),{x,0,60}],x] (* Vincenzo Librandi, Feb 25 2012 *)
a[ n_] := Quotient[ n^2, 3] + n + 1; (* Michael Somos, Aug 23 2015 *)
LinearRecurrence[{2,-1,1,-2,1},{1,2,4,7,10},60] (* Harvey P. Dale, Aug 24 2016 *)

{a(n) = if( n<-1, a(-3-n), polcoeff( (1 + x^2) / ( (1 - x)^2 * (1 - x^3)) + x*O(x^n), n))}; /* Michael Somos, Jun 07 2003 */

{a(n) = n^2\3 + n+1}; /* Michael Somos, Aug 23 2015 */

a(n) = #partitions(2*n, ,[1,3]); \\ Michel Marcus, Feb 12 2016

a(n) = #partitions(2*n+3, ,[3,3]); \\ Michel Marcus, Feb 12 2016

G.f.: (1 + x^2) / ((1 - x)^2 * (1 - x^3)). - Michael Somos, Jun 07 2003
a(n) = a(n-1) + a(n-3) -a(n-4) + 2 = a(-3-n) for all n in Z. - Michael Somos, Jun 07 2003
a(n) = ceiling((n+1)*(n+2)/3). - Paul Boddington, Jan 26 2004
a(n) = A192736(n+1) / (n+1). - Reinhard Zumkeller, Jul 08 2011
From Bruno Berselli, Oct 22 2010: (Start)
a(n) = ((n+1)*(n+2)+(2*cos(2*Pi*n/3)+1)/3)/3 = Sum_{i=1..n+1} A004396(i).
a(n) = 2*a(n-1) - a(n-2) + a(n-3) - 2*a(n-4) + a(n-5) for n>4.
a(n) = A002378(n+1)/3 if 3 divides A002378(n+1), a(n) = (A002378(n)+1)/3 otherwise. (End)
a(n) = A001840(n+1) + A001840(n-1). - R. J. Mathar, Aug 23 2015
From Michael Somos, Aug 23 2015: (Start)
Euler transform of length 4 sequence [2, 1, 1, -1].
a(n) = A001399(2*n) = A008796(2*n) = A008796(2*n + 3) = A069905(2*n + 3) = A211540(2*n + 5).
a(2*n)     = A238705(n+1).
a(3*n - 1) = A049451(n).
a(3*n)     = A003215(n).
a(3*n + 1) = A049450(n+1).
2*a(3*n - 1)  = A005449(n).
2*a(3*n + 1)  = A000326(n+1).
a(n+1) - a(n) = A004396(n+2). (End)
a(n) = floor((n^2+3*n+3)/3). - Giacomo Guglieri, May 01 2019
a(n) = A000212(n) + n+1. - Yuchun Ji, Oct 12 2020
Sum_{n>=0} 1/a(n) = (tanh(Pi/(2*sqrt(3)))-1)*Pi/sqrt(3) + 3. - Amiram Eldar, May 20 2023

A118382 Primitive Orloj clock sequences; row n sums to 2n-1.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 1, 2, 3, 1, 1, 2, 3, 3, 1, 2, 1, 2, 4, 1, 1, 1, 1, 3, 2, 2, 3, 1, 2, 3, 4, 3, 2, 1, 1, 1, 1, 2, 4, 1, 4, 2, 1, 1, 1, 3, 1, 2, 1, 5, 2, 2, 1, 2, 3, 1, 3, 3, 2, 6, 1, 2, 2, 1, 3, 1, 3, 2, 5, 1, 1, 1, 1, 2, 2, 1, 4, 1, 4, 1, 4, 1, 4, 1, 2, 3, 3, 1, 2, 3, 3, 3, 3, 3, 1, 2, 1, 2, 1, 1, 2, 5, 1, 2, 2

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 primitive sequence whose values sum to each odd m; all other sequences can be obtained by repeating and refining these. Refining means splitting one or more terms into values summing to that term. The Orloj clock sequence is the one summing to 15: 1,2,3,4,3,2, with a beautiful up and down pattern.

These are known in some papers as Sindel sequences. It appears that this sequence was submitted prior to the first such publication.

			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;
  1,2;
  1,2,2;
  1,2,3,1;
  1,2,3,3;
  1,2,1,2,4,1;
  ...

Michal Krížek, Alena Šolcová and Lawrence Somer, Construction of Šindel sequences, Comment. Math. Univ. Carolin., 48 (2007), 373-388.

Cf. A028355, A118383. Length of row n is A117484(2n-1) = A000224(2n-1).

{Orloj(n) = local(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}

Let b(i),0<=i

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

A321225 Break the infinite word 1232112321... into chunks so that the sum of the digits in the n-th chunk is n.

Original entry on oeis.org

1, 2, 3, 211, 23, 2112, 3211, 2321, 12321, 123211, 232112, 321123, 21123211, 2321123, 211232112, 321123211, 232112321, 1232112321, 12321123211, 23211232112, 32112321123, 2112321123211, 232112321123, 21123211232112, 32112321123211, 23211232112321, 123211232112321

Offset: 1

Seiichi Manyama, Oct 31 2018

For every n=3*k, a(n) must be divisible by 3 and is therefore a palindrome. - Ivan N. Ianakiev, Nov 01 2018

For every n=3*k, the number of digits of a(n) equals he number of digits of a(n-9)+5 and the starting/ending digits of a(n) and a(n-9) are the same. For any possible natural number m, there are five possible candidate numbers for a(3*k) that are of length m, of which only one, the palindrome, is divisible by 3. - Ivan N. Ianakiev, Nov 02 2018

			1, 2, 3, 2+1+1=4, 2+3=5, 2+1+1+2=6, 3+2+1+1=7, 2+3+2+1=8, 1+2+3+2+1=9, ... .

Seiichi Manyama, Table of n, a(n) for n = 1..1800

Cf. A028355, A028359, A321232.

getd(n) = {my(m = n % 5); if (!m, m = 1); [1, 2, 3, 2, 1][m];}
lista(nn) = {my(k = 1); for (n=1, nn, my (s = 0, list = List(), d); while (s != n, d = getd(k); listput(list, d); s += d; k++;); print1(fromdigits(Vec(list)), ", "););} \\ Michel Marcus, Nov 11 2018

Conjectures from Chai Wah Wu, Apr 18 2024: (Start)
a(n) = 100001*a(n-9) - 100000*a(n-18) for n > 18.
G.f.: x*(10000*x^16 + 20000*x^15 + 30000*x^14 + 21100*x^13 + 23000*x^12 + 21120*x^11 + 32110*x^10 + 23210*x^9 + 12321*x^8 + 2321*x^7 + 3211*x^6 + 2112*x^5 + 23*x^4 + 211*x^3 + 3*x^2 + 2*x + 1)/(100000*x^18 - 100001*x^9 + 1). (End)

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 1-8 of 8 results.

cp's OEIS Frontend

A028359 Two-bell analog of A028355.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

Extensions

A068962 Number of successive terms of A028356 that add to n; or length of n-th term of A028355.

Views

Author

Keywords

References

Links

Crossrefs

Formula

Extensions

A007980 Expansion of (1+x^2)/((1-x)^2*(1-x^3)).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

PARI

PARI

Formula

A118382 Primitive Orloj clock sequences; row n sums to 2n-1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

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

A321225 Break the infinite word 1232112321... into chunks so that the sum of the digits in the n-th chunk is n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

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