A008620 -id:A008620 - cp's OEIS frontend

A175676 a(n) = binomial(n,3) mod n.

0, 0, 1, 0, 0, 2, 0, 0, 3, 0, 0, 4, 0, 0, 5, 0, 0, 6, 0, 0, 7, 0, 0, 8, 0, 0, 9, 0, 0, 10, 0, 0, 11, 0, 0, 12, 0, 0, 13, 0, 0, 14, 0, 0, 15, 0, 0, 16, 0, 0, 17, 0, 0, 18, 0, 0, 19, 0, 0, 20, 0, 0, 21, 0, 0, 22, 0, 0, 23, 0, 0, 24, 0, 0, 25, 0, 0, 26, 0, 0, 27, 0, 0, 28, 0, 0, 29, 0, 0, 30, 0, 0, 31, 0

Offset: 1

Zak Seidov, Aug 07 2010

Number of partitions of n+3 into 3 parts that are in arithmetic progression. - Wesley Ivan Hurt, Dec 07 2020

Altug Alkan, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,2,0,0,-1).

Cf. A007290.

List([1..100],n->Binomial(n,3) mod n); # Muniru A Asiru, Apr 05 2018

Mathematica
```
Table[Mod[Binomial[n,3],n],{n,150}]
```

a(n)=if(n%3, 0, n/3); \\ Charles R Greathouse IV, Sep 02 2015 [Corrected by Altug Alkan, Apr 02 2018]

a(n)=!(n%3)*(1-n)\-3; \\ Altug Alkan, Apr 02 2018

a(n) = n/3 if n==0 (mod 3) else a(n) = 0.
G.f.: x^3 / ( (x-1)^2*(1+x+x^2)^2 ). - R. J. Mathar, Mar 11 2011
a(n) = A008620(n-1)*A079978(n). - Bruno Berselli, Jun 22 2012
a(n) = (n + 2*n*cos((2*n*Pi)/3))/9. - Kritsada Moomuang, Apr 02 2018

A259094 From the Lecture Hall Theorem: array read by antidiagonals: T(n,k) = number of partitions of n into odd parts of size < 2k.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 2, 2, 3, 3, 1, 1, 1, 1, 2, 2, 3, 4, 3, 1, 1, 1, 1, 2, 2, 3, 4, 4, 3, 1, 1, 1, 1, 2, 2, 3, 4, 5, 5, 4, 1, 1, 1, 1, 2, 2, 3, 4, 5, 6, 6, 4, 1, 1, 1, 1, 2, 2, 3, 4, 5, 6, 7, 7, 4, 1, 1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 9, 8, 5, 1

Offset: 1

N. J. A. Sloane, Jun 19 2015

The Lecture Hall Theorem states that (the number of partitions (d1,d2,...,dn) of m such that 0 <= d1/1 <= d2/2 <= ... <= dn/n) equals (the number of partitions of m into odd parts less than 2n).

			The array begins:
  1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1 ...
  1,1,1,2,2,2,3,3,3,4,4,4,5,5,5,6,6,6,7,7,7,8,8,8,9,9,9,10,10,10,11,11,11,12,12,12,13,13,13,14 ...
  1,1,1,2,2,3,4,4,5,6,7,8,9,10,11,13,14,15,17,18,20,22,23,25,27,29,31,33,35,37,40,42,44,47,49,52,55 ...
  1,1,1,2,2,3,4,5,6,7,9,10,12,14,16,19,21,24,27,30,34,38,42,46,51,56,61,67,73,79,86,93,100,108,116 ...
  1,1,1,2,2,3,4,5,6,8,10,11,14,16,19,23,26,30,35,40,45,52,58,65,74,82,91,102,113,124,138,151,165,182 ...
  1,1,1,2,2,3,4,5,6,8,10,12,15,17,21,25,29,34,40,46,53,62,70,80,91,103,116,131,147,164,184,204,227 ...
  1,1,1,2,2,3,4,5,6,8,10,12,15,18,22,26,31,36,43,50,58,68,78,90,103,118,134,153,173,195,220,247,277 ...
  1,1,1,2,2,3,4,5,6,8,10,12,15,18,22,27,32,37,45,52,61,72,83,96,111,128,146,168,191,217,247,279,314 ...
  ...
The successive antidiagonals are:
  [1]
  [1, 1]
  [1, 1, 1]
  [1, 1, 1, 1]
  [1, 1, 1, 2, 1]
  [1, 1, 1, 2, 2, 1]
  [1, 1, 1, 2, 2, 2, 1]
  [1, 1, 1, 2, 2, 3, 3, 1]
  [1, 1, 1, 2, 2, 3, 4, 3, 1]
  [1, 1, 1, 2, 2, 3, 4, 4, 3, 1]
  [1, 1, 1, 2, 2, 3, 4, 5, 5, 4, 1]
  [1, 1, 1, 2, 2, 3, 4, 5, 6, 6, 4, 1]
  [1, 1, 1, 2, 2, 3, 4, 5, 6, 7, 7, 4, 1]
  [1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 9, 8, 5, 1]
  ...

Seiichi Manyama, Antidiagonals n = 1..140, flattened
M. Bousquet-Mélou, K. Eriksson, Lecture hall partitions, The Ramanujan J. 1 (1997) 101-111.
M. Bousquet-Mélou, K. Eriksson, Lecture hall partitions II, The Ramanujan J. 1 (1997) 165-185.
Mireille Bousquet-Mélou, Kimmo Eriksson, A Refinement of the Lecture Hall Theorem, Journal of Combinatorial Theory, Series A, Volume 86, Issue 1, April 1999, Pages 63-84
Niklas Eriksen, A simple bijection between lecture hall partitions and partitions into odd integers Formal Power Series and Algebraic Combinatorics. 2002.
Robin Whitty, The Lecture Hall Partition Theorem
A. J. Yee, On combinatorics of lecture hall partitions, The Ramanujan J. 5 (2001) 247-262.

Many rows of the array are already in the OEIS: A008620, A008672, A008673, A008674, A008675, A287997, A287998, A288000, A288001.

G:=n->mul(1/(1-q^(2*i-1)),i=1..n);
M:=41;
G2:=n->seriestolist(series(G(n),q,M));
for n from 1 to 10 do lprint(G2(n)); od:
H:=n->[seq(G2(n-i+1)[i],i=1..n)];
for n from 1 to 14 do lprint(H(n)); od:

G[n_] := Product[1/(1-q^(2*i-1)), {i, 1, n}];
M = 41;
G2[n_] := CoefficientList[Series[G[n], {q, 0, M}], q];
For[n = 1, n <= 10, n++; Print[G2[n]]];
H[n_] := Table[G2[n-i+1][[i]], {i, 1, n}];
Reap[For[n = 1, n <= 14, n++, Print[H[n]]; Sow[H[n]]]][[2, 1]] // Flatten (* Jean-François Alcover, Jun 04 2017, translated from Maple *)

A152828 Triangle read by rows, A007318 rows repeated three times .

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 1, 4, 6, 4, 1, 1, 4, 6, 4, 1, 1, 5, 10, 10, 5, 1, 1, 5, 10, 10, 5, 1, 1, 5, 10, 10, 5, 1, 1, 6, 15, 20, 15, 6, 1, 1, 6, 15, 20, 15, 6, 1, 1, 6, 15, 20, 15, 6, 1, 1, 7, 21, 35, 35, 21, 7, 1, 1, 7, 21

Offset: 0

Philippe Deléham, Dec 13 2008

Diagonal sums : A079398 . Lengths of row are : 1,1,1,2,2,2,3,3,3,4,4,4,5,5,5,... (A008620) .

			Triangle begins : 1 ; 1 ; 1 ; 1,1 ; 1,1 ; 1,1 ; 1,2,1 ; 1,2,1 ; 1,2,1 ; 1,3,3,1 ; 1,3,3,1 ; 1,3,3,1 ; 1,4,6,4,1 ; ...

Cf. A007318, A152198, A152815

{#,#,#}&/@Table[Binomial[n,k],{n,0,11},{k,0,n}]//Flatten (* Harvey P. Dale, Jul 22 2024 *)

A025767 Expansion of 1/((1-x)(1-x^3)(1-x^4)).

Original entry on oeis.org

1, 1, 1, 2, 3, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 15, 17, 18, 20, 22, 24, 26, 28, 30, 33, 35, 37, 40, 43, 45, 48, 51, 54, 57, 60, 63, 67, 70, 73, 77, 81, 84, 88, 92, 96, 100, 104, 108, 113, 117, 121, 126, 131, 135, 140, 145, 150, 155, 160, 165, 171, 176, 181, 187, 193, 198

Offset: 0

N. J. A. Sloane

Apply the Riordan array (1/(1-x^4),x) to floor((n+3)/3). - Paul Barry, Jan 20 2006
Number of partitions of n into parts 1, 3, and 4. - David Neil McGrath, Aug 30 2014
Also, a(n-4) is equal to the number of partitions mu of n of length 3 such that mu_1-mu_2 is even and mu_2-mu_3 is odd or vice versa (see below example). - John M. Campbell, Jan 29 2016
With four 0's prepended and offset 0, a(n) is the number of partitions of n into four parts whose 2nd and 3rd largest parts are equal. - Wesley Ivan Hurt, Jan 05 2021

			The a(4)=3 partitions of 4 into parts 1, 3, and 4 are (4), (3,1), and (1,1,1,1). - _David Neil McGrath_, Aug 30 2014
From _John M. Campbell_, Jan 29 2016: (Start)
Letting n=12, there are a(n-4)=a(8)=6 partitions mu of n=12 of length 3 such that mu_1-mu_2 is even and mu_2-mu_3 is odd or vice versa:
(10,1,1) |- n
(8,3,1) |- n
(7,3,2) |- n
(6,5,1) |- n
(6,3,3) |- n
(5,5,2) |- n
(End)

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,1,0,-1,0,-1,1).

A008621(n) = A002265(n+4) = a(n) - a(n-3).

[Floor(n^2/24 + n/3 + 1): n in [0..70]]; // Vincenzo Librandi, Aug 31 2014

A056594 := proc(n) op(1+(n mod 4),[1,0,-1,0]) ; end proc:
A061347 := proc(n) op(1+(n mod 3),[-2,1,1]) ; end proc:
A025767 := proc(n) n^2/24+n/3+83/144+(-1)^n/16 +A061347(n+1)/9 +A056594(n)/4 ; end proc: # R. J. Mathar, Mar 31 2011

Table[Floor[n^2/24 + n/3 + 1], {n, 0, 60}] (* Vincenzo Librandi, Aug 31 2014 *)

PARI
```
a(n)=if(n<0,0,(n^2+8*n)\24+1)
```

{a(n) = round( ((n + 4)^2 - 1) / 24 )}; /* Michael Somos, Nov 09 2007 */

Vec(1/((1-x)*(1-x^3)*(1-x^4)) + O(x^80)) \\ Michel Marcus, Jan 29 2016

G.f.: 1/((1-x)*(1-x^3)*(1-x^4)).
a(n) = floor(n^2/24+n/3+1).
a(n) = Sum_{k=0..floor(n/4)} floor((n-4*k+3)/3). - Paul Barry, Jan 20 2006
Euler transform of length 4 sequence [1, 0, 1, 1]. - Michael Somos, Nov 09 2007
a(n) = a(-8 - n) for all n in Z. - Michael Somos, Nov 09 2007
a(n) = n^2/24 + n/3 + 83/144 + (-1)^n/16 + A061347(n+1)/9 + A056594(n)/4. - R. J. Mathar, Mar 31 2011
a(n) = a(n-1)+a(n-3)-a(n-5)-a(n-7)+a(n-8). - David Neil McGrath, Aug 30 2014
a(n) = Sum_{k=1..floor((n+4)/4)} Sum_{j=k..floor((n+4-k)/3)} Sum_{i=j..floor((n+4-j-k)/2)} [j = i], where [ ] is the Iverson bracket. - Wesley Ivan Hurt, Jan 17 2021
a(n)-a(n-1) = A008679(n). - R. J. Mathar, Jun 23 2021
a(n)-a(n-4) = A008620(n). - R. J. Mathar, Jun 23 2021

A246689 Expansion of e.g.f. 1/(1 - x^3)^(1 + 1/x + 1/x^2).

Original entry on oeis.org

1, 1, 3, 13, 61, 381, 2791, 22513, 210393, 2183401, 24575851, 305067621, 4097726293, 58876485253, 910581818511, 15005958062761, 261751577640241, 4844661893762193, 94564968066402643, 1938366513866527741, 41760228574294689261, 941821175462309114701

Offset: 0

Peter Bala, Sep 01 2014

Compare with A193281.

Seiichi Manyama, Table of n, a(n) for n = 0..448

Cf. A008620, A193281.

seq(coeftayl(n!/(1-x^3)^(1+1/x+1/x^2), x = 0, n), n = 0..10);

CoefficientList[Series[1/(1 - x^3)^(1 + 1/x + 1/x^2), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Sep 01 2014 *)

my(x='x+O('x^66)); Vec(serlaplace(1/(1 - x^3)^(1 + 1/x + 1/x^2))) \\ Joerg Arndt, Sep 01 2014

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!*sum(j=1, i, j/((j+2)\3)*v[i-j+1]/(i-j)!)); v; \\ Seiichi Manyama, Apr 30 2022

E.g.f.: A(x) = 1/(1 - x^3)^(1 + 1/x + 1/x^2)  = exp( Sum_{n>=1} x^n/A008620(n-1) ) = 1 + x + 3*x^2/2! + 13*x^3/3! + 61*x^4/4! + ....
A(x) = Sum_{n>=0} (x^n/n!)*Product {k = 1..n} (1 + x + k*x^2).
It appears that a(n) == 1 (mod n*(n-1)).
a(n) ~ n! * (n^2 / 54) * (1 + 6*log(n)/n). - Vaclav Kotesovec, Sep 01 2014
a(0) = 1; a(n) = (n-1)! * Sum_{k=1..n} k/A008620(k-1) * a(n-k)/(n-k)!. - Seiichi Manyama, Apr 30 2022

A039946 Expansion of Molien series for 8-dimensional complex Clifford group of genus 3 and order 743178240.

Original entry on oeis.org

1, 1, 2, 5, 9, 16, 31, 53, 89, 152, 245, 384, 601, 911, 1351, 1986, 2856, 4037, 5653, 7791, 10592, 14268, 18990, 24999, 32643, 42218, 54112, 68869, 86971, 109014, 135812, 168101, 206769, 252990, 307849, 372616, 448934, 538348

Offset: 0

E. M. Rains

			G.f. = 1 + x^8 + 2*x^16 + 5*x^24 + 9*x^32 + 16*x^40 + 31*x^48 + ...

Jean-François Alcover, Table of n, a(n) for n = 0..499
Index entries for Molien series
Index entries for linear recurrences with constant coefficients, signature (1,1,1,-2,-1,0,1,-1,1,0,0,-1,1,2,1,-3,-2,0,2,1,-1,0,0,-1,1,2,0,-2,-3,1,2,1,-1,0,0,1,-1,1,0,-1,-2,1,1,1,-1).

Bisection of A027633. Cf. A008621, A008718, A024186, A008620, A028288, A043330, A051354.

f(x):= (1 +x^3 +3*x^4 +3*x^5 +6*x^6 +8*x^7 +12*x^8 +18*x^9 +25*x^10 +29*x^11 +40*x^12 +50*x^13 +58*x^14 +69*x^15 +80*x^16 +85*x^17 +96*x^18 +104*x^19 +107*x^20 +109*x^21 +112*x^22 +109*x^23+107*x^24 +104*x^25 +96*x^26 +85*x^27 +80*x^28 +69*x^29 +58*x^30 +50*x^31 +40*x^32 +29*x^33 +25*x^34 +18*x^35 +12*x^36 +8*x^37 +6*x^38 +3*x^39 +3*x^40 +x^41 +x^44) / ( (1-x)^2*(1-x^3)^4*(1-x^5)^2*(1 +x +2*x^3 +2*x^4 + x^5 +4*x^6 +2*x^7 +x^8 +5*x^9 +2*x^10 +2*x^11 +5*x^12 +x^13 +2*x^14 + 4*x^15 +x^16 +2*x^17 +2*x^18 +x^20 +x^21) ); seq(coeff(series(f(x), x, n+1), x, n), n = 0..40);

CoefficientList[Series[(1+x^3+3*x^4+3*x^5+6*x^6+8*x^7+12*x^8+18*x^9+25*x^10 + 29*x^11+40*x^12+50*x^13+58*x^14+69*x^15+80*x^16+85*x^17+96*x^18+104*x^19 + 107*x^20+109*x^21+112*x^22+109*x^23+107*x^24+104*x^25+96*x^26+85*x^27+80*x^28 +69*x^29+58*x^30+50*x^31+40*x^32+29*x^33+25*x^34+18*x^35+12*x^36 + 8*x^37 + 6*x^38+3*x^39+3*x^40+x^41+x^44)/((1-x)^2*(1-x^3)^4*(1-x^5)^2*(1+x+2*x^3+2*x^4 +x^5+4*x^6+2*x^7+x^8+5*x^9+2*x^10+2*x^11+5*x^12+x^13+2*x^14+4*x^15+x^16+2*x^17 +2*x^18+x^20+x^21)), {x,0,40}], x] (* G. C. Greubel, Feb 01 2020 *)
LinearRecurrence[{1,1,1,-2,-1,0,1,-1,1,0,0,-1,1,2,1,-3,-2,0,2,1,-1,0,0,-1,1,2,0,-2,-3,1,2,1,-1,0,0,1,-1,1,0,-1,-2,1,1,1,-1},{1,1,2,5,9,16,31,53,89,152,245,384,601,911,1351,1986,2856,4037,5653,7791,10592,14268,18990,24999,32643,42218,54112,68869,86971,109014,135812,168101,206769,252990,307849,372616,448934,538348,642630,764021,904658,1066943,1253876,1468340,1713529},40] (* Harvey P. Dale, Jul 04 2021 *)

G.f.: (1 +x^24 +3*x^32 +3*x^40 +6*x^48 +8*x^56 +12*x^64 +18*x^72 +25*x^80 +29*x^88 +40*x^96 +50*x^104 +58*x^112 +69*x^120 +80*x^128 +85*x^136 +96*x^144 +104*x^152 +107*x^160 +109*x^168 +112*x^176 +109*x^184 +107*x^192 +104*x^200 +96*x^208 +85*x^216 +80*x^224 +69*x^232 +58*x^240 +50*x^248 +40*x^256 +29*x^264 +25*x^272 +18*x^280 +12*x^288 +8*x^296 +6*x^304 +3*x^312 +3*x^320 +x^328 +x^352) / ( (1-x^8)^2*(1-x^24)^4*(1-x^40)^2*(1 +x^8 +2*x^24 +2*x^32 + x^40 +4*x^48 +2*x^56 +x^64 +5*x^72 +2*x^80 +2*x^88 +5*x^96 +x^104 +2*x^112 + 4*x^120 +x^128 +2*x^136 +2*x^144 +x^160 +x^168) ), nonzero terms.

G.f.: (1 +x^3 +3*x^4 +3*x^5 +6*x^6 +8*x^7 +12*x^8 +18*x^9 +25*x^10 +29*x^11 +40*x^12 +50*x^13 +58*x^14 +69*x^15 +80*x^16 +85*x^17 +96*x^18 +104*x^19 +107*x^20 +109*x^21 +112*x^22 +109*x^23+107*x^24 +104*x^25 +96*x^26 +85*x^27 +80*x^28 +69*x^29 +58*x^30 +50*x^31 +40*x^32 +29*x^33 +25*x^34 +18*x^35 +12*x^36 +8*x^37 +6*x^38 +3*x^39 +3*x^40 +x^41 +x^44) / ( (1-x)^2*(1-x^3)^4*(1-x^5)^2*(1 +x +2*x^3 +2*x^4 + x^5 +4*x^6 +2*x^7 +x^8 +5*x^9 +2*x^10 +2*x^11 +5*x^12 +x^13 +2*x^14 + 4*x^15 +x^16 +2*x^17 +2*x^18 +x^20 +x^21) ). - G. C. Greubel, Feb 01 2020

Typo in reduced g.f.s. corrected by Georg Fischer, Apr 18 2020

A051354 Expansion of Molien series for 16-dimensional complex Clifford group of genus 4 and order 97029351014400.

Original entry on oeis.org

1, 1, 2, 7, 19, 52, 172, 550, 1782, 5845, 18508, 56345, 164157, 454518, 1196924, 3003750, 7198311, 16523847, 36447873, 77478005, 159172517, 316874035, 612729396, 1153359711, 2117566545, 3798941401, 6670327291, 11479693332, 19390588953, 32185179449, 52553840336

Offset: 0

N. J. A. Sloane

Oura gives an explicit formula for the Molien series that produces A027672; the present sequence is the subsequence formed from the terms whose exponents are multiples of 8 (that is, every other term of A027672). In other words, the present Molien series is (f(x)+f(z*x))/2, where z = exp(2*Pi*I/8) and f(x) is the Molien series for the group H_4 given explicitly by Oura in Theorem 4.1.

			1 + t^8 + 2*t^16 + 7*t^24 + 19*t^32 + 52*t^40 + 172*t^48 + ...

Ray Chandler, Table of n, a(n) for n = 0..1000
Ray Chandler, Mathematica program
G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
M. Oura, The dimension formula for the ring of code polynomials in genus 4, Osaka J. Math. 34 (1997), 53-72.
Index entries for linear recurrences with constant coefficients, signature (3, -2, 0, -3, 5, -2, -1, -1, 8, -7, -2, 2, 7, -7, -3, 1, 9, -11, 4, 3, 5, -9, -1, 0, 13, -15, 0, 4, 6, -9, 0, 8, 9, -18, -2, 12, -4, -4, -3, 6, 6, -8, -7, 18, -1, -6, -13, 13, 6, -14, -10, 30, -10, -10, -4, 22, -5, -6, -15, 28, -15, -6, -5, 22, -4, -10, -10, 30, -10, -14, 6, 13, -13, -6, -1, 18, -7, -8, 6, 6, -3, -4, -4, 12, -2, -18, 9, 8, 0, -9, 6, 4, 0, -15, 13, 0, -1, -9, 5, 3, 4, -11, 9, 1, -3, -7, 7, 2, -2, -7, 8, -1, -1, -2, 5, -3, 0, -2, 3, -1).
Index entries for Molien series

Cf. A027672, A003956, A008621, A008718, A024186, A008620, A028288, A043330.

Mathematica
```
(* See link for Mathematica program. *)
```

a(n) = A027672(2*n).

Edited by Georg Fischer, Jan 24 2021

A330779 Lexicographically earliest sequence of positive integers such that for any v > 0, the value v appears up to v times, and the associate function f defined by f(n) = Sum_{k = 1..n} a(k) * i^k for n >= 0 is injective (where i denotes the imaginary unit).

Original entry on oeis.org

1, 2, 2, 3, 3, 3, 4, 4, 4, 5, 4, 6, 5, 5, 6, 5, 5, 7, 6, 6, 7, 6, 6, 8, 7, 7, 7, 8, 7, 7, 8, 8, 8, 10, 9, 8, 8, 9, 8, 10, 10, 9, 11, 9, 9, 11, 9, 9, 10, 9, 9, 12, 10, 11, 11, 11, 12, 10, 13, 10, 10, 13, 10, 10, 12, 11, 11, 12, 11, 11, 11, 14, 11, 13, 12, 13

Offset: 1

Rémy Sigrist, Dec 31 2019

nonn

The variant of this sequence where each value can only appear up to once, twice or three times corresponds to A000027, A008619 and A008620 respectively.

Graphically, the representation of f resembles a windmill; the variant of f where we allow the value v to appear 3*v times resembles a butterfly (see illustrations in Links section).

			The first terms, alongside the corresponding values of f(n), are:
  n   a(n)  f(n)
  --  ----  ------
   0  N/A        0
   1     1       i
   2     2    -2+i
   3     2    -2-i
   4     3     1-i
   5     3   1+2*i
   6     3  -2+2*i
   7     4  -2-2*i
   8     4   2-2*i
   9     4   2+2*i
  10     5  -3+2*i
  11     4  -3-2*i
  12     6   3-2*i
See also illustration in Links section.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Illustration of first steps
Rémy Sigrist, Colored representation of f(n) for n = 0..1000000 in the complex plane (where the color is function of n)
Rémy Sigrist, Colored representation of the variant where the value v can appear up to 3*v times
Rémy Sigrist, Colored representation of the variant where the value v can appear up to A000265(v) times
Rémy Sigrist, Colored representation of the variant where the value v can appear up to prime(v) times
Rémy Sigrist, PARI program for A330779

See A331002 and A331003 for the real and imaginary parts of f, respectively.
See A330780 for another variant.
Cf. A000265.

PARI
```
See Links section.
```

A005041 A self-generating sequence.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 17, 17, 18, 18, 18, 18

Offset: 0

N. J. A. Sloane, Jeffrey Shallit

See A008620 for run lengths: each k occurs A008620(k+2) times. - Reinhard Zumkeller, Mar 16 2012

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
James Propp, Problem 1047, Math. Mag., 52 (1979), 265.
Jeffrey Shallit, Letter to N. J. A. Sloane, Nov 10 1979. Attached: James Propp, Problem 1047, Math. Mag., 52 (1979), 265. [Annotated scanned copy]
Aaron Snook, Augmented Integer Linear Recurrences, Thesis, 2012. - From _N. J. A. Sloane_, Dec 19 2012

Cf. A005038, A005039, A005040, A005043, A005044, A055086, A001462, A082462, A024417, A084500.

a005041 n = a005041_list !! n
a005041_list = 1 : f 1 1 (tail ts) where
   f y i gs'@((j,a):gs) | i < j  = y : f y (i+1) gs'
                        | i == j = a : f a (i+1) gs
   ts = [(6*k + 3*k*(k-1) `div` 2 + r*(k+2), 3*k+r+1) |
         k <- [0..], r <- [0,1,2]]
-- Reinhard Zumkeller, Mar 16 2012

Table[n+1, {n, 0, 20}, {Ceiling[(n+1)/3]+1}] // Flatten (* Jean-François Alcover, Dec 10 2014 *)

For any k in {0, 1, 2, ...} and r in {0, 1, 2}, we have: if n = 6*k + (3/2)*k*(k-1) + r*(k+2), then a(n) = 3*k + r + 1. E.g., for k=3 and r=1, we have n = 6*3 + (3/2)*3*(3-1) + 1*(3+2) = 32 and so a(32) = 3*3 + 1 + 1 = 11. - Francois Jooste (phukraut(AT)hotmail.com), Mar 12 2002

More terms from Samuel Hilliard (sam_spade1977(AT)hotmail.com), Apr 11 2004

A086161 Number of monomial ideals in two variables x, y that are Artinian, integrally closed, of colength n and contain x^2.

Original entry on oeis.org

1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 7, 8, 8, 8, 9, 9, 9, 10, 10, 10, 11, 11, 11, 12, 12, 12, 13, 13, 13, 14, 14, 14, 15, 15, 15, 16, 16, 16, 17, 17, 17, 18, 18, 18, 19, 19, 19, 20, 20, 20, 21, 21, 21, 22, 22, 22, 23, 23, 23, 24, 24, 24, 25, 25, 25

Offset: 0

Jan Snellman, Aug 25 2003

Alternatively, "concave partitions" of n with at most 2 parts, where a concave partition is defined by demanding that the monomial ideal, generated by the monomials whose exponents do not lie in the Ferrers diagram of the partition, is integrally closed.

G. E. Andrews, The Theory of Partitions, Addison-Wesley Publishing Company, 1976.
M. Paulsen and J. Snellman, Enumerativa egenskaper hos konkava partitioner (in Swedish), Department of Mathematics, Stockholm University.

V. Crispin Quinonez, Integrally closed monomial ideals and powers of ideals, Research Reports in Mathematics Number 7 2002, Department of Mathematics, Stockholm University.
Jan Snellman and Michael Paulsen, Enumeration of Concave Integer Partitions, J. Integer Seqs., Vol. 7, 2004.
Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

Cf. A008620, A084913, A086162, A086163.

Vec((1+x^2-x^3)/((1-x)*(1-x^3)) + O(x^80)) \\ Michel Marcus, May 22 2015

G.f.: (1 + x^2 - x^3)/((1 - x)*(1 - x^3)).
a(n) = A008620(n+1). - R. J. Mathar, Sep 12 2008
E.g.f.: (3*exp(x)*(3 + x) - 2*sqrt(3)*exp(-x/2)*sin(sqrt(3)*x/2))/9. - Stefano Spezia, Feb 11 2023

cp's OEIS Frontend

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A259094 From the Lecture Hall Theorem: array read by antidiagonals: T(n,k) = number of partitions of n into odd parts of size < 2k.

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