A008621 -id:A008621 - cp's OEIS frontend

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

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

A047227 Numbers that are congruent to {1, 2, 3, 4} mod 6.

Original entry on oeis.org

1, 2, 3, 4, 7, 8, 9, 10, 13, 14, 15, 16, 19, 20, 21, 22, 25, 26, 27, 28, 31, 32, 33, 34, 37, 38, 39, 40, 43, 44, 45, 46, 49, 50, 51, 52, 55, 56, 57, 58, 61, 62, 63, 64, 67, 68, 69, 70, 73, 74, 75, 76, 79, 80, 81, 82, 85, 86, 87, 88, 91, 92, 93, 94, 97, 98

Offset: 1

N. J. A. Sloane

a(k)^m is a term for k and m in N. - Jerzy R Borysowicz, Apr 18 2023

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

Cf. A008621, A047235, A047241, A047245, A047246, A047247, A093148.

Complement of A047264. Equals A203016 divided by 3.

[n: n in [0..100] | n mod 6 in [1..4]]; // Vincenzo Librandi, Jan 06 2013

A047227:=n->(6*n-5-I^(2*n)+(1+I)*I^(1-n)+(1-I)*I^(n-1))/4: seq(A047227(n), n=1..100); # Wesley Ivan Hurt, May 20 2016

Complement[Range[100], Flatten[Table[{6n - 1, 6n}, {n, 0, 15}]]] (* Alonso del Arte, Jul 07 2011 *)
Select[Range[100], MemberQ[{1, 2, 3, 4}, Mod[#, 6]]&] (* Vincenzo Librandi, Jan 06 2013 *)

a(n)=([0,1,0,0,0; 0,0,1,0,0; 0,0,0,1,0; 0,0,0,0,1; -1,1,0,0,1]^(n-1)*[1;2;3;4;7])[1,1] \\ Charles R Greathouse IV, May 03 2023

From Johannes W. Meijer, Jul 07 2011: (Start)
a(n) = floor((n+2)/4) + floor((n+1)/4) + floor(n/4) + 2*floor((n-1)/4) + floor((n+3)/4).
G.f.: x*(1 + x + x^2 + x^3 + 2*x^4)/(x^5 - x^4 - x + 1). (End)
From Wesley Ivan Hurt, May 20 2016: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
a(n) = (6n - 5 - i^(2n) + (1+i)*i^(1-n) + (1-i)*i^(n-1))/4 where i=sqrt(-1).
a(2n) = A047235(n), a(2n-1) = A047241(n). (End)
E.g.f.: (4 + sin(x) - cos(x) + (3*x - 2)*sinh(x) + 3*(x - 1)*cosh(x))/2. - Ilya Gutkovskiy, May 21 2016
From Wesley Ivan Hurt, May 21 2016: (Start)
a(n) = A047246(n) + 1.
a(n+2) - a(n+1) = A093148(n) for n>0.
a(1-n) = - A047247(n). (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt(3)*Pi/12 + 2*log(2)/3 - log(3)/4. - Amiram Eldar, Dec 17 2021

A028288 Molien series for complex 4-dimensional Clifford group of order 92160 and genus 2. Also Molien series of ring of biweight enumerators of Type II self-dual binary codes.

Original entry on oeis.org

1, 1, 1, 3, 4, 5, 8, 10, 12, 17, 21, 24, 31, 37, 42, 52, 60, 67, 80, 91, 101, 117, 131, 144, 164, 182, 198, 222, 244, 264, 293, 319, 343, 377, 408, 437, 476, 512, 546, 591, 633, 672, 723, 771, 816, 874, 928, 979, 1044, 1105, 1163, 1235, 1303, 1368

Offset: 0

N. J. A. Sloane

T. D. Noe, Table of n, a(n) for n = 0..1000
A. R. Calderbank, E. M. Rains, P. W. Shor and N. J. A. Sloane, Quantum error correction via codes over GF(4), IEEE Trans. Inform. Theory, 44 (1998), 1369-1387.
W. Duke, On codes and Siegel modular forms, Int. Math. Res. Notes 1993, No. 5, Theorem 2.
W. C. Huffman, The biweight enumerator of self-orthogonal binary codes, Discr. Math. Vol. 26 1979, pp. 129-143.
G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
E. M. Rains and N. J. A. Sloane, Self-dual codes, pp. 177-294 of Handbook of Coding Theory, Elsevier, 1998 (Abstract, pdf, ps).
Index entries for Molien series
Index entries for linear recurrences with constant coefficients, signature (1,0,2,-2,1,-2,1,-2,2,0,1,-1).

Cf. A008621, A008718, A024186, A039946, A051263.

R:=PowerSeriesRing(Integers(), 60); Coefficients(R!( (1+x^4)/((1-x)*(1-x^3)^2*(1-x^5)) )); // G. C. Greubel, Feb 01 2020

seq(coeff(series((1+x^4)/((1-x)*(1-x^3)^2*(1-x^5)), x, n+1), x, n), n = 0..60); # G. C. Greubel, Feb 01 2020

LinearRecurrence[{1,0,2,-2,1,-2,1,-2,2,0,1,-1}, {1,1,1,3,4,5,8,10,12,17,21,24}, 60] (* Jean-François Alcover, Jan 27 2015 *)
CoefficientList[Series[(1+x^4)/((1-x)(1-x^3)^2(1-x^5)),{x,0,60}],x] (* Harvey P. Dale, Jul 10 2019 *)

Vec((1+x^4)/((1-x)*(1-x^3)^2*(1-x^5)) + O('x^60)) \\ G. C. Greubel, Feb 01 2020

def A028288_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x^4)/((1-x)*(1-x^3)^2*(1-x^5)) ).list()
A028288_list(60) # G. C. Greubel, Feb 01 2020

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

a(n) ~ 1/135*n^3. - Ralf Stephan, Apr 29 2014

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

A110655 a(n) = A110654(A110654(n)).

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 10, 10, 10, 10, 11, 11, 11, 11, 12, 12, 12, 12, 13, 13, 13, 13, 14, 14, 14, 14, 15, 15, 15, 15, 16, 16, 16, 16, 17, 17, 17, 17, 18, 18, 18, 18, 19, 19, 19, 19, 20, 20, 20, 20, 21

Offset: 0

Reinhard Zumkeller, Aug 05 2005

a(n) = A008621(n+1) = A002265(n+3).

A110656(n) = A110654(a(n)) = a(A110654(n)).

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

Cf. A002265, A008621, A110654, A110656, A110657.

A110655:=n->ceil(n/4): seq(A110655(n), n=0..100); # Wesley Ivan Hurt, May 29 2016

CoefficientList[Series[x/(x^5 - x^4 - x + 1), {x, 0, 100}], x] (* Wesley Ivan Hurt, May 29 2016 *)
LinearRecurrence[{1, 0, 0, 1, -1}, {0, 1, 1, 1, 1}, 50] (* G. C. Greubel, May 29 2016 *)

a(n) = ceiling(n/4).
From Chai Wah Wu, May 29 2016: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5) for n>4.
G.f.: x/(x^5 - x^4 - x + 1). (End)

A047530 Numbers that are congruent to {0, 1, 3, 7} mod 8.

Original entry on oeis.org

0, 1, 3, 7, 8, 9, 11, 15, 16, 17, 19, 23, 24, 25, 27, 31, 32, 33, 35, 39, 40, 41, 43, 47, 48, 49, 51, 55, 56, 57, 59, 63, 64, 65, 67, 71, 72, 73, 75, 79, 80, 81, 83, 87, 88, 89, 91, 95, 96, 97, 99, 103, 104, 105, 107, 111, 112, 113, 115, 119, 120, 121, 123

Offset: 1

N. J. A. Sloane

Numbers n such that the n-th homotopy group of the topological group O(oo) does not vanish [see Baez]. Cf. A195679.

The a(n+1) determine the maximal number of linearly independent smooth nowhere zero vector fields on a (2n+1)-sphere, see A053381. - Johannes W. Meijer, Jun 07 2011

John C. Baez, The Octonions, Bull. Amer. Math. Soc., Vol. 39, No. 2 (2002), pp. 145-205; Errata, ibid., Vol. 42, No. 2 (2005), p. 213; alternative link.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A008621. - Johannes W. Meijer, Jun 07 2011

Cf. A047470, A047522, A053381, A195679.

A047530 := proc(n): ceil(n/4) + 2*ceil((n-1)/4) + 4*ceil((n-2)/4) + ceil((n-3)/4) end: seq(A047530(n), n=0..47); # Johannes W. Meijer, Jun 07 2011
A047530:=n->(8*n-9+I^(2*n)+(2+I)*I^(-n)+(2-I)*I^n)/4: seq(A047530(n), n=1..100); # Wesley Ivan Hurt, May 21 2016

Table[(8n-9+I^(2n)+(2+I)*I^(-n)+(2-I)*I^n)/4, {n, 80}] (* Wesley Ivan Hurt, May 21 2016 *)

a(n)=n>>2<<3+[-1,0,1,3][n%4+1] \\ Charles R Greathouse IV, Jun 09 2011

From Johannes W. Meijer, Jun 07 2011: (Start)
a(n) = ceiling(n/4) + 2*ceiling((n-1)/4) + 4*ceiling((n-2)/4) + ceiling((n-3)/4).
a(n+1) = A053381(2^p). (End)
G.f.: x^2*(1+2*x+4*x^2+x^3) / ((1+x)*(x^2+1)*(x-1)^2). - R. J. Mathar, Oct 08 2011
From Wesley Ivan Hurt, May 21 2016: (Start)
a(n) = a(n-1)+a(n-4)-a(n-5) for n>5.
a(n) = (8n-9+i^(2n)+(2+i)*i^(-n)+(2-i)*i^n)/4, where i=sqrt(-1).
a(2n) = A047522(n), a(2n-1) = A047470(n). (End)
E.g.f.: (2 + sin(x) + 2*cos(x) + (4*x - 5)*sinh(x) + 4*(x - 1)*cosh(x))/2. - Ilya Gutkovskiy, May 21 2016
Sum_{n>=2} (-1)^n/a(n) = (8-3*sqrt(2))*log(2)/16 + 3*sqrt(2)*log(2+sqrt(2))/8 - (sqrt(2)-1)*Pi/16. - Amiram Eldar, Dec 20 2021

More terms from Wesley Ivan Hurt, May 21 2016

A195079 Fractalization of (1+[n/4]), where [ ]=floor.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Sep 08 2011

nonn

See A194959 for a discussion of fractalization and the interspersion fractally induced by a sequence. The sequence (1+[n/4]) is A008621. See A194959 for a discussion of fractalization and the interspersion fractally induced by a sequence.

Cf. A008621, A195080, A195081.

r = 4; p[n_] := 1 + Floor[n/r]
Table[p[n], {n, 1, 90}]  (* A008621 *)
g[1] = {1}; g[n_] := Insert[g[n - 1], n, p[n]]
f[1] = g[1]; f[n_] := Join[f[n - 1], g[n]]
f[20] (* A195079 *)
row[n_] := Position[f[30], n];
u = TableForm[Table[row[n], {n, 1, 5}]]
v[n_, k_] := Part[row[n], k];
w = Flatten[Table[v[k, n - k + 1], {n, 1, 13},
{k, 1, n}]] (* A195080 *)
q[n_] := Position[w, n]; Flatten[Table[q[n],
{n, 1, 80}]] (* A195081 *)

A246720 Number A(n,k) of partitions of n into parts of the k-th list of distinct parts in the order given by A246688; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 2, 0, 1, 0, 1, 1, 0, 2, 1, 1, 0, 1, 0, 1, 1, 3, 0, 1, 0, 1, 1, 0, 2, 0, 3, 1, 1, 0, 1, 0, 1, 0, 2, 0, 4, 0, 1, 0, 1, 0, 1, 1, 1, 2, 1, 4, 1, 1, 0, 1, 1, 0, 1, 2, 0, 3, 0, 5, 0, 1, 0, 1, 1, 2, 0, 1, 2, 0, 3, 0, 5, 1, 1, 0

Offset: 0

Alois P. Heinz, Sep 02 2014

The first lists of distinct parts in the order given by A246688 are: 0:[], 1:[1], 2:[2], 3:[1,2], 4:[3], 5:[1,3], 6:[4], 7:[1,4], 8:[2,3], 9:[5], 10:[1,2,3], 11:[1,5], 12:[2,4], 13:[6], 14:[1,2,4], 15:[1,6], 16:[2,5], 17:[3,4], 18:[7], 19:[1,2,5], 20:[1,3,4], ... .

			Square array A(n,k) begins:
  1, 1, 1, 1, 1, 1, 1, 1, 1, 1,  1, 1, 1, 1,  1, ...
  0, 1, 0, 1, 0, 1, 0, 1, 0, 0,  1, 1, 0, 0,  1, ...
  0, 1, 1, 2, 0, 1, 0, 1, 1, 0,  2, 1, 1, 0,  2, ...
  0, 1, 0, 2, 1, 2, 0, 1, 1, 0,  3, 1, 0, 0,  2, ...
  0, 1, 1, 3, 0, 2, 1, 2, 1, 0,  4, 1, 2, 0,  4, ...
  0, 1, 0, 3, 0, 2, 0, 2, 1, 1,  5, 2, 0, 0,  4, ...
  0, 1, 1, 4, 1, 3, 0, 2, 2, 0,  7, 2, 2, 1,  6, ...
  0, 1, 0, 4, 0, 3, 0, 2, 1, 0,  8, 2, 0, 0,  6, ...
  0, 1, 1, 5, 0, 3, 1, 3, 2, 0, 10, 2, 3, 0,  9, ...
  0, 1, 0, 5, 1, 4, 0, 3, 2, 0, 12, 2, 0, 0,  9, ...
  0, 1, 1, 6, 0, 4, 0, 3, 2, 1, 14, 3, 3, 0, 12, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Columns k=0-11, 13-20, 23 give: A000007, A000012, A059841, A008619, A079978, A008620, A121262, A008621, A103221, A079998, A001399, A002266(n+5), A079979, A008642, A097992, A008616, A008679, A082784, A000115, A025767, A008676.
Main diagonal gives A246721.
Cf. A246688, A246690 (the same for compositions).

b:= proc(n, i) b(n, i):= `if`(n=0, [[]], `if`(i>n, [],
      [map(x->[i, x[]], b(n-i, i+1))[], b(n, i+1)[]]))
    end:
f:= proc() local i, l; i, l:=0, [];
      proc(n) while n>=nops(l)
        do l:=[l[], b(i, 1)[]]; i:=i+1 od; l[n+1]
      end
    end():
g:= proc(n, l) option remember; `if`(n=0, 1, `if`(l=[], 0,
      add(g(n-l[-1]*j, subsop(-1=NULL, l)), j=0..n/l[-1])))
    end:
A:= (n, k)-> g(n, f(k)):
seq(seq(A(n, d-n), n=0..d), d=0..16);

b[n_, i_] := b[n, i] = If[n == 0, {{}}, If[i > n, {}, Join[Prepend[#, i]& /@ b[n - i, i + 1], b[n, i + 1]]]];
f = Module[{i = 0, l = {}}, Function[n, While[ n >= Length[l], l = Join[l, b[i, 1]]; i++ ]; l[[n + 1]]]];
g[n_, l_] := g[n, l] = If[n == 0, 1, If[l == {}, 0, Sum[g[n - l[[-1]] j, ReplacePart[l, -1 -> Nothing]], {j, 0, n/l[[-1]]}]]];
A[n_, k_] := g[n, f[k]];
Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 16}] // Flatten (* Jean-François Alcover, Dec 07 2020, after Alois P. Heinz *)

A349839 Triangle T(n,k) built by placing all ones on the left edge, [1,0,0,0] repeated on the right edge, and filling the body using the Pascal recurrence T(n,k) = T(n-1,k) + T(n-1,k-1).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 3, 1, 1, 1, 4, 6, 4, 2, 0, 1, 5, 10, 10, 6, 2, 0, 1, 6, 15, 20, 16, 8, 2, 0, 1, 7, 21, 35, 36, 24, 10, 2, 1, 1, 8, 28, 56, 71, 60, 34, 12, 3, 0, 1, 9, 36, 84, 127, 131, 94, 46, 15, 3, 0, 1, 10, 45, 120, 211, 258, 225, 140, 61, 18, 3, 0, 1, 11, 55, 165, 331, 469, 483, 365, 201, 79, 21, 3, 1

Offset: 0

Michael A. Allen, Dec 01 2021

This is the m=4 member in the sequence of triangles A007318, A059259, A118923, A349839, A349841 which have all ones on the left side, ones separated by m-1 zeros on the other side, and whose interiors obey Pascal's recurrence.
T(n,k) is the (n,n-k)-th entry of the (1/(1-x^4),x/(1-x)) Riordan array.
For n>0, T(n,n-1) = A008621(n-1).
For n>1, T(n,n-2) = A001972(n-2).
For n>2, T(n,n-3) = A122046(n).
Sums of rows give A115451.
Sums of antidiagonals give A349840.

			Triangle begins:
  1;
  1,   0;
  1,   1,   0;
  1,   2,   1,   0;
  1,   3,   3,   1,   1;
  1,   4,   6,   4,   2,   0;
  1,   5,  10,  10,   6,   2,   0;
  1,   6,  15,  20,  16,   8,   2,   0;
  1,   7,  21,  35,  36,  24,  10,   2,   1;
  1,   8,  28,  56,  71,  60,  34,  12,   3,   0;
  1,   9,  36,  84, 127, 131,  94,  46,  15,   3,   0;
  1,  10,  45, 120, 211, 258, 225, 140,  61,  18,   3,   0;
  1,  11,  55, 165, 331, 469, 483, 365, 201,  79,  21,   3,   1;

Michael A. Allen and Kenneth Edwards, On Two Families of Generalizations of Pascal's Triangle, J. Int. Seq. 25 (2022) Article 22.7.1.

Other members of sequence of triangles: A007318, A059259, A118923, A349841.
Columns: A000012, A001477, A000217, A000292, A145126, A051745.
Diagonals: A121262, A008621, A001972, A122046.

Flatten[Table[CoefficientList[Series[(1-x*y)/((1-(x*y)^4)(1 - x - x*y)), {x, 0, 24}, {y, 0, 12}], {x, y}][[n+1,k+1]],{n,0,12},{k,0,n}]]

G.f.: (1-x*y)/((1-(x*y)^4)(1-x-x*y)) in the sense that T(n,k) is the coefficient of x^n*y^k in the series expansion of the g.f.
T(n,0) = 1.
T(n,n) = delta(n mod 4,0).
T(n,1) = n-1 for n>0.
T(n,2) = (n-1)*(n-2)/2 for n>1.
T(n,3) = (n-1)*(n-2)*(n-3)/6 for n>2.
T(n,4) = C(n-1,4) + 1 for n>3.
T(n,5) = C(n-1,5) + n - 5 for n>4.
For 0 <= k < n, T(n,k) = (n-k)*Sum_{j=0..floor(k/4)} binomial(n-4*j,n-k)/(n-4*j).
The g.f. of the n-th subdiagonal is 1/((1-x^4)(1-x)^n).

cp's OEIS Frontend

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