A240871 Number of partitions p of n into distinct parts such that max(p) = 3 + min(p).

0, 0, 0, 0, 0, 1, 0, 2, 1, 1, 2, 2, 0, 2, 2, 1, 1, 2, 1, 2, 1, 1, 2, 2, 0, 2, 2, 1, 1, 2, 1, 2, 1, 1, 2, 2, 0, 2, 2, 1, 1, 2, 1, 2, 1, 1, 2, 2, 0, 2, 2, 1, 1, 2, 1, 2, 1, 1, 2, 2, 0, 2, 2, 1, 1, 2, 1, 2, 1, 1, 2, 2, 0, 2, 2, 1, 1, 2, 1, 2, 1, 1, 2, 2, 0, 2, 2, 1, 1, 2, 1, 2, 1, 1, 2, 2, 0, 2, 2, 1, 1, 2, 1, 2, 1, 1, 2, 2, 0

Offset

0,8

Links

Antti Karttunen, Table of n, a(n) for n = 0..20000

Index entries for sequences related to partitions

Index entries for linear recurrences with constant coefficients, signature (-1,-1,0,1,1,1).

Formula

From Alois P. Heinz, Jan 13 2025: (Start)

G.f.: -x^5*(x^7+2*x^6+3*x^5+3*x^4+3*x^3+3*x^2+x+1)/((x-1)*(x+1)*(x^2+1)*(x^2+x+1)).

a(n) = a(n-12) for n>=19. (End)

Example

a(7) counts these 2 partitions: 5+2, 4+2+1.

Mathematica

z = 40; f[n_] := f[n] = Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 &]; Table[Count[f[n], p_ /; Max[p] == 2 + Min[p]], {n, 0, z}] (* A171182 *)
    Table[Count[f[n], p_ /; Max[p] == 3 + Min[p]], {n, 0, z}] (* A240871 *)
    Table[Count[f[n], p_ /; Max[p] == 4 + Min[p]], {n, 0, z}] (* A240872 *)
    Table[Count[f[n], p_ /; Max[p] == 5 + Min[p]], {n, 0, z}] (* A240873 *)

Prog

(PARI)
A240871aux(n, minp=0, maxp=0) = if(0==n, (minp>0 && (maxp==3+minp)), sum(i=1+maxp, min(n, 3+minp), A240871aux(n-i, if(!minp, i, minp), i)));
A240871(n) = sum(i=1, floor(n/2), A240871aux(n-i, i, i)); \\ Antti Karttunen, Jan 13 2025

Crossrefs

Cf. A128508, A171182, A240872, A240873.

Sequence in context: A357458 A349277 A307014 * A236765 A171531 A171532

Adjacent sequences: A240868 A240869 A240870 * A240872 A240873 A240874

Keyword

nonn,easy

Author

Clark Kimberling, Apr 15 2014

Extensions

More terms from Antti Karttunen, Jan 13 2025

Status

approved