cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A241549 Number of partitions p of n such that (number of numbers of the form 5k in p) is a part of p.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 8, 12, 17, 25, 35, 48, 67, 91, 122, 163, 215, 283, 369, 478, 615, 786, 1004, 1270, 1604, 2014, 2521, 3139, 3902, 4824, 5954, 7314, 8970, 10957, 13362, 16232, 19691, 23804, 28737, 34581, 41559, 49802, 59596, 71139, 84799
Offset: 0

Views

Author

Clark Kimberling, Apr 26 2014

Keywords

Comments

Each number in p is counted once, regardless of its multiplicity.

Examples

			a(6) counts this single partition:  51.

Crossrefs

Cf. A241550, A241551, A241552, A241553.

Programs

  • Mathematica
    z = 30; f[n_] := f[n] = IntegerPartitions[n]; s[k_, p_] := Count[Mod[DeleteDuplicates[p], 5], k]
Table[Count[f[n], p_ /; MemberQ[p, s[0, p]]], {n, 0, z}]  (* A241549 *)
Table[Count[f[n], p_ /; MemberQ[p, s[1, p]]], {n, 0, z}]  (* A241550 *)
Table[Count[f[n], p_ /; MemberQ[p, s[2, p]]], {n, 0, z}]  (* A241551 *)
Table[Count[f[n], p_ /; MemberQ[p, s[3, p]]], {n, 0, z}]  (* A241552 *)
Table[Count[f[n], p_ /; MemberQ[p, s[4, p]]], {n, 0, z}]  (* A241553 *)

A241550 Number of partitions p of n such that (number of numbers of the form 5k + 1 in p) is a part of p.

Original entry on oeis.org

0, 1, 1, 2, 3, 5, 7, 10, 14, 21, 28, 39, 51, 70, 92, 122, 158, 206, 265, 343, 432, 554, 695, 879, 1098, 1373, 1703, 2115, 2607, 3218, 3937, 4831, 5882, 7175, 8699, 10541, 12733, 15358, 18464, 22184, 26548, 31774, 37891, 45166, 53681, 63743, 75529, 89381
Offset: 0

Views

Author

Clark Kimberling, Apr 26 2014

Keywords

Comments

Each number in p is counted once, regardless of its multiplicity.

Examples

			a(6) counts these 7 partitions:  51, 411, 321, 3111, 2211, 21111, 111111.

Crossrefs

Cf. A241549, A241551, A241552, A241553.

Programs

  • Mathematica
    z = 30; f[n_] := f[n] = IntegerPartitions[n]; s[k_, p_] := Count[Mod[DeleteDuplicates[p], 5], k]
Table[Count[f[n], p_ /; MemberQ[p, s[0, p]]], {n, 0, z}]  (* A241549 *)
Table[Count[f[n], p_ /; MemberQ[p, s[1, p]]], {n, 0, z}]  (* A241550 *)
Table[Count[f[n], p_ /; MemberQ[p, s[2, p]]], {n, 0, z}]  (* A241551 *)
Table[Count[f[n], p_ /; MemberQ[p, s[3, p]]], {n, 0, z}]  (* A241552 *)
Table[Count[f[n], p_ /; MemberQ[p, s[4, p]]], {n, 0, z}]  (* A241553 *)

A241552 Number of partitions p of n such that (number of numbers of the form 5k + 3 in p) is a part of p.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 2, 3, 5, 8, 12, 17, 23, 34, 47, 64, 87, 115, 154, 204, 266, 346, 444, 573, 731, 933, 1174, 1479, 1855, 2320, 2884, 3578, 4411, 5443, 6678, 8185, 9977, 12157, 14753, 17886, 21608, 26058, 31326, 37631, 45066, 53911, 64300, 76609, 91061
Offset: 0

Views

Author

Clark Kimberling, Apr 26 2014

Keywords

Comments

Each number in p is counted once, regardless of its multiplicity.

Examples

			a(6) counts these 2 partitions:  321, 3111.

Crossrefs

Cf. A241549, A241550, A241551, A241553.

Programs

  • Mathematica
    z = 30; f[n_] := f[n] = IntegerPartitions[n]; s[k_, p_] := Count[Mod[DeleteDuplicates[p], 5], k]
Table[Count[f[n], p_ /; MemberQ[p, s[0, p]]], {n, 0, z}]  (* A241549 *)
Table[Count[f[n], p_ /; MemberQ[p, s[1, p]]], {n, 0, z}]  (* A241550 *)
Table[Count[f[n], p_ /; MemberQ[p, s[2, p]]], {n, 0, z}]  (* A241551 *)
Table[Count[f[n], p_ /; MemberQ[p, s[3, p]]], {n, 0, z}]  (* A241552 *)
Table[Count[f[n], p_ /; MemberQ[p, s[4, p]]], {n, 0, z}]  (* A241553 *)

A241553 Number of partitions p of n such that (number of numbers of the form 5k + 4 in p) is a part of p.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 8, 12, 17, 25, 33, 49, 65, 90, 119, 159, 210, 277, 358, 466, 593, 766, 968, 1231, 1548, 1942, 2427, 3026, 3747, 4642, 5704, 7022, 8587, 10498, 12775, 15519, 18799, 22730, 27394, 32981, 39558, 47426, 56676, 67650, 80564, 95781
Offset: 0

Views

Author

Clark Kimberling, Apr 26 2014

Keywords

Comments

Each number in p is counted once, regardless of its multiplicity.

Examples

			a(6) counts this single partition:  411.

Crossrefs

Cf. A241549, A241550, A241551, A241552.

Programs

  • Mathematica
    z = 30; f[n_] := f[n] = IntegerPartitions[n]; s[k_, p_] := Count[Mod[DeleteDuplicates[p], 5], k]
Table[Count[f[n], p_ /; MemberQ[p, s[0, p]]], {n, 0, z}]  (* A241549 *)
Table[Count[f[n], p_ /; MemberQ[p, s[1, p]]], {n, 0, z}]  (* A241550 *)
Table[Count[f[n], p_ /; MemberQ[p, s[2, p]]], {n, 0, z}]  (* A241551 *)
Table[Count[f[n], p_ /; MemberQ[p, s[3, p]]], {n, 0, z}]  (* A241552 *)
Table[Count[f[n], p_ /; MemberQ[p, s[4, p]]], {n, 0, z}]  (* A241553 *)

A341312 a(n) = a(n-1) + a(n-3) unless a(n-1) and a(n-3) are both even in which case a(n) = (a(n-1) + a(n-3))/2, with a(0) = a(1) = a(2) = 1.

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 3, 6, 5, 8, 7, 12, 10, 17, 29, 39, 56, 85, 124, 90, 175, 299, 389, 564, 863, 1252, 908, 1771, 3023, 3931, 5702, 8725, 12656, 9179, 17904, 15280, 24459, 42363, 57643, 82102, 124465, 182108, 132105, 256570, 219339, 351444, 304007, 523346, 437395, 741402, 632374
Offset: 0

Views

Author

N. J. A. Sloane, Feb 16 2021

Keywords

Comments

A sequence intermediate between Narayana's A000930 and Reed Kelly's A214551.
It will be interesting to compare the growth rates of A000930 (well-understood), A241551 (a mystery), the present sequence, and A341313.
It appears that the equation log(a(n)) = 0.296869*n - 4.69131 is a good fit to the data (see the figures). - Hugo Pfoertner, Feb 17 2021

Links

Crossrefs

Cf. A000930, A214551, A341313.

Programs

  • Maple
    RK2:=proc(n) local t1; option remember;
if n <= 2 then 1 else t1:=RK2(n-3)+RK2(n-1);
   if (RK2(n-3) mod 2) = 0 and (RK2(n-1) mod 2) = 0 then t1:=t1/2; fi;
t1; fi; end;
[seq(RK2(n),n=0..60)];
  • PARI
    a341312(nterms)={my(a=vector(nterms));a[1]=a[2]=1;a[3]=2;for(n=4,nterms,a[n]=if(a[n-1]%2==0&&a[n-3]%2==0,(a[n-1]+a[n-3])/2,a[n-1]+a[n-3]));concat([1],a)};
a341312(60) \\ Hugo Pfoertner, Feb 17 2021

A341313 a(n) = (a(n-1) + a(n-3))/2^m, where 2^m is the highest power of 2 that divides both a(n-1) and a(n-3), with a(0) = a(1) = a(2) = 1.

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 3, 6, 5, 8, 7, 12, 5, 12, 6, 11, 23, 29, 40, 63, 92, 33, 96, 47, 80, 11, 58, 69, 80, 69, 138, 109, 178, 158, 267, 445, 603, 870, 1315, 1918, 1394, 2709, 4627, 6021, 8730, 13357, 19378, 14054, 27411, 46789, 60843, 88254, 135043, 195886, 142070, 277113, 472999
Offset: 0

Views

Author

N. J. A. Sloane, Feb 16 2021

Keywords

Comments

A sequence intermediate between Narayana's A000930 and Reed Kelly's A214551.
It will be interesting to compare the growth rates of A000930 (well-understood), A241551 (a mystery), the present sequence, and A341312.
It appears that the equation log(a(n)) = 0.265986*n + 1.56445 is a good fit to the data (see the figures). - Hugo Pfoertner, Feb 17 2021

Links

Crossrefs

Cf. A000930, A214551, A341312.

Programs

  • Maple
    RK3:=proc(n) local t1,t2; option remember;
if n <= 2 then 1 else t1:=RK3(n-3)+RK3(n-1);
t2 := min( padic[ordp](RK3(n-3), 2), padic[ordp](RK3(n-1), 2) );
t1/2^t2;
fi;
end;
[seq(RK3(n),n=0..60)];
  • PARI
    a341313(nterms)={my(a=vector(nterms));a[1]=a[2]=1;a[3]=2;for(n=4,nterms,a[n]=(a[n-1]+a[n-3])/2^min(valuation(a[n-1],2),valuation(a[n-3],2)));concat([1],a)};
a341313(60) \\ Hugo Pfoertner, Feb 16 2021
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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