A199936 -id:A199936 - cp's OEIS frontend

A144115 Total number of Fibonacci parts in all partitions of n.

1, 3, 6, 11, 19, 32, 49, 77, 114, 169, 241, 345, 480, 667, 910, 1237, 1656, 2213, 2918, 3840, 5003, 6497, 8368, 10751, 13711, 17441, 22052, 27806, 34879, 43645, 54355, 67535, 83571, 103171, 126907, 155766, 190554, 232629, 283158, 343969, 416716, 503900, 607807

Offset: 1

Omar E. Pol, Sep 11 2008

nonn

a(n) is also the sum of the differences between the sum of f-th largest and the sum of (f+1)-st largest elements in all partitions of n for all Fibonacci parts f. - Omar E. Pol, Oct 27 2012

			From _Omar E. Pol_, Nov 20 2011 (Start):
For n = 6 we have:
--------------------------------------
.                        Number of
Partitions            Fibonacci parts
--------------------------------------
6 .......................... 0
3 + 3 ...................... 2
4 + 2 ...................... 1
2 + 2 + 2 .................. 3
5 + 1 ...................... 2
3 + 2 + 1 .................. 3
4 + 1 + 1 .................. 2
2 + 2 + 1 + 1 .............. 4
3 + 1 + 1 + 1 .............. 4
2 + 1 + 1 + 1 + 1 .......... 5
1 + 1 + 1 + 1 + 1 + 1 ...... 6
------------------------------------
Total ..................... 32
So a(6) = 32. (End)

Alois P. Heinz, Table of n, a(n) for n = 1..8000

Cf. A000045, A006128, A037032, A144116, A144117, A144118, A199936, A309537.

b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, 0,
      b(n, i-1)+ (p-> p+`if`((t-> issqr(t+4) or issqr(t-4)
      )(5*i^2), [0, p[1]], 0))(b(n-i, min(n-i, i)))))
    end:
a:= n-> b(n$2)[2]:
seq(a(n), n=1..60); # Alois P. Heinz, Jun 24 2009, revised Aug 06 2019

Clear[b]; b[] = False; l = {0, 1}; For[k=1, k <= 100, k++, b[l[[1]]] = True; l = {l[[2]], l[[1]] + l[[2]]}]; aa[n, i_] := aa[n, i] = Module[{g, h}, If[n==0, {1, 0}, If[i==0 || n<0, {0, 0}, g = aa[n, i-1]; h = aa[n-i, i]; {g[[1]] + h[[1]], g[[2]] + h[[2]] + If[b[i], h[[1]], 0]}]]]; a[n_] := aa[n, n][[2]]; Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Jul 30 2015, after Alois P. Heinz *)

G.f.: Sum_{i>=2} x^Fibonacci(i)/(1 - x^Fibonacci(i)) / Product_{j>=1} (1 - x^j). - Ilya Gutkovskiy, Jan 24 2017

More terms from Alois P. Heinz, Jun 24 2009

A326982 Total sum of composite parts in all partitions of n.

Original entry on oeis.org

0, 0, 0, 0, 4, 4, 14, 18, 44, 67, 117, 166, 283, 391, 603, 848, 1250, 1702, 2442, 3280, 4565, 6094, 8266, 10878, 14566, 18970, 24953, 32255, 41909, 53619, 68983, 87542, 111496, 140561, 177436, 222125, 278425, 346293, 430951, 533083, 659268, 810948, 997322

Offset: 0

Omar E. Pol, Aug 09 2019

nonn

			For n = 6 we have:
--------------------------------------
Partitions                Sum of
of 6                  composite parts
--------------------------------------
6 .......................... 6
3 + 3 ...................... 0
4 + 2 ...................... 4
2 + 2 + 2 .................. 0
5 + 1 ...................... 0
3 + 2 + 1 .................. 0
4 + 1 + 1 .................. 4
2 + 2 + 1 + 1 .............. 0
3 + 1 + 1 + 1 .............. 0
2 + 1 + 1 + 1 + 1 .......... 0
1 + 1 + 1 + 1 + 1 + 1 ...... 0
--------------------------------------
Total ..................... 14
So a(6) = 14.

Alois P. Heinz, Table of n, a(n) for n = 0..8000

Cf. A000041, A002808, A066186, A073118, A194544, A194545, A199936, A326958, A326981.

b:= proc(n, i) option remember; `if`(n=0 or i=1, [1, 0], b(n, i-1)+
      (p-> p+[0, `if`(isprime(i), 0, p[1]*i)])(b(n-i, min(n-i, i))))
    end:
a:= n-> b(n$2)[2]:
seq(a(n), n=0..50);  # Alois P. Heinz, Aug 13 2019

Table[Total[Select[Flatten[IntegerPartitions[n]],CompositeQ]],{n,0,50}] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Apr 19 2020 *)
b[n_, i_] := b[n, i] = If[n == 0 || i == 1, {1, 0}, b[n, i - 1] +
     With[{p = b[n-i, Min[n-i, i]]}, p+{0, If[PrimeQ[i], 0, p[[1]]*i]}]];
a[n_] := b[n, n][[2]];
a /@ Range[0, 50] (* Jean-François Alcover, Jun 07 2021, after Alois P. Heinz *)

a(n) = A194545(n) - A000070(n-1), n >= 1.

a(n) = A066186(n) - A326958(n).

A326958 Total sum of noncomposite parts in all partitions of n.

Original entry on oeis.org

0, 1, 4, 9, 16, 31, 52, 87, 132, 203, 303, 450, 641, 922, 1287, 1792, 2446, 3347, 4488, 6030, 7975, 10538, 13778, 17987, 23234, 29980, 38383, 49015, 62195, 78766, 99137, 124560, 155672, 194158, 241104, 298780, 368747, 454276, 557619, 683132, 834252, 1016955

Offset: 0

Omar E. Pol, Aug 08 2019

nonn

			For n = 6 we have:
--------------------------------------
.                         Sum of
Partitions             noncomposite
of 6                       parts
--------------------------------------
6 .......................... 0
3 + 3 ...................... 6
4 + 2 ...................... 2
2 + 2 + 2 .................. 6
5 + 1 ...................... 6
3 + 2 + 1 .................. 6
4 + 1 + 1 .................. 2
2 + 2 + 1 + 1 .............. 6
3 + 1 + 1 + 1 .............. 6
2 + 1 + 1 + 1 + 1 .......... 6
1 + 1 + 1 + 1 + 1 + 1 ...... 6
------------------------------------
Total ..................... 52
So a(6) = 52.

Cf. A000041, A000070, A008578 (noncomposites), A066186, A073118, A194545, A199936, A326957, A326982.

b:= proc(n, i) option remember; `if`(n=0 or i=1, [1, n], b(n, i-1)+
      (p-> p+[0, `if`(isprime(i), p[1]*i, 0)])(b(n-i, min(n-i, i))))
    end:
a:= n-> b(n$2)[2]:
seq(a(n), n=0..50);  # Alois P. Heinz, Aug 13 2019

b[n_, i_] := b[n, i] = If[n==0 || i==1, {1, n}, b[n, i-1] + # + {0, If[PrimeQ[i], #[[1]] i, 0]}&[b[n-i, Min[n-i, i]]]];
a[n_] := b[n, n][[2]];
a /@ Range[0, 50] (* Jean-François Alcover, Nov 17 2020, after Alois P. Heinz *)

a(n) = A073118(n) + A000070(n-1), n >= 1.

a(n) = A066186(n) - A326982(n).

cp's OEIS Frontend

A144115 Total number of Fibonacci parts in all partitions of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A326982 Total sum of composite parts in all partitions of n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A326958 Total sum of noncomposite parts in all partitions of n.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Mathematica

Formula