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 1-8 of 8 results.

A144116 Number of non-Fibonacci parts in all partitions of n.

Original entry on oeis.org

0, 0, 0, 1, 1, 3, 5, 9, 14, 23, 34, 54, 76, 113, 158, 226, 309, 431, 580, 790, 1049, 1402, 1838, 2423, 3140, 4081, 5242, 6739, 8574, 10918, 13780, 17392, 21795, 27291, 33969, 42248, 52258, 64572, 79429, 97577, 119388, 145891, 177630, 215986, 261789
Offset: 1

Views

Author

Omar E. Pol, Sep 11 2008

Keywords

Links

Crossrefs

Cf. A001690, A006128, A144115, A144117, A144118, A144119.

Programs

  • Maple
    b:= proc(n) option remember; true end: l:= [0, 1]: for k to 100 do b(l[1]):= false; l:= [l[2], l[1]+l[2]] od: aa:= proc(n, i) option remember; local g, h; if n=0 then [1, 0] elif i=0 or n<0 then [0, 0] else g:= aa(n, i-1); h:= aa(n-i, i); [g[1]+h[1], g[2]+h[2] +`if`(b(i), h[1], 0)] fi end: a:= n-> aa(n, n)[2]: seq(a(n), n=1..60); # Alois P. Heinz, Jul 28 2009
  • Mathematica
    Clear[b]; b[] = True; l = {0, 1}; For[k=1, k <= 100, k++, b[l[[1]]] = False; 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 *)

Formula

a(n) = A006128(n)-A144115(n).

Extensions

More terms from Alois P. Heinz, Jul 28 2009

A144117 Number of Fibonacci parts in the last section of the set of partitions of n.

Original entry on oeis.org

1, 2, 3, 5, 8, 13, 17, 28, 37, 55, 72, 104, 135, 187, 243, 327, 419, 557, 705, 922, 1163, 1494, 1871, 2383, 2960, 3730, 4611, 5754, 7073, 8766, 10710, 13180, 16036, 19600, 23736, 28859, 34788, 42075, 50529, 60811, 72747, 87184, 103907, 124019, 147330
Offset: 1

Views

Author

Omar E. Pol, Sep 11 2008

Keywords

Comments

First differences of A144115.
Also number of Fibonacci parts in the n-th section of the set of partitions of any positive integer >= n. - Omar E. Pol, Jul 30 2015

Links

Crossrefs

Cf. A000045, A135010, A138121, A138137, A144115, A144116, A144118, A144120.

Programs

  • Maple
    b:= proc(n) option remember; false end: l:= [0, 1]: for k to 100 do b(l[1]):= true; l:= [l[2], l[1]+l[2]] od: aa:= proc(n, i) option remember; local g, h; if n=0 then [1, 0] elif i=0 or n<0 then [0, 0] else g:= aa(n, i-1); h:= aa(n-i, i); [g[1]+h[1], g[2]+h[2] +`if`(b(i), h[1], 0)] fi end: a:= n-> aa(n, n)[2] -aa(n-1, n-1)[2]: seq(a(n), n=1..60); # Alois P. Heinz, Jul 28 2009
  • Mathematica
    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]] - aa[n - 1, n - 1][[2]]; Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Jul 30 2015, after Alois P. Heinz *)

Formula

a(n) = A144115(n) - A144115(n-1).

Extensions

More terms from Alois P. Heinz, Jul 28 2009

A144118 Number of non-Fibonacci parts in the last section of the set of partitions of n.

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 2, 4, 5, 9, 11, 20, 22, 37, 45, 68, 83, 122, 149, 210, 259, 353, 436, 585, 717, 941, 1161, 1497, 1835, 2344, 2862, 3612, 4403, 5496, 6678, 8279, 10010, 12314, 14857, 18148, 21811, 26503, 31739, 38356, 45803, 55066, 65553, 78488, 93129
Offset: 1

Views

Author

Omar E. Pol, Sep 11 2008

Keywords

Comments

First differences of A144116.

Crossrefs

Cf. A001690, A135010, A138121, A138137, A144115, A144116, A144117, A144121.

Programs

  • Maple
    b:= proc(n) option remember; true end: l:= [0, 1]: for k to 100 do b(l[1]):= false; l:= [l[2], l[1]+l[2]] od: aa:= proc(n, i) option remember; local g, h; if n=0 then [1, 0] elif i=0 or n<0 then [0, 0] else g:= aa(n, i-1); h:= aa(n-i, i); [g[1]+h[1], g[2]+h[2] +`if`(b(i), h[1], 0)] fi end: a:= n-> aa(n, n)[2] -aa(n-1, n-1)[2]: seq(a(n), n=1..60); # Alois P. Heinz, Jul 28 2009
  • Mathematica
    Clear[b]; b[] = True; l = {0, 1}; For[k = 1, k <= 100, k++, b[l[[1]]] = False; l = {l[[2]], l[[1]] + l[[2]]}]; a[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]] - aa[n-1, n-1][[2]]; Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Dec 05 2016 after Alois P. Heinz *)

Formula

a(n) = A138137(n)-A144117(n) = A144116(n)-A144116(n-1).

Extensions

More terms from Alois P. Heinz, Jul 28 2009

A199936 Total sum of Fibonacci parts in all partitions of n.

Original entry on oeis.org

0, 1, 4, 9, 16, 31, 52, 80, 133, 197, 298, 428, 621, 879, 1230, 1696, 2329, 3142, 4231, 5619, 7447, 9781, 12771, 16553, 21391, 27440, 35089, 44600, 56510, 71232, 89538, 112011, 139759, 173679, 215279, 265840, 327527, 402162, 492703, 601830, 733550, 891634
Offset: 0

Views

Author

Omar E. Pol, Nov 21 2011

Keywords

Examples

			For n = 6 we have:
--------------------------------------
.                         Sum of
Partitions            Fibonacci 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.

Links

Crossrefs

Cf. A000045, A066186, A073118, A144115, A194544, A194545.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, 0,
      `if`(i>n, 0, ((p, m)-> p +`if`(issqr(m+4) or issqr(m-4),
      [0, p[1]*i], 0))(b(n-i, i), 5*i^2)) +b(n, i-1)))
    end:
a:= n-> b(n$2)[2]:
seq(a(n), n=0..50);  # Alois P. Heinz, Feb 01 2017
  • Mathematica
    max = 42; F = Fibonacci; gf = Sum[F[i]*x^F[i]/(1-x^F[i]), {i, 2, max}] / Product[1-x^j, {j, 1, max}] + O[x]^max; CoefficientList[gf, x] (* Jean-François Alcover, Feb 21 2017, after Ilya Gutkovskiy *)
b[n_, i_] := b[n, i] = If[n==0, {1, 0}, If[i<1, 0, If[i>n, 0, Function[{p, m}, p+If[IntegerQ @ Sqrt[m+4] || IntegerQ @ Sqrt[m-4], {0, p[[1]]*i}, 0] ][b[n-i, i], 5*i^2]]+b[n, i-1]]]; a[n_] := b[n, n][[2]]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Feb 21 2017, after Alois P. Heinz *)

Formula

G.f.: Sum_{i>=2} Fibonacci(i)*x^Fibonacci(i)/(1 - x^Fibonacci(i)) / Product_{j>=1} (1 - x^j). - Ilya Gutkovskiy, Feb 01 2017

Extensions

More terms from Alois P. Heinz, Nov 21 2011

A183088 Total number of parts that are partition numbers A000041 in all partitions of n.

Original entry on oeis.org

0, 1, 3, 6, 11, 19, 32, 50, 77, 115, 170, 244, 348, 485, 674, 922, 1251, 1678, 2241, 2959, 3892, 5076, 6592, 8497, 10915, 13930, 17719, 22417, 28267, 35474, 44395, 55312, 68730, 85082, 105049, 129261, 158675, 194171, 237077, 288651
Offset: 0

Views

Author

Omar E. Pol, Aug 05 2011

Keywords

Examples

			a(5) = 19 because the 7 partitions of 5 are [5], [4, 1], [3, 2], [3, 1, 1], [2, 2, 1], [2, 1, 1, 1], [1, 1, 1, 1, 1] and we can see that there are 19 parts that are partition numbers A000041. Note that there are 20 parts but the 4 is not a partition number, so a(5) = 20 - 1 = 19.

Links

Crossrefs

Cf. A006128, A037032, A144115.

Programs

  • Mathematica
    A000041 = Table[PartitionsP[n], {n, 0, 45}]; Table[Length[Select[Flatten[IntegerPartitions[n]], MemberQ[A000041, #] &]], {n, 40}] (* Alonso del Arte, Aug 05 2011 *)

A309537 Total number of Fibonacci parts in all compositions of n.

Original entry on oeis.org

0, 1, 3, 8, 19, 46, 106, 241, 541, 1198, 2629, 5724, 12380, 26625, 56978, 121413, 257740, 545308, 1150272, 2419856, 5078336, 10633921, 22222338, 46353669, 96525324, 200686620, 416645184, 863834256, 1788756288, 3699688128, 7643727360, 15776156928, 32529718272
Offset: 0

Views

Author

Alois P. Heinz, Aug 06 2019

Keywords

Links

Crossrefs

Cf. A000045, A102291, A124091, A144115.

Programs

  • Maple
    a:= proc(n) option remember; add(a(n-j)+`if`((t->issqr(t+4)
      or issqr(t-4))(5*j^2), ceil(2^(n-j-1)), 0), j=1..n)
    end:
seq(a(n), n=0..33);
  • Mathematica
    a[n_] := a[n] = Sum[a[n - j] + With[{t = 5 j^2}, If[IntegerQ@Sqrt[t + 4] || IntegerQ@Sqrt[t - 4], Ceiling[2^(n - j - 1)], 0]], {j, 1, n}];
a /@ Range[0, 33] (* Jean-François Alcover, Dec 29 2020, after Alois P. Heinz *)

Formula

G.f.: Sum_{k>=2} x^Fibonacci(k)*(1-x)^2/(1-2*x)^2.
a(n) ~ c * 2^n * n, where c = 0.22756969930196647294851075611776578612085598114... - Vaclav Kotesovec, Aug 18 2019
c = A124091/4 - 3/8. - Vaclav Kotesovec, Mar 17 2024

A326957 Total number of noncomposite parts in all partitions of n.

Original entry on oeis.org

0, 1, 3, 6, 11, 19, 32, 50, 77, 115, 170, 244, 348, 486, 675, 923, 1253, 1682, 2246, 2968, 3904, 5094, 6616, 8533, 10962, 13997, 17808, 22538, 28426, 35689, 44670, 55678, 69199, 85692, 105826, 130261, 159935, 195778, 239092, 291191, 353854, 428925, 518848
Offset: 0

Views

Author

Omar E. Pol, Aug 08 2019

Keywords

Examples

			For n = 6 we have:
--------------------------------------
.                        Number of
Partitions             noncomposite
of 6                       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.

Crossrefs

First differs from A183088 at a(13).
Cf. A000041, A000070, A006128, A008578 (noncomposites), A037032, A144115, A144116, A144119, A326958, A326981.

Programs

  • Maple
    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], 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
  • Mathematica
    b[n_] := Sum[PrimeNu[k] PartitionsP[n-k], {k, 1, n}];
c[n_] := SeriesCoefficient[Product[1/(1-x^k), {k, 1, n}]/(1-x), {x, 0, n}];
a[n_] := b[n] + c[n-1];
a /@ Range[0, 50] (* Jean-François Alcover, Nov 15 2020 *)

Formula

a(n) = A037032(n) + A000070(n-1), n >= 1.
a(n) = A006128(n) - A326981(n).

A326981 Total number of composite parts in all partitions of n.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 3, 4, 9, 13, 22, 31, 51, 70, 105, 145, 210, 283, 398, 530, 726, 958, 1283, 1673, 2212, 2854, 3714, 4756, 6119, 7764, 9893, 12457, 15728, 19674, 24636, 30615, 38079, 47034, 58109, 71396, 87692, 107179, 130943, 159278, 193619, 234486, 283720
Offset: 0

Views

Author

Omar E. Pol, Aug 09 2019

Keywords

Examples

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

Crossrefs

Cf. A000041, A002808, A006128, A037032, A144115, A144116, A144119, A326957, A326982.

Programs

  • Maple
    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])])(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
  • Mathematica
    b[n_, i_] := b[n, i] = If[n==0 || i==1, {1, 0}, b[n, i-1] + # + {0, If[PrimeQ[i], 0, #[[1]]]}&[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 *)

Formula

a(n) = A144119(n) - A000070(n-1), n >= 1.
a(n) = A006128(n) - A326957(n).
Showing 1-8 of 8 results.

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