A296338 -id:A296338 - cp's OEIS frontend

A307666 Number of partitions of n into consecutive positive triangular numbers.

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

Offset: 1

Ilya Gutkovskiy, Apr 20 2019

nonn

Equivalently, number of ways n can be expressed as the difference between two tetrahedral numbers. - Charlie Neder, Apr 24 2019

Records: a(10)=2, a(2180)=3, a(10053736)=4. - Robert Israel, Aug 20 2019

			10 = 1 + 3 + 6, so a(10) = 2.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000217, A001227, A007294, A024940, A034706, A296338, A307614, A309783.

N:= 100:
V:= Vector(N):
for i from 1 while i*(i+1)/2 <= N do
  s:= i*(i+1)*(i+2)/6;
  for j from i-1 to 0 by -1 do
    t:= j*(j+1)*(j+2)/6;
    if s-t > N then break fi;
    V[s-t]:= V[s-t]+1
  od;
od:
convert(V,list); # Robert Israel, Aug 20 2019

G.f.: Sum_{i>=1} Sum_{j>=i} Product_{k=i..j} x^(k*(k+1)/2).

A297199 a(n) = number of partitions of n into consecutive positive cubes.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Seiichi Manyama, Jan 15 2018

nonn

			    1 = 1^3,                   so   a(1) = 1.
    8 = 2^3,                   so   a(8) = 1.
    9 = 1^3 + 2^3,             so   a(9) = 1.
   27 = 3^3,                   so  a(27) = 1.
   35 = 2^3 + 3^3,             so  a(35) = 1.
   36 = 1^3 + 2^3 + 3^3,       so  a(36) = 1.
   64 = 4^3,                   so  a(64) = 1.
   91 = 3^3 + 4^3,             so  a(91) = 1.
   99 = 2^3 + 3^3 + 4^3,       so  a(99) = 1.
  100 = 1^3 + 2^3 + 3^3 + 4^3, so a(100) = 1.

Antti Karttunen, Table of n, a(n) for n = 1..65537 (terms 1..10000 from Robert Israel)

Cf. A000578, A001227, A062682, A217843, A265845, A296338, A307609.

N:= 200: # to get a(1)..a(N)
F:= (a, b) -> (b^2*(b+1)^2-a^2*(a-1)^2)/4:
A:= Vector(N):
for b from 1 to floor(N^(1/3)) do
  for a from b to 1 by -1 do
     v:= F(a,b);
     if v > N then break fi;
     A[v]:= A[v]+1;
od od:
convert(A,list); # Robert Israel, Jan 15 2018, corrected Jan 29 2018

A297199(n) = { my(s=0, k=1, c); while((c=k^3) <= n, my(u=n-c, i=k); while(u>0, i++; c = i^3; u=u-c); s += (!u); k++); (s); }; \\ Antti Karttunen, Aug 22 2019

a(A217843(n)) >= 1 for n > 1.
a(n) >= 2 for n in A265845. - Robert Israel, Jan 15 2018
G.f.: Sum_{i>=1} Sum_{j>=i} Product_{k=i..j} x^(k^3). - Ilya Gutkovskiy, Apr 18 2019
a(A000578(n)) = A307609(n). - Antti Karttunen, Aug 22 2019

A299173 a(n) is the maximum number of squared consecutive positive integers into which the integer n can be partitioned.

Original entry on oeis.org

1, 0, 0, 1, 2, 0, 0, 0, 1, 0, 0, 0, 2, 3, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 3, 4, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 3, 0, 0, 0, 4, 5, 0, 0, 0, 0, 0, 2, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 1, 0, 0, 0, 2, 4, 0, 0, 0, 5, 6, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Jean-François Alcover, Feb 04 2018

a(k^2)>=1, the inequality being strict if k is in A097812.

			25 = 5^2 = 3^2 + 4^2 and no such partition is longer, so a(25) = 2.
30 = 1^2 + 2^2 + 3^2 + 4^2 and no such partition is longer, so a(30) = 4.
2018 = 7^2 + 8^2 + 9^2 + 10^2 + 11^2 + 12^2 + 13^2 + 14^2 + 15^2 + 16^2 + 17^2 + 18^2 and no such partition is longer, so a(2018) = 12. (This special example is due to _Seiichi Manyama_.) - _Jean-François Alcover_, Feb 05 2018

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A034705, A097812, A130052, A111044, A234304, A234311, A296338, A298467.

N:= 200: # to get a(1)..a(N)
A:= Vector(N):
S:= n -> n*(n+1)*(2*n+1)/6:
M:= floor(sqrt(N)):
for d from 1 to M do
  for b from d to M do
    s:= S(b) - S(b-d);
    if s > N then break fi;
    A[s]:= d
od od:
convert(A,list); # Robert Israel, Feb 04 2018

terms = 100; jmax = Ceiling[Sqrt[terms]]; kmax = Ceiling[(3*terms)^(1/3)]; Clear[a]; a[_] = 0; Do[r = Range[j, j + k - 1]; n = r . r; If[k > a[n], a[n] = k], {j, jmax}, {k, kmax}]; Array[a, terms]

A307608 Number of partitions of n^2 into consecutive positive squares.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1

Offset: 1

Ilya Gutkovskiy, Apr 18 2019

nonn

			29^2 = 20^2 + 21^2, so a(29) = 2.

Index entries for sequences related to sums of squares

Cf. A000290, A030273, A034705, A037444, A097812, A151557, A296338.

a(n) = [x^(n^2)] Sum_{i>=1} Sum_{j>=i} Product_{k=i..j} x^(k^2).
a(n) = A296338(A000290(n)).
a(n) >= 2 for n in A097812.

A368073 Number of representations of n as the difference of two positive square pyramidal numbers.

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Dec 10 2023

nonn

			a(25) = 2: 25 = 30 - 5 = 55 - 30.

Eric Weisstein's World of Mathematics, Square Pyramidal Number

Cf. A000330, A100073, A296338, A368072.

cp's OEIS Frontend

A307666 Number of partitions of n into consecutive positive triangular numbers.

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A297199 a(n) = number of partitions of n into consecutive positive cubes.

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A299173 a(n) is the maximum number of squared consecutive positive integers into which the integer n can be partitioned.

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Maple

Mathematica

A307608 Number of partitions of n^2 into consecutive positive squares.

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Examples

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A368073 Number of representations of n as the difference of two positive square pyramidal numbers.

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