A063890 -id:A063890 - cp's OEIS frontend

A348165 Number of solutions to +-1^2 +- 2^2 +- 3^2 +- ... +- n^2 = n.

1, 1, 0, 0, 1, 2, 0, 0, 2, 4, 0, 0, 19, 29, 0, 0, 127, 208, 0, 0, 1121, 1917, 0, 0, 10479, 19360, 0, 0, 113213, 204121, 0, 0, 1290968, 2363982, 0, 0, 15303057, 28397538, 0, 0, 187446097, 351339307, 0, 0, 2355979330, 4455357992, 0, 0, 30360404500, 57630025172

Offset: 0

Ilya Gutkovskiy, Jan 28 2022

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..200

Cf. A063890, A158092, A348892.

b:= proc(n, i) option remember; (m-> `if`(n>m, 0, `if`(n=m, 1,
      b(abs(n-i^2), i-1)+b(n+i^2, i-1))))((1+(3+2*i)*i)*i/6)
    end:
a:= n-> `if`(irem(n, 4)>1, 0, b(n$2)):
seq(a(n), n=0..50);  # Alois P. Heinz, Jan 28 2022

b[n_, i_] := b[n, i] = Function[m, If[n > m, 0, If[n == m, 1, b[Abs[n - i^2], i - 1] + b[n + i^2, i - 1]]]][(1 + (3 + 2*i)*i)*i/6];
a[n_] := If[Mod[n, 4] > 1, 0, b[n, n]];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Feb 26 2022, after Alois P. Heinz *)

from functools import lru_cache
@lru_cache(maxsize=None)
def b(n, i):
    if n > i*(i+1)*(2*i+1)//6: return 0
    if i == 0: return 1
    return b(n+i**2, i-1) + b(abs(n-i**2), i-1)
def a(n): return b(n, n)
print([a(n) for n in range(50)]) # Michael S. Branicky, Jan 28 2022

a(n) = [x^n] Product_{k=1..n} (x^(k^2) + 1/x^(k^2)).

A348892 Number of solutions to +-1^3 +- 2^3 +- 3^3 +- ... +- n^3 = n.

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 2, 4, 0, 0, 83, 69, 0, 0, 353, 414, 0, 0, 7800, 12496, 0, 0, 48162, 56870, 0, 0, 733392, 1253467, 0, 0, 4892337, 10022277, 0, 0, 45859303, 149422926, 0, 0, 623257759, 1339056922, 0, 0, 7453502893, 13446831198

Offset: 0

Ilya Gutkovskiy, Jan 28 2022

nonn

Cf. A063890, A158118, A348165.

from functools import lru_cache
@lru_cache(maxsize=None)
def b(n, i):
    if n > (i*(i+1)//2)**2: return 0
    if i == 0: return 1
    return b(n+i**3, i-1) + b(abs(n-i**3), i-1)
def a(n): return b(n, n)
print([a(n) for n in range(54)]) # Michael S. Branicky, Jan 28 2022

a(n) = [x^n] Product_{k=1..n} (x^(k^3) + 1/x^(k^3)).

A368243 Number of solutions to +- 1^2 +- 2^2 +- 3^2 +- ... +- n^2 = n^2.

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 0, 0, 2, 3, 0, 0, 5, 15, 0, 0, 127, 184, 0, 0, 819, 1382, 0, 0, 9441, 18176, 0, 0, 96562, 172371, 0, 0, 1192142, 2252342, 0, 0, 13869696, 25741462, 0, 0, 177056022, 334176492, 0, 0, 2207693292, 4182801839, 0, 0, 28966597122, 55125154468

Offset: 0

Ilya Gutkovskiy, Jan 22 2024

nonn

Cf. A063890, A158092, A348165.

b:= proc(n, i) option remember; (m-> `if`(n>m, 0, `if`(n=m, 1,
      b(abs(n-i^2), i-1) +b(n+i^2, i-1))))((1+(3+2*i)*i)*i/6)
    end:
a:= n-> `if`(irem(n, 4)>1, 0, b(n^2, n)):
seq(a(n), n=0..49);  # Alois P. Heinz, Jan 22 2024

a(n) = [x^(n^2)] Product_{k=1..n} (x^(k^2) + 1/x^(k^2)).

A368845 Number of solutions to +- 1^3 +- 2^3 +- 3^3 +- ... +- n^3 = n^3.

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 6, 4, 0, 0, 8, 187, 0, 0, 831, 1086, 0, 0, 7127, 3983, 0, 0, 20086, 120445, 0, 0, 674006, 1056938, 0, 0, 6983613, 5964500, 0, 0, 40031490, 142694311, 0, 0, 853687222, 1622335105, 0, 0, 10288998770, 12509111104

Offset: 0

Ilya Gutkovskiy, Jan 22 2024

nonn

Cf. A063890, A158118, A348892.

b:= proc(n, i) option remember; (m-> `if`(n>m, 0, `if`(n=m, 1,
      b(abs(n-i^3), i-1) +b(n+i^3, i-1))))((i*(i+1)/2)^2)
    end:
a:= n-> `if`(irem(n, 4)>1, 0, b(n^3, n)):
seq(a(n), n=0..53);  # Alois P. Heinz, Jan 22 2024

a(n) = [x^(n^3)] Product_{k=1..n} (x^(k^3) + 1/x^(k^3)).

A369390 a(n) = [x^prime(n)] Product_{k=1..n} (x^prime(k) + 1 + 1/x^prime(k)).

Original entry on oeis.org

1, 1, 2, 4, 6, 13, 31, 77, 188, 449, 1191, 3014, 7920, 21498, 57833, 154073, 412733, 1141274, 3106771, 8576977, 24015471, 66489615, 185886699, 517837152, 1435964205, 4034697191, 11438332340, 32395341851, 92396549863, 263233759500, 736127855014, 2093027604453

Offset: 1

Ilya Gutkovskiy, Jan 22 2024

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..400

Cf. A063890, A316706, A350880.

s:= proc(n) s(n):= `if`(n<1, 0, ithprime(n)+s(n-1)) end:
b:= proc(n, i) option remember; `if`(n>s(i), 0, `if`(i=0, 1,
      b(n, i-1)+b(n+ithprime(i), i-1)+b(abs(n-ithprime(i)), i-1)))
    end:
a:= n-> b(ithprime(n), n):
seq(a(n), n=1..40);  # Alois P. Heinz, Jan 22 2024

Table[Coefficient[Product[(x^Prime[k] + 1 + 1/x^Prime[k]), {k, 1, n}], x, Prime[n]], {n, 1, 32}]

A350695 Number of solutions to +-2 +- 3 +- 5 +- 7 +- ... +- prime(n-1) = n.

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 1, 1, 4, 5, 9, 15, 26, 45, 77, 137, 243, 434, 774, 1408, 2554, 4667, 8627, 15927, 29559, 54867, 101688, 189425, 355315, 668598, 1264180, 2395462, 4506221, 8507311, 16084405, 30545142, 57898862, 110199367, 209957460, 400430494, 765333684

Offset: 0

Ilya Gutkovskiy, Jan 29 2022

nonn

Cf. A000040, A022894, A058377, A063890, A113040, A261061, A350404.

Table[SeriesCoefficient[Product[x^Prime[k] + 1/x^Prime[k], {k, n - 1}], {x, 0, n}], {n, 0, 40}] (* Stefano Spezia, Jan 30 2022 *)

from sympy import sieve, primerange
from functools import cache
@cache
def b(n, i):
    maxsum = 0 if i < 2 else sum(p for p in primerange(2, sieve[i-1]+1))
    if n > maxsum: return 0
    if i < 2: return 1
    return b(n+sieve[i-1], i-1) + b(abs(n-sieve[i-1]), i-1)
def a(n): return b(n, n)
print([a(n) for n in range(41)]) # Michael S. Branicky, Jan 29 2022

a(n) = [x^n] Product_{k=1..n-1} (x^prime(k) + 1/x^prime(k)).

A351002 Number of solutions to +-1 +- 3 +- 6 +- 10 +- ... +- n*(n + 1)/2 = n.

Original entry on oeis.org

1, 1, 1, 0, 0, 1, 3, 0, 4, 3, 9, 0, 27, 43, 71, 0, 190, 318, 604, 0, 1846, 3127, 5664, 0, 19048, 34065, 62045, 0, 205713, 378243, 705836, 0, 2403370, 4434940, 8276125, 0, 28980680, 54167797, 101541048, 0, 358095372, 674776903, 1274888645, 0, 4551828850, 8612421500

Offset: 0

Ilya Gutkovskiy, Jan 29 2022

nonn

Cf. A000217, A058498, A063890, A158380, A348165, A350287.

from functools import lru_cache
@lru_cache(maxsize=None)
def b(n, i):
    if n > i*(i+1)*(i+2)//6: return 0
    if i == 0: return 1
    return b(n+i*(i+1)//2, i-1) + b(abs(n-i*(i+1)//2), i-1)
def a(n): return b(n, n)
print([a(n) for n in range(50)]) # Michael S. Branicky, Jan 29 2022

a(n) = [x^n] Product_{k=1..n} (x^(k*(k+1)/2) + 1/x^(k*(k+1)/2)).

A368206 a(n) = [x^n] Product_{k=1..n} (x^(k^4) + 1/x^(k^4)).

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 8, 13, 0, 0, 272, 0, 0, 0, 5400, 8915, 0, 0, 30433, 1590, 0, 0, 68638, 73470, 0, 0, 90808, 6638072, 0, 0, 127356, 319803, 0, 0, 20130146, 559282596, 0, 0, 1507066936, 3820244957, 0, 0

Offset: 0

Ilya Gutkovskiy, Jan 25 2024

nonn

Cf. A000583, A063890, A158465, A348165, A348892.

b:= proc(n, i) option remember; (m-> `if`(n>m, 0, `if`(n=m, 1,
      b(abs(n-i^4), i-1)+b(n+i^4, i-1))))(i*(i+1)*(2*i+1)*(3*i^2+3*i-1)/30)
    end:
a:= n-> `if`(irem(n, 4)>1, 0, b(n, n)):
seq(a(n), n=0..43);  # Alois P. Heinz, Jan 25 2024

a(46)-a(59) from Alois P. Heinz, Jan 25 2024

A367416 Triangle read by rows: T(n,k) = number of solutions to +- 1^k +- 2^k +- 3^k +- ... +- n^k is a k-th power, n >= 2.

Original entry on oeis.org

4, 8, 1, 16, 1, 32, 0, 2, 64, 6, 128, 8, 256, 16, 4, 512, 26, 1024, 17, 10, 2048, 67, 4, 3, 4096, 100, 10, 8192, 137, 34, 6, 16384, 426, 28, 1, 32768, 661, 96, 6, 65536, 1351, 146, 16, 8, 131072, 2637, 230, 15, 262144, 3831, 258, 40, 524288, 8095, 1130, 50

Offset: 2

Jean-Marc Rebert, Jan 26 2024

In the case of n = 1, there are solutions for all k. In particular, 1^k is always a k-th power and -(1^k) is a k-th power for odd k. As a formula: T(1,k) = 1 + (k mod 2). This row is not included in the sequence.

			Triangle begins:
            k = 1      2     3   4  5
  n= 2:         4;
  n= 3:         8,     1;
  n= 4:        16,     1;
  n= 5:        32,     0,    2;
  n= 6:        64,     6;
  n= 7:       128,     8;
  n= 8:       256,    16,    4;
  n= 9:       512,    26;
  n=10:      1024,    17,   10;
  n=11:      2048,    67,    4,  3;
  n=12:      4096,   100,   10;
  n=13:      8192,   137,   34,  6;
  n=14:     16384,   426,   28,  1;
  n=15:     32768,   661,   96,  6;
  n=16:     65536,  1351,  146, 16, 8;
  n=17:    131072,  2637,  230, 15;
  n=18:    262144,  3831,  258, 40;
  n=19:    524288,  8095, 1130, 50;
  n=20:   1048576, 15241,  854, 77, 6;
  ...
The T(6,2) = 6 solutions are:
  - 1^2 - 2^2 + 3^2 - 4^2 + 5^2 + 6^2 = 49 = 7^2,
  - 1^2 - 2^2 + 3^2 + 4^2 + 5^2 - 6^2 =  9 = 3^2,
  - 1^2 - 2^2 + 3^2 + 4^2 + 5^2 + 6^2 = 81 = 9^2,
  + 1^2 - 2^2 + 3^2 - 4^2 - 5^2 + 6^2 =  1 = 1^2,
  + 1^2 + 2^2 - 3^2 + 4^2 + 5^2 - 6^2 =  1 = 1^2,
  + 1^2 + 2^2 + 3^2 - 4^2 - 5^2 + 6^2 =  9 = 3^2.

Cf. A063890, A215083, A368243, A368845, A369629.

f(k,u)=my(x=0,v=vector(#u));for(i=1,#u,u[i]=if(u[i]==0,-1,1);v[i]=i^k);u*v~
is(k,u)=my(x=f(k,u));ispower(x,k)
T(n,k)=my(u=vector(n,i,[0,1]),nbsol=0);if(k%2==1,u[1]=[1,1]);forvec(X=u,if(is(k,X),nbsol++));if(k%2==1,nbsol*=2);nbsol

A367736 a(0) = 1; for n > 0, a(n) is the coefficient of x^a(n-1) in the expansion of Product_{k=0..n-1} (x^a(k) + 1 + 1/x^a(k)).

Original entry on oeis.org

1, 1, 2, 4, 6, 11, 19, 32, 58, 97, 163, 290, 501, 856, 1483, 2561, 4424, 7652, 13273, 23024, 39784, 69001, 119614, 207042, 358746, 621117, 1075865, 1864050, 3227724, 5590548, 9682402, 16770033, 29049713, 50310453, 87142439, 150939346, 261424583, 452810957

Offset: 0

Ilya Gutkovskiy, Jan 24 2024

Cf. A063890, A265853, A316706.

a[0] = 1; a[n_] := a[n] = Coefficient[Product[x^a[k] + 1 + 1/x^a[k], {k, 0, n - 1}], x, a[n - 1]]; Table[a[n], {n, 0, 28}]

from itertools import islice
from collections import Counter
def A367736_gen(): # generator of terms
    c, b = {0:1}, 1
    while True:
        yield b
        d = Counter(c)
        for k in c:
            e = c[k]
            d[k+b] += e
            d[k-b] += e
        c = d
        b = c[b]
A367736_list = list(islice(A367736_gen(),20)) # Chai Wah Wu, Feb 05 2024

a(29)-a(37) from Chai Wah Wu, Feb 05 2024

cp's OEIS Frontend

A348165 Number of solutions to +-1^2 +- 2^2 +- 3^2 +- ... +- n^2 = n.

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A348892 Number of solutions to +-1^3 +- 2^3 +- 3^3 +- ... +- n^3 = n.

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A368243 Number of solutions to +- 1^2 +- 2^2 +- 3^2 +- ... +- n^2 = n^2.

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A368845 Number of solutions to +- 1^3 +- 2^3 +- 3^3 +- ... +- n^3 = n^3.

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A369390 a(n) = [x^prime(n)] Product_{k=1..n} (x^prime(k) + 1 + 1/x^prime(k)).

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A350695 Number of solutions to +-2 +- 3 +- 5 +- 7 +- ... +- prime(n-1) = n.

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A351002 Number of solutions to +-1 +- 3 +- 6 +- 10 +- ... +- n*(n + 1)/2 = n.

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A368206 a(n) = [x^n] Product_{k=1..n} (x^(k^4) + 1/x^(k^4)).

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A367416 Triangle read by rows: T(n,k) = number of solutions to +- 1^k +- 2^k +- 3^k +- ... +- n^k is a k-th power, n >= 2.

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A367736 a(0) = 1; for n > 0, a(n) is the coefficient of x^a(n-1) in the expansion of Product_{k=0..n-1} (x^a(k) + 1 + 1/x^a(k)).