A111253 -id:A111253 - cp's OEIS frontend

A158465 Number of solutions to +-1+-2^4+-3^4+-4^4...+-n^4=0.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 16, 18, 0, 0, 32, 100, 0, 0, 424, 510, 0, 0, 2792, 5988, 0, 0, 29058, 45106, 0, 0, 276828, 473854, 0, 0, 2455340, 4777436, 0, 0, 27466324, 46429640, 0, 0, 280395282, 526489336, 0, 0, 3193589950, 5661226928, 0, 0

Offset: 1

Pietro Majer, Mar 19 2009

nonn

Constant term in the expansion of (x + 1/x)(x^16 + 1/x^16)..(x^n^4 + 1/x^n^4).
a(n)=0 for any n=1 (mod 4) or n=2 (mod 4).
Andrica & Tomescu give a more general integral formula than the one below. The asymptotic formula below is a conjecture by Andrica & Ionascu; it remains unproven. - Jonathan Sondow, Nov 11 2013

			For n=16 the a(16) = 2 solutions are +1 +16 +81 +256 -625 -1296 -2401 +4096 +6561 +10000 +14641 +20736 -28561 -38416 -50625 +65536 = 0 and the opposite.

D. Andrica and E. J. Ionascu, Variations on a result of Erdős and SurányiINTEGERS 2013 slides.
Dorin Andrica and Ioan Tomescu, On an Integer Sequence Related to a Product of Trigonometric Functions, and Its Combinatorial Relevance, J. Integer Sequences, 5 (2002), Article 02.2.4.

Cf. A063865, A158092, A158118, A158380, A019568.

A111253(n) = a(n)/2. - Alois P. Heinz, Oct 31 2011

N:=32: p:=1 a:=[]: for n from 32 to N do p:=expand
(p*(x^(n^4)+x^(-n^4))): a:=[op(a), coeff(p,x,0)]: od:a;

Integral representation: a(n) = ((2^n)/Pi)*int_0^pi prod_{k=1}^n cos(x*k^4) dx.

Asymptotic formula: a(n) = (2^n)*sqrt(18/(Pi*n^9))*(1+o(1)) as n->infinity; n=-1 or 0 (mod 4).

a(35)-a(58) from Alois P. Heinz, Oct 31 2011

A369629 Number of solutions to +- 1^4 +- 2^4 +- 3^4 +- ... +- n^4 = 0 or 1.

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 5, 0, 16, 18, 0, 21, 32, 100, 126, 0, 424, 510, 0, 1428, 2792, 5988, 9786, 7, 29058, 45106, 22, 150437, 276828, 473854, 836737, 1838, 2455340, 4777436, 15847, 14696425, 27466324, 46429640, 83010230, 738627

Offset: 0

Ilya Gutkovskiy, Jan 28 2024

nonn

Chai Wah Wu, Table of n, a(n) for n = 0..73

Cf. A000583, A025591, A111253, A158465, A350403, A350861.

from itertools import count, islice
from collections import Counter
def A369629_gen(): # generator of terms
    ccount = Counter({0:1})
    yield 1
    for i in count(1):
        bcount = Counter()
        for a in ccount:
            bcount[a+(j:=i**4)] += ccount[a]
            bcount[a-j] += ccount[a]
        ccount = bcount
        yield(ccount[0]+ccount[1])
A369629_list = list(islice(A369629_gen(),20)) # Chai Wah Wu, Jan 29 2024

a(46)-a(50) from Chai Wah Wu, Jan 29 2024

A369732 Number of solutions to +- 1^4 +- 2^4 +- 3^4 +- ... +- n^4 = 1.

Original entry on oeis.org

0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 21, 0, 0, 126, 0, 0, 0, 0, 1428, 0, 0, 9786, 7, 0, 0, 22, 150437, 0, 0, 836737, 1838, 0, 0, 15847, 14696425, 0, 0, 83010230, 738627, 0, 0, 6024822, 1640652994, 0, 0, 10271377082, 226033104, 0, 0

Offset: 0

Ilya Gutkovskiy, Jan 30 2024

nonn

Cf. A000583, A063866, A111253, A158465, A368206, A369629, A369730, A369731.

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

a(46)-a(60) from Alois P. Heinz, Jan 30 2024

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A158465 Number of solutions to +-1+-2^4+-3^4+-4^4...+-n^4=0.

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A369629 Number of solutions to +- 1^4 +- 2^4 +- 3^4 +- ... +- n^4 = 0 or 1.

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A369732 Number of solutions to +- 1^4 +- 2^4 +- 3^4 +- ... +- n^4 = 1.

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