A002704 -id:A002704 - cp's OEIS frontend

A262570 a(n) = A002704(n)/2.

1, 13, 469, 21400, 1065229, 55778797, 3020636341, 167544629317, 9461343754981, 541786421337805, 31372013730812824, 1833216802356228733, 107939805322734748117, 6396432908013911687437, 381139174656828902370421, 22819171231066917509661277

Offset: 0

N. J. A. Sloane, Oct 20 2015

nonn

Alexander Rosa and Štefan Znám, A combinatorial problem in the theory of congruences (Russian with English summary), Mat.-Fys. Casopis Sloven. Akad. Vied 15 1965 49-59. [Annotated scanned copy.]
Alexander Rosa and Štefan Znám, A remark on a combinatorial problem (Russian with English summary), Mat.-Fyz. Casopis Sloven. Akad. Vied 15 1965 313-316. [Annotated scanned copy] See Table 3.

Cf. A002704, A262583, A262584.

More terms from R. J. Mathar, Oct 21 2015

A262583 a(n) = A002704(n)-2.

Original entry on oeis.org

0, 24, 936, 42798, 2130456, 111557592, 6041272680, 335089258632, 18922687509960, 1083572842675608, 62744027461625646, 3666433604712457464, 215879610645469496232, 12792865816027823374872, 762278349313657804740840, 45638342462133835019322552

Offset: 0

N. J. A. Sloane, Oct 20 2015

nonn

Alexander Rosa and Štefan Znám, A combinatorial problem in the theory of congruences (Russian with English summary), Mat.-Fys. Casopis Sloven. Akad. Vied 15 1965 49-59. [Annotated scanned copy.]
Alexander Rosa and Štefan Znám, A remark on a combinatorial problem (Russian with English summary), Mat.-Fyz. Casopis Sloven. Akad. Vied 15 1965 313-316. [Annotated scanned copy] See Table 3.

Cf. A002704, A262570, A262584.

More terms from R. J. Mathar, Oct 21 2015

a(15) corrected by Georg Fischer, Jun 27 2020

A262584 a(n) = (A002704(n) - 2)/2.

Original entry on oeis.org

0, 12, 468, 21399, 1065228, 55778796, 3020636340, 167544629316, 9461343754980, 541786421337804, 31372013730812823, 1833216802356228732, 107939805322734748116, 6396432908013911687436, 381139174656828902370420

Offset: 0

N. J. A. Sloane, Oct 20 2015

nonn

Alexander Rosa and Štefan Znám, A combinatorial problem in the theory of congruences (Russian with English summary), Mat.-Fys. Casopis Sloven. Akad. Vied 15 1965 49-59. [Annotated scanned copy.]
Alexander Rosa and Štefan Znám, A remark on a combinatorial problem (Russian with English summary), Mat.-Fyz. Casopis Sloven. Akad. Vied 15 1965 313-316. [Annotated scanned copy] See Table 3.

Cf. A002704, A262570, A262583.

More terms from R. J. Mathar, Oct 21 2015

A002703 Sets with a congruence property.

Original entry on oeis.org

0, 0, 0, 2, 6, 14, 24, 46, 88, 162, 300, 562, 1056, 1982, 3742, 7082, 13438, 25574, 48768, 93198, 178480, 342392, 657918, 1266202, 2440318, 4709374, 9099504, 17602322, 34087010, 66076414, 128207976, 248983550, 483939976, 941362694, 1832519262, 3569842946, 6958934352

Offset: 3

N. J. A. Sloane

nonn

a(n) is the sequence k(n) in Table 3 of the first 1965 paper. - N. J. A. Sloane, Oct 20 2015

See English summary at the end of the first 1965 paper, which is repeated in the Zentralblatt review. - Jonathan Sondow, Nov 02 2013

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alexander Rosa and Štefan Znám, A combinatorial problem in the theory of congruences (Russian with English summary), Mat.-Fys. Casopis Sloven. Akad. Vied 15 (1965) 49-59. [Annotated scanned copy.] This is the sequence k(n) in Table 3. Tables 1 and 2 are A053632 and A178666 respectively.
Alexander Rosa and Štefan Znám, A remark on a combinatorial problem (Russian with English summary), Mat.-Fyz. Casopis Sloven. Akad. Vied 15 (1965) 313-316. [Annotated scanned copy]
Zentralblatt, Review of Rosa and Znám, A combinatorial problem in the theory of congruences.

Cf. A002704, A002705.
See A262567, A262568, A262569 for other versions.
Tables 1 and 2 of the first Rosa-Znám 1965 paper are A053632 and A178666 respectively.

A002703 := proc(n)
    A262568(n)-2 ;
end proc: # R. J. Mathar, Oct 21 2015

A178666[r_, s_] := SeriesCoefficient[Product[ (1 + x^(2i+1)), {i, 0, Floor[(s-1)/2]}], {x, 0, r}];
kstart[n_, m_] := Ceiling[Binomial[n+1, 2]/m];
kend[n_, m_] := Floor[Binomial[3n+1, 2]/3/m];
A262568[n_] := Module[{s = 2n-1, m = 2n+1, Q=0, vi, k}, For[k = kstart[n, m], k <= kend[n, m], k++, vi = m k - Binomial[n+1, 2]; Q += A178666[vi, s] ]; Q];
a[n_] := A262568[n] - 2;
a /@ Range[3, 39] (* Jean-François Alcover, Mar 24 2020, after R. J. Mathar in A262568 *)

More terms from R. J. Mathar, Oct 21 2015

A002705 Sets with a congruence property.

Original entry on oeis.org

0, 4, 40, 468, 5828, 76260, 1032444, 14316584, 202116108, 2893451652, 41886157564, 611902123284, 9007199254740, 133439988963012, 1987795697598012, 29752813022112180, 447193795726343004, 6746237832670921768, 102105221251235572188

Offset: 0

N. J. A. Sloane

nonn

The values for k=1, Q(n,1) in table 1 on page 315 for n = 3,5,7,9,... are 0, 2, 6, 18, 62, 210, 728, 2570, 9198, 33288, 121574, 447394, 1657008, 6170930, 23091222, 86767016, 327235610, 1238188770, 4698767640 ... (see A262590), - R. J. Mathar, Oct 21 2015

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alexander Rosa and Štefan Znám, A combinatorial problem in the theory of congruences (Russian with English summary), Mat.-Fys. Casopis Sloven. Akad. Vied 15 1965 49-59. [Annotated scanned copy.]
Alexander Rosa and Štefan Znám, A remark on a combinatorial problem (Russian with English summary), Mat.-Fyz. Casopis Sloven. Akad. Vied 15 1965 313-316. [Annotated scanned copy] See Table 2.

Cf. A002703, A002704, A262585, A262590.

p := proc(r,s,k)
    option remember;
    if r = 0 then
        1;
    elif r < 0 then
        0;
    elif s < 0 then
        0;
    elif igcd(s,2*k+1) > 1 then
        procname(r,s-1,k) ;
    else
        procname(r,s-1,k)+procname(r-s,s-1,k) ;
    end if;
end proc:
Q := proc(n,k)
    local q,knrat,alpha,m ;
    q := 0 ;
    knrat := (2*k*n^2+n^2+k^2)/4/k ;
    if type(knrat,'integer') then
        for alpha from 0 to knrat do
            m := 2*n+n/k ;
            if modp(2*alpha,m) = modp(knrat,m) then
                q := q+p(alpha,n+(n-k)/2/k,k) ;
            end if;
        end do:
    end if;
    q ;
end proc:
A002705 := proc(n)
    nloc := 2+4*n ;
    Q(nloc,2) ;
end proc: # R. J. Mathar, Oct 21 2015

p[r_, s_, k_] := p[r, s, k] = Which[r == 0, 1, r < 0, 0, s < 0, 0, GCD[s, 2 k + 1] > 1, p[r, s - 1, k], True, p[r, s - 1, k] + p[r - s, s - 1, k]];
Q[n_, k_] := Module[{q = 0, knrat, alpha, m}, knrat = (2 k n^2 + n^2 + k^2)/4/k; If[IntegerQ[knrat], For[alpha = 0, alpha <= knrat, alpha++, m = 2 n + n/k; If[Mod[2 alpha, m] == Mod[knrat, m], q += p[alpha, n + (n - k)/2/k, k]]]]; q];
a[n_] := Q[4 n + 2, 2];
a /@ Range[0, 18] (* Jean-François Alcover, Mar 27 2020, after R. J. Mathar *)

See Maple code!

More terms from R. J. Mathar, Oct 21 2015

cp's OEIS Frontend

A262570 a(n) = A002704(n)/2.

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A262583 a(n) = A002704(n)-2.

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A262584 a(n) = (A002704(n) - 2)/2.

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A002703 Sets with a congruence property.

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A002705 Sets with a congruence property.

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