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 1-5 of 5 results.

A036820 Number of partitions satisfying (cn(2,5) = cn(3,5) = 0).

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 4, 4, 5, 7, 10, 12, 14, 16, 21, 27, 33, 37, 44, 54, 68, 80, 92, 106, 129, 155, 182, 207, 240, 283, 337, 389, 444, 508, 594, 692, 797, 902, 1030, 1187, 1373, 1564, 1770, 2004, 2295, 2624, 2978, 3349, 3783, 4293, 4880, 5501, 6174, 6932, 7830, 8834
Offset: 0

Views

Author

Olivier Gérard

Keywords

Comments

For a given partition cn(i,n) means the number of its parts equal to i modulo n.
Short: (2=3 := 0).
It appears that this sequence is related to the generalized heptagonal numbers A085787 in the same way as the partition numbers A000041 are related to the generalized pentagonal numbers A001318. (See the table in comments section of A195825.) Conjecture: Column 1 of triangle A195837. Also 1 together with the row sums of triangle A195837. Also column 3 of the square array A195825. - Omar E. Pol, Oct 08 2011
Note that this sequence contains two plateaus: [1, 1, 1, 1] and [4, 4]. For more information see A195825 and A210843. - Omar E. Pol, Jun 23 2012

Examples

			G.f. = 1 + x + x^2 + x^3 + 2*x^4 + 3*x^5 + 4*x^6 + 4*x^7 + 5*x^8 + 7*x^9 + 10*x^10 + ...
G.f. = q^-9 + q^31 + q^71 + q^111 + 2*q^151 + 3*q^191 + 4*q^231 + 4*q^271 + 5*q^311 + ... - _Michael Somos_, Sep 08 2012

Links

Crossrefs

Cf. A113429.

Programs

  • Maple
    with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*[1, 1, 0, 0, 1]
      [1+irem(d, 5)], d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Apr 04 2014
  • Mathematica
    a[n_] := a[n] = If[n == 0, 1, Sum[ Sum[ d*{1, 1, 0, 0, 1}[[1 + Mod[d, 5]]], {d, Divisors[j]}] * a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Apr 15 2015, after Alois P. Heinz *)
  • PARI
    {a(n) = if( n<0, 0, polcoeff( 1 / prod( k=1, (n+4)\5, (1 - x^(5*k - 4)) * (1 - x^(5*k - 1)) * (1 - x^(5*k)), 1 + x * O(x^n)), n))}; /* Michael Somos, Feb 09 2012 */
(GW-BASIC)' A program with two A-numbers:
10 Dim A085787(100), A057077(100), a(100): a(0)=1
20 For n = 1 to 56: For j = 1 to n
30 If A085787(j) <= n then a(n) = a(n) + A057077(j-1)*a(n - A085787(j))
40 Next j: Print a(n-1);: Next n ' Omar E. Pol, Jun 10 2012

Formula

Euler transform of period 5 sequence [1, 0, 0, 1, 1, ...]. - Michael Somos, Feb 09 2012
Expansion of 1 / f(-x, -x^4) in powers of x where f(, ) is Ramanujan's general theta function. - Michael Somos, Sep 08 2012
Convolution inverse of A113429. - Michael Somos, Feb 09 2012
G.f.: 1 / (Product_{k>0} (1 - x^(5*k)) * (1 - x^(5*k - 1)) * (1 - x^(5*k - 4))). - Michael Somos, Sep 08 2012
G.f.: 1 / (Sum_{k in Z} (-1)^k * x^(k * (5*k + 3) / 2)). - Michael Somos, Sep 08 2012
a(n) ~ sqrt(1+sqrt(5)) * exp(sqrt(2*n/5)*Pi) / (2^(5/2)*5^(1/4)*n). - Vaclav Kotesovec, Oct 06 2015
a(n) = (1/n)*Sum_{k=1..n} A284361(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 25 2017

A113429 Expansion of f(-x, -x^4) in powers of x where f(, ) is Ramanujan's general theta function.

Original entry on oeis.org

1, -1, 0, 0, -1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 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, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0
Offset: 0

Views

Author

Michael Somos, Oct 31 2005

Keywords

Comments

For the g.f. identity see the Hardy-Wright reference, Theorem 355 on p. 284. - Wolfdieter Lang, Oct 28 2016

Examples

			G.f. = 1 - x - x^4 + x^7 + x^13 - x^18 - x^27 + x^34 + x^46 - x^55 - x^70 + ...
G.f. = q^9 - q^49 - q^169 + q^289 + q^529 - q^729 - q^1089 + q^1369 + q^1849 + ...

References

  • G. H. Hardy, Ramanujan, AMS Chelsea Publ., Providence, RI, 2002, p. 93.
  • G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fifth ed., Clarendon Press, Oxford, 2003, p. 284.

Links

Crossrefs

Cf. A085787, A113428.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^5] QPochhammer[ x^4, x^5] QPochhammer[ x^5], {x, 0, n}]; (* Michael Somos, Jun 26 2017 *)
a[ n_] := Module[{m = 40 n + 9, k}, If[IntegerQ[k = Sqrt[m]], If[Mod[k, 10] == 7, k = -k]; (-1)^Quotient[k, 10], 0]]; (* Michael Somos, Jun 26 2017 *)
  • PARI
    {a(n) = if( n<0, 0, polcoeff( prod(k=1, n, 1 - x^k*[1, 1, 0, 0, 1][k%5 + 1], 1 + x * O(x^n)), n))};
  • PARI
    {a(n) = my(m, k); if( n<0, 0, issquare(m = 40*n + 9, &k), if( k%10==7, k=-k); (-1)^(k\10), 0)}; /* Michael Somos, Oct 29 2016 */

Formula

Euler transform of period 5 sequence [-1, 0, 0, -1, -1, ...].
|a(n)| is the characteristic function of A085787.
G.f.: Product_{k>0} (1 - x^(5*k)) * (1 - x^(5*k-1)) * (1 - x^(5*k-4)) = Sum_{k in Z} (-1)^k * x^((5*k^2+3*k)/2).
f(a, b) = Sum_{k in Z} a^((k^2+k)/2) * b^((k^2-k)/2) is Ramanujan's general theta function.
G.f.: Sum_{n>=0} (x^(n*(n+1)) * Product_{k>=n+1} (1-x^k)). - Joerg Arndt, Apr 07 2011
From Wolfdieter Lang, Oct 30 2016: (Start)
a(n) = (-1)^k if n = b(2*k) for k >= 0, a(n) = (-1)^k if n = b(2*k-1), for k >= 1, and a(n) = 0 otherwise, where b(n) = A085787(n). See the second formula.
G.f.: Sum_{n>=0} (-1)^n*x^(n*(5*n+3)/2)*(1-x^(2*n+1)). See the Hardy reference, p. 93, G_1(x,x) from eq. (6.11.1) with C_n(x,x) = 1.
(End)
G.f.: Sum_{n>=0} (-1)^n*x^(n*(5*n-3)/2)*(1-x^(4*(2*n+1))). Reordered G_1(x,x) from the preceding formula. This is G_4(x,x) from Hardy, p. 93, eq. (6.11.1) with C_n(x,x) = 1. Note that Hardy uses only G_0, G_1 and G_2. - Wolfdieter Lang, Nov 01 2016
a(n) = -(1/n)*Sum_{k=1..n} A284361(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 25 2017

A284362 a(n) = Sum_{d|n, d = 0, 1, or 5 mod 6} d.

Original entry on oeis.org

1, 1, 1, 1, 6, 7, 8, 1, 1, 6, 12, 19, 14, 8, 6, 1, 18, 25, 20, 6, 8, 12, 24, 43, 31, 14, 1, 8, 30, 42, 32, 1, 12, 18, 48, 73, 38, 20, 14, 6, 42, 56, 44, 12, 6, 24, 48, 91, 57, 31, 18, 14, 54, 79, 72, 8, 20, 30, 60, 114, 62, 32, 8, 1, 84, 84, 68, 18, 24, 48, 72
Offset: 1

Views

Author

Seiichi Manyama, Mar 25 2017

Keywords

Examples

			From _Peter Bala_, Dec 11 2020: (Start)
n = 20: n is not of the form m*(3*m +- 2), so e(n) = 0 and a(20) = a(19) + a(15) - a(12) - a(4) = 20 + 6 - 19 - 1 = 6;
n = 21: n = m*(3*m - 2) for m = 3, so e(n) = 21 and a(21) = 21 + a(20) + a(16) - a(13) - a(5) = 21 + 6 + 1 - 14 - 6 = 8;
n = 40: n = m*(3*m - 2) for m = 4, so e(n) = -40 and a(4) = -40 + a(39) + a(35) - a(32) - a(24) + a(19) + a(7) = -40 + 14 + 48  - 1 - 43 + 20 + 8 = 6. (End)

Links

Crossrefs

Cf. A089802 (f(-x, -x^5)), A195848 (1/f(-x, -x^5)), A222171.
Cf. Sum_{d|n, d = 0, 1, or k-1 mod k} d: A000203 (k=3), A284361 (k=5), this sequence (k=6), A284363 (k=7), A284372 (k=12).

Programs

  • Mathematica
    Table[Sum[If[Mod[d, 6] <2 || Mod[d, 6]==5, d, 0], {d, Divisors[n]}], {n, 80}] (* Indranil Ghosh, Mar 25 2017 *)
  • PARI
    a(n) = sumdiv(n, d, ((d + 1) % 6 < 3) * d); \\ Amiram Eldar, Apr 12 2024

Formula

From Peter Bala, Dec 11 2020: (Start)
O.g.f.: Sum_{k >= 1} ( (6*k)*x^(6*k)/(1 - x^(6*k)) + (6*k-1)*x^(6*k-1)/(1 - x^(6*k-1)) + (6*k-5)*x^(6*k-5)/(1 - x^(6*k-5)) ).
Define a(n) = 0 for n < 1. Then a(n) = e(n) + a(n-1) + a(n-5) - a(n-8) - a(n-16) + + - -, where [1, 5, 8, 16, ...] is the sequence of generalized octagonal numbers A001082, and e(n) = (-1)^(m+1)*n if n is a generalized octagonal number of the form m*(3*m+-2); otherwise e(n) = 0. Examples of this recurrence are given below. (End)
Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^2/24 = A222171 = 0.411233... . - Amiram Eldar, Apr 12 2024

A284363 a(n) = Sum_{d|n, d = 0, 1, or 6 mod 7} d.

Original entry on oeis.org

1, 1, 1, 1, 1, 7, 8, 9, 1, 1, 1, 7, 14, 22, 16, 9, 1, 7, 1, 21, 29, 23, 1, 15, 1, 14, 28, 50, 30, 22, 1, 9, 1, 35, 43, 43, 1, 1, 14, 29, 42, 91, 44, 23, 16, 1, 1, 63, 57, 51, 1, 14, 1, 34, 56, 114, 58, 30, 1, 42, 1, 63, 92, 73, 14, 29, 1, 35, 70, 127, 72, 51, 1, 1
Offset: 1

Views

Author

Seiichi Manyama, Mar 25 2017

Keywords

Links

Crossrefs

Cf. A232714.
Cf. Sum_{d|n, d = 0, 1, or k-1 mod k} d: A000203 (k=3), A284361 (k=5), A284362 (k=6), this sequence (k=7), A284372 (k=12).

Programs

  • Mathematica
    Table[Sum[If[Mod[d, 7] <2 || Mod[d, 7]==6, d, 0], {d, Divisors[n]}], {n, 80}] (* Indranil Ghosh, Mar 25 2017 *)
  • PARI
    a(n) = sumdiv(n, d, ((d + 1) % 7 < 3) * d); \\ Amiram Eldar, Apr 12 2024

Formula

Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^2/28 = 0.352485... . - Amiram Eldar, Apr 12 2024

A284372 a(n) = Sum_{d|n, d = 0, 1, or 11 mod 12} d.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 12, 13, 14, 1, 1, 1, 1, 1, 1, 1, 1, 12, 24, 37, 26, 14, 1, 1, 1, 1, 1, 1, 12, 1, 36, 49, 38, 1, 14, 1, 1, 1, 1, 12, 1, 24, 48, 85, 50, 26, 1, 14, 1, 1, 12, 1, 1, 1, 60, 73, 62, 1, 1, 1, 14, 12, 1, 1, 24, 36, 72, 145, 74, 38, 26, 1
Offset: 1

Views

Author

Seiichi Manyama, Mar 25 2017

Keywords

Examples

			From _Peter Bala_, Dec 11 2020: (Start)
n = 24: n is not of the form m*(6*m +- 5), so e(n) = 0 and a(24) = a(23) + a(13) - a(10)  = 24 + 14 - 1  = 37;
n = 39: n = m*(6*m - 5) for m = 3, so e(n) = 39 and a(39) = 39 + a(38) + a(28) - a(25) - a(5) = 39 + 1 + 1 - 26 - 1 = 14;
n = 76: n = m*(6*m - 5) for m = 4, so e(n) = -76 and a(4) = -76 + a(75) + a(65) - a(62) - a(42) + a(37) + a(7) = -76 + 26 + 14  - 1 - 1 + 38 + 1 = 1. (End)

Links

Crossrefs

Cf. A210964 (1/f(-x, -x^11)), A245058.
Cf. Sum_{d|n, d = 0, 1, or k-1 mod k} d: A000203 (k=3), A113184(k=4), A284361 (k=5), A284362 (k=6), A284363 (k=7), this sequence (k=12).

Programs

  • Mathematica
    Table[Sum[If[Mod[d, 12]<2 || Mod[d, 12]==11, d, 0], {d, Divisors[n]}], {n, 80}] (* Indranil Ghosh, Mar 25 2017 *)
sd12[n_]:=Total[Select[Divisors[n],MemberQ[{0,1,11},Mod[#,12]]&]]; Array[sd12,80] (* Harvey P. Dale, Aug 29 2024 *)
  • PARI
    a(n) = sumdiv(n, d, ((d + 1) % 12 < 3) * d); \\ Amiram Eldar, Apr 12 2024

Formula

From Peter Bala, Dec 11 2020: (Start)
O.g.f.: Sum_{k >= 1, k == 0, 1 or 11 (mod 12)} k*x^k/(1 - x^k).
Define a(n) = 0 for n < 1. Then a(n) = e(n) + a(n-1) + a(n-11) - a(n-14) - a(n-34) + + - -, where [1, 11, 14, 34, ...] is the sequence of generalized 14-gonal numbers A195818, and e(n) = (-1)^(m+1)*n if n is a generalized 14-gonal number of the form m*(6*m+-5); otherwise e(n) = 0. Examples of this recurrence are given below. (End)
Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^2/48 = -A245058 = 0.205616... . - Amiram Eldar, Apr 12 2024
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