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 1-4 of 4 results.

A210964 Column 10 of square array A195825. Also column 1 of triangle A210954. Also 1 together with the row sums of triangle A210954.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 7, 10, 12, 13, 13, 13, 13, 13, 13, 13, 14, 16, 21, 27, 32, 34, 35, 35, 35, 35, 35, 36, 38, 44, 54, 67, 77, 83, 85, 86, 86, 86, 87, 89, 95, 107, 128, 152, 173, 185, 191, 193, 194, 195
Offset: 0

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Author

Omar E. Pol, Jun 16 2012

Keywords

Comments

Note that this sequence contains five plateaus: [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], [4, 4, 4, 4, 4, 4, 4, 4, 4], [13, 13, 13, 13, 13, 13, 13], [35, 35, 35, 35, 35], [86, 86, 86]. For more information see A210843 and other sequences of this family. - Omar E. Pol, Jun 29 2012

Links

Crossrefs

Cf. A000041, A006950, A036820, A057077, A195818, A195825, A195848, A195849, A195850, A195851, A195852, A196933, A210944, A210954, A211970, A211971.
Cf. A247133.

Programs

  • Mathematica
    nmax = 100; CoefficientList[Series[Product[1 / ((1 - x^(12*k)) * (1 - x^(12*k-1)) * (1 - x^(12*k-11))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 08 2015 *)

Formula

Expansion of 1 / f(-x, -x^11) in powers of x where f() is a Ramanujan theta function. - Michael Somos, Jan 10 2015
Partitions of n into parts of the form 12*k, 12*k+1, 12*k+11. - Michael Somos, Jan 10 2015
Euler transform of period 12 sequence [ 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, ...]. - Michael Somos, Jan 10 2015
G.f.: Product_{k>0} 1 / ((1 - x^(12*k)) * (1 - x^(12*k - 1)) * (1 - x^(12*k - 11))).
Convolution inverse of A247133.
a(n) ~ sqrt(2)*(1+sqrt(3)) * exp(Pi*sqrt(n/6)) / (8*n). - Vaclav Kotesovec, Nov 08 2015
a(n) = (1/n)*Sum_{k=1..n} A284372(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 25 2017
a(n) = a(n-1) + a(n-11) - a(n-14) - a(n-34) + + - - (with the convention a(n) = 0 for negative n), where 1, 11, 14, 34, ... is the sequence of generalized 14-gonal numbers A195818. - Peter Bala, Dec 10 2020

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

A284361 a(n) = Sum_{d|n, d = 0, 1, or 4 mod 5} d.

Original entry on oeis.org

1, 1, 1, 5, 6, 7, 1, 5, 10, 16, 12, 11, 1, 15, 21, 21, 1, 16, 20, 40, 22, 12, 1, 35, 31, 27, 10, 19, 30, 67, 32, 21, 12, 35, 41, 56, 1, 20, 40, 80, 42, 42, 1, 60, 75, 47, 1, 51, 50, 91, 52, 31, 1, 70, 72, 75, 20, 30, 60, 151, 62, 32, 31, 85, 71, 84, 1, 39, 70, 135
Offset: 1

Views

Author

Seiichi Manyama, Mar 25 2017

Keywords

Links

Crossrefs

Cf. A036820 (1/f(-x, -x^4)), A113429 (f(-x, -x^4)), A102753.
Cf. Sum_{d|n, d = 0, 1, or k-1 mod k} d: A000203 (k=3), this sequence (k=5), A284362 (k=6), A284363 (k=7), A284372 (k=12).

Programs

  • Mathematica
    Table[Sum[If[Mod[d, 5]<2 || Mod[d, 5]==4, d, 0], {d, Divisors[n]}], {n, 80}] (* Indranil Ghosh, Mar 25 2017 *)
Table[Total[Select[Divisors[n],MemberQ[{0,1,4},Mod[#,5]]&]],{n,70}] (* Harvey P. Dale, Aug 02 2020 *)
  • PARI
    a(n) = sumdiv(n, d, ((d + 1) % 5 < 3) * d); \\ Amiram Eldar, Apr 12 2024

Formula

Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^2/20 = A102753 / 10 = 0.4934802... . - Amiram Eldar, Apr 12 2024
Showing 1-4 of 4 results.

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