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.

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A247133 Expansion of f(-x, -x^11) in powers of x where f() is a Ramanujan theta function.

Original entry on oeis.org

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

Views

Author

Michael Somos, Jan 10 2015

Keywords

Examples

			G.f. = 1 - x - x^11 + x^14 + x^34 - x^39 - x^69 + x^76 + x^116 - x^125 + ...
G.f. = q^25 - q^49 - q^289 + q^361 + q^841 - q^961 - q^1681 + q^1849 + ...

Links

Crossrefs

Cf. A210964.

Programs

  • PARI
    {a(n) = my(m = 24*n + 25); if( issquare(m, &m) && (m%12==5 || m%12==7), (-1)^((m+6) \ 12))};

Formula

Euler transform of period 12 sequence [ -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, -1, ...].
G.f.: Product_{k>0} (1 - x^(12*k)) * (1 - x^(12*k - 1)) * (1 - x^(12*k - 11)).
Convolution inverse of A210964.

A210954 Triangle read by rows which arises from A210944 in the same way as A175003 arises from A195310. Column k starts at row A195818(k).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 3, 1, 4, 1, -1, 4, 1, -1, 4, 1, -1, 4, 1, -1, 4, 1, -1, 4, 1, -1, 4, 1, -1, 4, 1, -1, 4, 2, -1, 5, 3, -1, 7, 4, -1, 10, 4, -2, 12, 4, -3, 13, 4, -4, 13, 4, -4, 13, 4, -4, 13, 4, -4, 13, 4, -4, 13, 4, -4, 13, 5, -4, 14, 7, -4, -1
Offset: 1

Views

Author

Omar E. Pol, Jun 16 2012

Keywords

Comments

The sum of terms of row n is equal to the leftmost term of row n+1. Also 1 together with the row sums give A210964. This sequence is related to the generalized 14-gonal numbers A195818, A210954 and A210964 in the same way as A175003 is related to the generalized pentagonal numbers A001318, A195310 and A000041. See comments in A195825.

Examples

			Written as an irregular triangle:
1;
1;
1;
1;
1;
1;
1;
1;
1;
1;
1,  1;
2,  1;
3,  1;
4,  1, -1;
4,  1, -1;
4,  1, -1;
4,  1, -1;
4,  1, -1;
4,  1, -1;
4,  1, -1;
4,  1, -1;
4,  2, -1;
5,  3, -1;
7,  4, -1;
10, 4, -2;
12, 4, -3;
13, 4, -4;
13, 4, -4;
13, 4, -4;
13, 4, -4;
13, 4, -4;
13, 4, -4;
13, 5, -4;
14, 7, -4, -1;

Crossrefs

Cf. A175003, A195818, A195825, A195836, A195837, A195838, A195839, A195840, A195841, A195842, A195843, A210944, A210964.

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

A210944 Triangle read by rows with T(n,k) = n - A195818(k), n>=1, k>=1, if (n - A195818(k))>=0.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 0, 11, 1, 12, 2, 13, 3, 0, 14, 4, 1, 15, 5, 2, 16, 6, 3, 17, 7, 4, 18, 8, 5, 19, 9, 6, 20, 10, 7, 21, 11, 8, 22, 12, 9, 23, 13, 10, 24, 14, 11, 25, 15, 12, 26, 16, 13, 27, 17, 14, 28, 18, 15, 29, 19, 16, 30, 20, 17, 31, 21, 18
Offset: 1

Views

Author

Omar E. Pol, Jun 16 2012

Keywords

Comments

Also triangle read by rows in which column k lists the nonnegative integers A001477 starting at the row A195818(k).
This sequence is related to the generalized 14-gonal numbers A195818, A210954 and A210964 in the same way as A195310 is related to the generalized pentagonal numbers A001318, A175003 and A000041. See comments in A195825.

Examples

			Written as an irregular triangle:
0;
1;
2;
3;
4;
5;
6;
7;
8;
9;
10, 0;
11, 1;
12, 2;
13, 3,  0;
14, 4,  1;
15, 5,  2;
16, 6,  3;
17, 7,  4;
18, 8,  5;
19, 9,  6;

Crossrefs

Cf. A195310, A195825, A195826, A195827, A195828, A195829, A195830, A195831, A195832, A195833, A195818, A210954, A210964.

A308400 Expansion of 1 / Sum_{k=-oo..oo} (-x)^(k*(6*k + 1)).

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 2, 0, 1, 1, 0, 3, 0, 3, 1, 1, 3, 0, 6, 1, 3, 3, 1, 8, 1, 8, 3, 3, 9, 2, 14, 3, 9, 9, 4, 19, 4, 19, 9, 10, 21, 6, 32, 10, 22, 22, 12, 42, 12, 43, 23, 25, 48, 18, 67, 25, 51, 51, 31, 88, 31, 90, 54, 59, 101, 44, 137, 60, 108, 109, 73, 177, 73
Offset: 0

Views

Author

Ilya Gutkovskiy, May 24 2019

Keywords

Comments

Number of partitions of n into parts congruent to {0, 5, 7} mod 12.
Convolution inverse of A247223.

Crossrefs

Cf. A006950, A036498, A049453, A210964, A247223, A263536, A308399.

Programs

  • Mathematica
    nmax = 78; CoefficientList[Series[1/Sum[(-x)^(k (6 k + 1)), {k, -nmax, nmax}], {x, 0, nmax}], x]
nmax = 78; CoefficientList[Series[Product[1/((1 - x^(12 k - 7)) * (1 - x^(12 k - 5)) * (1 - x^(12 k))), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: 1 / Sum_{k>=1} (-x)^A036498(k).
G.f.: Product_{k>=1} 1 / ((1 - x^(12*k - 7)) * (1 - x^(12*k - 5)) * (1 - x^(12*k))).
a(n) ~ (sqrt(3) - 1) * exp(sqrt(n/6)*Pi) / (2^(5/2)*n). - Vaclav Kotesovec, May 25 2019
Previous Showing 11-15 of 15 results.

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