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.

Previous Showing 11-16 of 16 results.

A195829 Triangle read by rows with T(n,k) = n - A118277(k), n>=1, k>=1, if (n - A118277(k))>=0.

Original entry on oeis.org

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

Views

Author

Omar E. Pol, Sep 24 2011

Keywords

Comments

Also triangle read by rows in which column k lists the nonnegative integers A001477 starting at the row A118277(k).
This sequence is related to the generalized enneagonal numbers A118277, A195839 and A195849 in the same way as A195310 is related to the generalized pentagonal numbers A001318, A175003 and A000041. See comments in A195825.

Examples

			Written as a triangle:
.  0;
.  1;
.  2;
.  3;
.  4;
.  5,  0;
.  6,  1;
.  7,  2;
.  8,  3,  0;
.  9,  4,  1;
. 10,  5,  2;
. 11,  6,  3;
. 12,  7,  4;
. 13,  8,  5;
. 14,  9,  6;
. 15, 10,  7;
. 16, 11,  8;
. 17, 12,  9;
. 18, 13, 10,  0;
. 19, 14, 11,  1;
. 20, 15, 12,  2;

Crossrefs

Cf. A118277, A195310, A195825, A195826, A195827, A195828, A195830, A195831, A195832, A195833, A195839, A195849.

A195839 Triangle read by rows which arises from A195829, in the same way as A175003 arises from A195310. Column k starts at row A118277(k).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 3, 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, 5, -4, 14, 7, -4, -1, 16, 10, -4, -1, 21, 12, -5, -1, 27, 13, -7, -1, 32, 13, -10, -1, 34, 14, -12, -1, 36, 16, -13, -2, 1
Offset: 1

Views

Author

Omar E. Pol, Sep 24 2011

Keywords

Comments

The sum of terms of row n is equal to the leftmost term of row n+1. This sequence is related to the generalized enneagonal numbers A118277, A195829 and A195849 in the same way as A175003 is related to the generalized pentagonal numbers A001318, A195310 and A000041. See comments in A195825.

Examples

			Written as a triangle:
.  1;
.  1;
.  1;
.  1;
.  1;
.  1,  1;
.  2,  1;
.  3,  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,  5, -4;
. 14,  7, -4, -1;
. 16, 10, -4, -1;
. 21, 12, -5, -1;

Crossrefs

Row sums give A195849.
Cf. A001082, A175003, A195828, A195836, A195837, A195838.

A232714 Expansion of f(-x, -x^6) in powers of x where f is Ramanujan's two-variable theta function.

Original entry on oeis.org

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

Views

Author

Michael Somos, Nov 28 2013

Keywords

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Examples

			G.f. = 1 - x - x^6 + x^9 + x^19 - x^24 - x^39 + x^46 + x^66 - x^75 - x^100 + ...
G.f. = q^25 - q^81 - q^361 + q^529 + q^1089 - q^1369 - q^2209 + q^2601 + q^3721 + ...

Links

Crossrefs

Cf. A195849, A274179.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ QPochhammer[ q, q^7] QPochhammer[ q^6, q^7] QPochhammer[ q^7], {q, 0, n}];
  • PARI
    {a(n) = my(m); if( issquare( 56*n + 25, &m), (-1)^round( m / 14), 0)};

Formula

Euler transform of period 7 sequence [ -1, 0, 0, 0, 0, -1, -1, ...].
G.f.: Sum_{k in Z} (-1)^k * x^(k * (7*k + 5) / 2).
G.f.: Product_{k>0} (1 - x^(7*k-6)) * (1 - x^(7*k-1)) * (1 - x^(7*k)).
a(3*n + 2) = a(5*n + 2) = a(5*n + 3) = 0.
Convolution inverse of A195849.
abs(a(n)) = A274179(n). - Michael Somos, Jan 28 2017
a(n) = -(1/n)*Sum_{k=1..n} A284363(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 25 2017

A346797 Number of partitions of n into parts congruent to 0, 2 or 5 (mod 7).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 1, 2, 1, 3, 2, 3, 4, 3, 7, 4, 9, 6, 10, 11, 11, 17, 13, 22, 19, 25, 29, 28, 42, 34, 53, 46, 61, 67, 69, 92, 83, 115, 109, 133, 149, 152, 198, 182, 243, 233, 282, 309, 324, 398, 385, 485, 483, 563, 621, 648, 784, 768, 944, 947, 1096, 1194, 1262
Offset: 0

Views

Author

Ludovic Schwob, Aug 04 2021

Keywords

Examples

			For n=17 the a(17)=6 solutions are 2+2+2+2+2+2+5, 2+2+2+2+2+7, 2+2+2+2+9, 2+5+5+5, 5+5+7 and 5+12.

Links

Crossrefs

Cf. A000041, A047386, A274830, A006950, A036820, A036822, A195848, A035363, A195849, A346798.

Programs

  • Mathematica
    CoefficientList[Series[Product[1/((1 - x^(7*k))(1 - x^(7*k-2))(1 - x^(7*k-5))),{k,52}],{x,0,52}],x] (* Stefano Spezia, Aug 04 2021 *)

Formula

G.f.: Product_{k>=1} 1/((1 - x^(7*k))*(1 - x^(7*k-2))*(1 - x^(7*k-5))).
a(n) = a(n-2) + a(n-5) - a(n-11) - a(n-17) + + - - (with a(0)=1 and a(n) = 0 for negative n), where 2, 5, 11, 17, ... is the sequence A274830.
a(n) ~ exp(Pi*sqrt(2*n/7)) / (8*cos(3*Pi/14)*n). - Vaclav Kotesovec, Aug 05 2021

A346798 Number of partitions of n into parts congruent to 0, 3 or 4 (mod 7).

Original entry on oeis.org

1, 0, 0, 1, 1, 0, 1, 2, 1, 1, 3, 3, 2, 3, 6, 4, 4, 8, 9, 6, 10, 15, 12, 12, 21, 22, 18, 25, 36, 30, 32, 48, 52, 45, 60, 78, 72, 75, 105, 113, 105, 130, 166, 156, 166, 218, 236, 224, 274, 332, 325, 345, 436, 469, 462, 544, 649, 644, 688, 839, 907, 903, 1051
Offset: 0

Views

Author

Ludovic Schwob, Aug 04 2021

Keywords

Examples

			For n=19 the a(19)=6 solutions are 3+3+3+3+3+4, 3+3+3+3+7, 3+3+3+10, 3+4+4+4+4, 4+4+4+7, and 4+4+11.

Links

Crossrefs

Cf. A000041, A047342, A057570, A006950, A036820, A036822, A195848, A035363, A195849, A346797.

Programs

  • Mathematica
    CoefficientList[Series[Product[1/((1 - x^(7*k))(1 - x^(7*k-3))(1 - x^(7*k-4))),{k,55}],{x,0,55}],x] (* Stefano Spezia, Aug 04 2021 *)

Formula

G.f.: Product_{k>=1} 1/((1 - x^(7*k))*(1 - x^(7*k-3))*(1 - x^(7*k-4))).
a(n) = a(n-3) + a(n-4) - a(n-13) - a(n-15) + + - - (with a(0)=1 and a(n) = 0 for negative n), where 3, 4, 13, 15, ... is the sequence A057570.
a(n) ~ exp(Pi*sqrt(2*n/7)) / (8*cos(Pi/14)*n). - Vaclav Kotesovec, Aug 05 2021

A287325 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Sum_{j=-inf..inf} (-1)^j*x^(k*j*(j-1)/2 + j^2).

Original entry on oeis.org

1, 1, -2, 1, -1, 0, 1, -1, -1, 0, 1, -1, 0, 0, 2, 1, -1, 0, -1, 0, 0, 1, -1, 0, 0, 0, 1, 0, 1, -1, 0, 0, -1, 0, 0, 0, 1, -1, 0, 0, 0, 0, 1, 1, 0, 1, -1, 0, 0, 0, -1, 0, 0, 0, -2, 1, -1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, -1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, -1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, -1, 0
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 13 2017

Keywords

Examples

			Square array begins:
   1,   1,   1,   1,   1,   1, ...
  -2,  -1,  -1,  -1,  -1,  -1, ...
   0,  -1,   0,   0,   0,   0, ...
   0,   0,  -1,   0,   0,   0, ...
   2,   0,   0,  -1,   0,   0, ...
   0,   1,   0,   0,  -1,   0, ...

Links

Crossrefs

Columns k=0..10 give A002448, A010815, A106459, A113429, A089802, A232714, A244525, A281814 (absolute values), A205988 (absolute values), A281815 (absolute values), A247133.
Cf. A000041, A000290, A000326, A000384, A000566, A000567, A001106, A001107, A006950, A015128, A036820, A051682, A051624, A051865, A195825, A195848, A195849, A195850, A195851, A195852, A196933, A211970.

Programs

  • Mathematica
    Table[Function[k, SeriesCoefficient[Sum[(-1)^i x^(k i (i - 1)/2 + i^2), {i, -n, n}], {x, 0, n}]][j - n], {j, 0, 13}, {n, 0, j}] // Flatten
Table[Function[k, SeriesCoefficient[Product[(1 - x^((k + 2) i)) (1 - x^((k + 2) i - 1)) (1 - x^((k + 2) i - k - 1)), {i, 1, n}], {x, 0, n}]][j - n], {j, 0, 13}, {n, 0, j}] // Flatten
Table[Function[k, SeriesCoefficient[(x^(2 + k) QPochhammer[1/x, x^(2 + k)] QPochhammer[x^(-1 - k), x^(2 + k)] QPochhammer[x^(2 + k), x^(2 + k)])/((-1 + x) (-1 + x^(1 + k))), {x, 0, n}]][j - n], {j, 0, 13}, {n, 0, j}] // Flatten

Formula

G.f. of column 0: Sum_{j=-inf..inf} (-1)^j*x^A000290(j) = Product_{i>=1} (1 + x^i)/(1 - x^i) (convolution inverse of A015128).
G.f. of column 1: Sum_{j=-inf..inf} (-1)^j*x^A000326(j) = Product_{i>=1} (1 - x^i) (convolution inverse of A000041).
G.f. of column 2: Sum_{j=-inf..inf} (-1)^j*x^A000384(j) = Product_{i>=1} (1 - x^(2*i))/(1 + x^(2*i-1)) (convolution inverse of A006950).
G.f. of column 3: Sum_{j=-inf..inf} (-1)^j*x^A000566(j) = Product_{i>=1} (1 - x^(5*i))*(1 - x^(5*i-1))*(1 - x^(5*i-4)) (convolution inverse of A036820).
G.f. of column 4: Sum_{j=-inf..inf} (-1)^j*x^A000567(j) = Product_{i>=1} (1 - x^(6*i))*(1 - x^(6*i-1))*(1 - x^(6*i-5)) (convolution inverse of A195848).
G.f. of column 5: Sum_{j=-inf..inf} (-1)^j*x^A001106(j) = Product_{i>=1} (1 - x^(7*i))*(1 - x^(7*i-1))*(1 - x^(7*i-6)) (convolution inverse of A195849).
G.f. of column 6: Sum_{j=-inf..inf} (-1)^j*x^A001107(j) = Product_{i>=1} (1 - x^(8*i))*(1 - x^(8*i-1))*(1 - x^(8*i-7)) (convolution inverse of A195850).
G.f. of column 7: Sum_{j=-inf..inf} (-1)^j*x^A051682(j) = Product_{i>=1} (1 - x^(9*i))*(1 - x^(9*i-1))*(1 - x^(9*i-8)) (convolution inverse of A195851).
G.f. of column 8: Sum_{j=-inf..inf} (-1)^j*x^A051624(j) = Product_{i>=1} (1 - x^(10*i))*(1 - x^(10*i-1))*(1 - x^(10*i-9)) (convolution inverse of A195852).
G.f. of column 9: Sum_{j=-inf..inf} (-1)^j*x^A051865(j) = Product_{i>=1} (1 - x^(11*i))*(1 - x^(11*i-1))*(1 - x^(11*i-10)) (convolution inverse of A196933).
G.f. of column k: Sum_{j=-inf..inf} (-1)^j*x^(k*j*(j-1)/2+j^2) = Product_{i>=1} (1 - x^((k+2)*i))*(1 - x^((k+2)*i-1))*(1 - x^((k+2)*i-k-1)).
Previous Showing 11-16 of 16 results.

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