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 1-6 of 6 results.

A344817 a(n) = Sum_{k=1..n} floor(n/k) * (-2)^(k-1).

Original entry on oeis.org

1, 0, 5, -4, 13, -16, 49, -88, 173, -324, 701, -1384, 2713, -5416, 10989, -21916, 43621, -87224, 174921, -349872, 698773, -1397356, 2796949, -5593872, 11183361, -22366976, 44742149, -89483716, 178951741, -357903312, 715838513, -1431678040, 2863290285
Offset: 1

Views

Author

Seiichi Manyama, May 29 2021

Keywords

Links

Crossrefs

Column k=2 of A344824.
Cf. A081295.
Sums of the form Sum_{k=1..n} q^(k-1)*floor(n/k): A344820 (q=-n), A344819 (q=-4), A344818 (q=-3), this sequence (q=-2), A059851 (q=-1), A006218 (q=1), A268235 (q=2), A344814 (q=3), A344815 (q=4), A344816 (q=5), A332533 (q=n).

Programs

  • Magma
    A344817:= func< n | (&+[Floor(n/k)*(-2)^(k-1): k in [1..n]]) >;
[A344817(n): n in [1..40]]; // G. C. Greubel, Jun 26 2024
  • Mathematica
    a[n_] := Sum[(-2)^(k - 1) * Quotient[n, k], {k, 1, n}]; Array[a, 30] (* Amiram Eldar, May 29 2021 *)
  • PARI
    a(n) = sum(k=1, n, n\k*(-2)^(k-1));
  • PARI
    a(n) = sum(k=1, n, sumdiv(k, d, (-2)^(d-1)));
  • PARI
    my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, x^k/(1+2*x^k))/(1-x))
  • PARI
    my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, (-2)^(k-1)*x^k/(1-x^k))/(1-x))
  • SageMath
    def A344817(n): return sum((n//k)*(-2)^(k-1) for k in range(1,n+1))
[A344817(n) for n in range(1,41)] # G. C. Greubel, Jun 26 2024

Formula

a(n) = Sum_{k=1..n} Sum_{d|k} (-2)^(d-1).
G.f.: (1/(1 - x)) * Sum_{k>=1} x^k/(1 + 2*x^k).
G.f.: (1/(1 - x)) * Sum_{k>=1} (-2)^(k-1) * x^k/(1 - x^k).
a(n) ~ -(-1)^n * 2^n / 3. - Vaclav Kotesovec, Jun 05 2021

A344820 a(n) = Sum_{k=1..n} floor(n/k) * (-n)^(k-1).

Original entry on oeis.org

1, 0, 9, -52, 520, -6636, 102984, -1864600, 38741463, -909081740, 23775986069, -685854111804, 21633935838489, -740800448012044, 27368368159530285, -1085102592823737200, 45957792326631241516, -2070863582899905915336, 98920982783031811482920, -4993219047619535240997780
Offset: 1

Views

Author

Seiichi Manyama, May 29 2021

Keywords

Links

Crossrefs

Diagonal of A344824.
Sums of the form Sum_{k=1..n} q^(k-1)*floor(n/k): this sequence (q=-n), A344819 (q=-4), A344818 (q=-3), A344817 (q=-2), A059851 (q=-1), A006218 (q=1), A268235 (q=2), A344814 (q=3), A344815 (q=4), A344816 (q=5), A332533 (q=n).

Programs

  • Magma
    A344820:= func< n | (&+[Floor(n/k)*(-n)^(k-1): k in [1..n]]) >;
[A344820(n): n in [1..40]]; // G. C. Greubel, Jun 26 2024
  • Mathematica
    a[n_] := Sum[(-n)^(k - 1) * Quotient[n, k], {k, 1, n}]; Array[a, 20] (* Amiram Eldar, May 29 2021 *)
  • PARI
    a(n) = sum(k=1, n, n\k*(-n)^(k-1));
  • PARI
    a(n) = sum(k=1, n, sumdiv(k, d, (-n)^(d-1)));
  • SageMath
    def A344820(n): return sum((n//k)*(-n)^(k-1) for k in range(1,n+1))
[A344820(n) for n in range(1,41)] # G. C. Greubel, Jun 26 2024

Formula

a(n) = Sum_{k=1..n} Sum_{d|k} (-n)^(d-1).
a(n) = [x^n] (1/(1 - x)) * Sum_{k>=1} x^k/(1 + n*x^k).
a(n) = [x^n] (1/(1 - x)) * Sum_{k>=1} (-n)^(k-1) * x^k/(1 - x^k).
a(n) ~ -(-1)^n * n^(n-1). - Vaclav Kotesovec, Jun 05 2021

A344818 a(n) = Sum_{k=1..n} floor(n/k) * (-3)^(k-1).

Original entry on oeis.org

1, -1, 9, -20, 62, -174, 556, -1660, 4911, -14693, 44357, -133053, 398389, -1195207, 3587853, -10763270, 32283452, -96850386, 290570104, -871710994, 2615074146, -7845220010, 23535839600, -70607518824, 211822017739, -635466060265, 1906399774635, -5719199303975
Offset: 1

Views

Author

Seiichi Manyama, May 29 2021

Keywords

Links

Crossrefs

Column k=3 of A344824.
Cf. A101561.
Sums of the form Sum_{k=1..n} q^(k-1)*floor(n/k): A344820 (q=-n), A344819 (q=-4), this sequence (q=-3), A344817 (q=-2), A059851 (q=-1), A006218 (q=1), A268235 (q=2), A344814 (q=3), A344815 (q=4), A344816 (q=5), A332533 (q=n).

Programs

  • Magma
    A344818:= func< n | (&+[Floor(n/k)*(-3)^(k-1): k in [1..n]]) >;
[A344818(n): n in [1..40]]; // G. C. Greubel, Jun 26 2024
  • Mathematica
    a[n_] := Sum[(-3)^(k - 1) * Quotient[n, k], {k, 1, n}]; Array[a, 30] (* Amiram Eldar, May 29 2021 *)
  • PARI
    a(n) = sum(k=1, n, n\k*(-3)^(k-1));
  • PARI
    a(n) = sum(k=1, n, sumdiv(k, d, (-3)^(d-1)));
  • PARI
    my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, x^k/(1+3*x^k))/(1-x))
  • PARI
    my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, (-3)^(k-1)*x^k/(1-x^k))/(1-x))
  • SageMath
    def A344818(n): return sum((n//k)*(-3)^(k-1) for k in range(1,n+1))
[A344818(n) for n in range(1,41)] # G. C. Greubel, Jun 26 2024

Formula

a(n) = Sum_{k=1..n} Sum_{d|k} (-3)^(d-1).
G.f.: (1/(1 - x)) * Sum_{k>=1} x^k/(1 + 3*x^k).
G.f.: (1/(1 - x)) * Sum_{k>=1} (-3)^(k-1) * x^k/(1 - x^k).
a(n) ~ -(-1)^n * 3^n / 4. - Vaclav Kotesovec, Jun 05 2021

A344819 a(n) = Sum_{k=1..n} floor(n/k) * (-4)^(k-1).

Original entry on oeis.org

1, -2, 15, -52, 205, -806, 3291, -13160, 52393, -209498, 839079, -3356300, 13420917, -53683854, 214751875, -859006400, 3435960897, -13743843762, 54975632975, -219902535924, 879609095965, -3518436366566, 14073749677851, -56294998711576, 225179977999337, -900719912066074
Offset: 1

Views

Author

Seiichi Manyama, May 29 2021

Keywords

Links

Crossrefs

Column k=4 of A344824.
Cf. A059851, A101562, A268235, A332533, A344814, A344815, A344816, A344817, A344818, A344820.

Programs

  • Magma
    A344819:= func< n | (&+[(-4)^(k-1)*Floor(n/k): k in [1..n]]) >;
[A344819(n): n in [1..40]]; // G. C. Greubel, Jun 25 2024
  • Mathematica
    a[n_] := Sum[(-4)^(k - 1) * Quotient[n, k], {k, 1, n}]; Array[a, 30] (* Amiram Eldar, May 29 2021 *)
  • PARI
    a(n) = sum(k=1, n, n\k*(-4)^(k-1));
  • PARI
    a(n) = sum(k=1, n, sumdiv(k, d, (-4)^(d-1)));
  • PARI
    my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, x^k/(1+4*x^k))/(1-x))
  • PARI
    my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, (-4)^(k-1)*x^k/(1-x^k))/(1-x))
  • SageMath
    def A344819(n): return sum((-4)^(k-1)*int(n//k) for k in range(1,n+1))
[A344819(n) for n in range(1,41)] # G. C. Greubel, Jun 25 2024

Formula

a(n) = Sum_{k=1..n} Sum_{d|k} (-4)^(d-1).
G.f.: (1/(1 - x)) * Sum_{k>=1} x^k/(1 + 4*x^k).
G.f.: (1/(1 - x)) * Sum_{k>=1} (-4)^(k-1) * x^k/(1 - x^k).
a(n) ~ -(-1)^n * 4^n / 5. - Vaclav Kotesovec, Jun 05 2021

A344821 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals downwards, where A(n,k) = Sum_{j=1..n} floor(n/j) * k^(j-1).

Original entry on oeis.org

1, 1, 2, 1, 3, 3, 1, 4, 5, 4, 1, 5, 9, 8, 5, 1, 6, 15, 20, 10, 6, 1, 7, 23, 46, 37, 14, 7, 1, 8, 33, 92, 128, 76, 16, 8, 1, 9, 45, 164, 349, 384, 141, 20, 9, 1, 10, 59, 268, 790, 1394, 1114, 280, 23, 10, 1, 11, 75, 410, 1565, 3946, 5491, 3332, 541, 27, 11
Offset: 1

Views

Author

Seiichi Manyama, May 29 2021

Keywords

Examples

			Square array, A(n, k), begins:
  1,  1,  1,   1,    1,    1,    1, ...
  2,  3,  4,   5,    6,    7,    8, ...
  3,  5,  9,  15,   23,   33,   45, ...
  4,  8, 20,  46,   92,  164,  268, ...
  5, 10, 37, 128,  349,  790, 1565, ...
  6, 14, 76, 384, 1394, 3946, 9384, ...
Antidiagonal triangle, T(n, k), begins:
  1;
  1,  2;
  1,  3,  3;
  1,  4,  5,   4;
  1,  5,  9,   8,   5;
  1,  6, 15,  20,  10,   6;
  1,  7, 23,  46,  37,  14,   7;
  1,  8, 33,  92, 128,  76,  16,  8;
  1,  9, 45, 164, 349, 384, 141, 20,  9;

Links

Crossrefs

Columns k=0..5 give A000027, A006218, A268235, A344814, A344815, A344816.
A(n,n) gives A332533.
Cf. A308813, A344824.

Programs

  • Magma
    A:= func< n,k | k eq n select n else (&+[Floor(n/j)*k^(j-1): j in [1..n]]) >;
A344821:= func< n,k | A(k+1, n-k-1) >;
[A344821(n,k): k in [0..n-1], n in [1..12]]; // G. C. Greubel, Jun 27 2024
  • Mathematica
    A[n_, k_] := Sum[If[k == 0 && j == 1, 1, k^(j - 1)] * Quotient[n, j], {j, 1, n}]; Table[A[k, n - k], {n, 1, 10}, {k, 1, n}] // Flatten (* Amiram Eldar, May 29 2021 *)
  • PARI
    A(n, k) = sum(j=1, n, n\j*k^(j-1));
  • PARI
    A(n, k) = sum(j=1, n, sumdiv(j, d, k^(d-1)));
  • SageMath
    def A(n,k): return n if k==n else sum((n//j)*k^(j-1) for j in range(1,n+1))
def A344821(n,k): return A(k+1, n-k-1)
flatten([[A344821(n,k) for k in range(n)] for n in range(1,13)]) # G. C. Greubel, Jun 27 2024

Formula

G.f. of column k: (1/(1 - x)) * Sum_{j>=1} x^j/(1 - k*x^j).
G.f. of column k: (1/(1 - x)) * Sum_{j>=1} k^(j-1) * x^j/(1 - x^j).
A(n, k) = Sum_{j=1..n} Sum_{d|j} k^(d - 1).
T(n, k) = Sum_{j=1..k+1} floor((k+1)/j) * (n-k-1)^(j-1), for n >= 1, 0 <= k <= n-1 (antidiagonal triangle). - G. C. Greubel, Jun 27 2024

A345033 Square array T(n,k), n >= 1, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=1..n} (-k)^(floor(n/j) - 1).

Original entry on oeis.org

1, 1, 1, 1, 0, 2, 1, -1, 3, 2, 1, -2, 6, 0, 3, 1, -3, 11, -8, 3, 3, 1, -4, 18, -28, 17, 2, 4, 1, -5, 27, -66, 81, -27, 5, 4, 1, -6, 38, -128, 255, -234, 70, 0, 5, 1, -7, 51, -220, 623, -1009, 739, -136, 5, 5, 1, -8, 66, -348, 1293, -3102, 4112, -2216, 255, 4, 6, 1, -9, 83, -518, 2397, -7743, 15649, -16452, 6545, -491, 7, 6
Offset: 1

Views

Author

Seiichi Manyama, Jun 06 2021

Keywords

Examples

			Square array begins:
  1, 1,   1,    1,     1,     1,     1, ...
  1, 0,  -1,   -2,    -3,    -4,    -5, ...
  2, 3,   6,   11,    18,    27,    38, ...
  2, 0,  -8,  -28,   -66,  -128,  -220, ...
  3, 3,  17,   81,   255,   623,  1293, ...
  3, 2, -27, -234, -1009, -3102, -7743, ...

Crossrefs

Columns k=0..3 give A110654, A271860, A345034, A345035.
T(n,n) gives A345036.
Cf. A344824, A345032.

Programs

  • Mathematica
    T[n_, 0] := Floor[(n + 1)/2]; T[n_, k_] := Sum[(-k)^(Floor[n/j] - 1), {j, 1, n}]; Table[T[k, n - k], {n, 1, 12}, {k, 1, n}] // Flatten (* Amiram Eldar, Jun 06 2021 *)
  • PARI
    T(n, k) = sum(j=1, n, (-k)^(n\j-1));

Formula

G.f. of column k: (1/(1 - x)) * Sum_{j>=1} x^j * (1 - x^j)/(1 + k*x^j).
Showing 1-6 of 6 results.

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