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.

A344814 a(n) = Sum_{k=1..n} floor(n/k) * 3^(k-1).

Original entry on oeis.org

1, 5, 15, 46, 128, 384, 1114, 3332, 9903, 29671, 88721, 266151, 797593, 2392649, 7175709, 21526834, 64573556, 193720536, 581141026, 1743422288, 5230207428, 15690619684, 47071679294, 141215037738, 423644574301, 1270933715189, 3812799550089, 11438398630159
Offset: 1

Views

Author

Seiichi Manyama, May 29 2021

Keywords

Comments

Partial sums of A034730.

Links

Crossrefs

Column k=3 of A344821.
Cf. A059851, A268235, A332533, A344815, A344816, A344817, A344818, A344819, A344820.

Programs

  • Magma
    A344814:= func< n | (&+[Floor(n/k)*3^(k-1): k in [1..n]]) >;
[A344814(n): n in [1..40]]; // G. C. Greubel, Jun 26 2024
  • Maple
    seq(add(3^(k-1)*floor(n/k), k=1..n), n=1..60); # Ridouane Oudra, Feb 05 2023
  • 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 A344814(n): return sum((n//k)*3^(k-1) for k in range(1,n+1))
[A344814(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) ~ 3^n / 2. - Vaclav Kotesovec, Jun 05 2021
a(n) = (1/2) * Sum_{k=1..n} (3^floor(n/k) - 1). - Ridouane Oudra, Feb 05 2023

A344815 a(n) = Sum_{k=1..n} floor(n/k) * 4^(k-1).

Original entry on oeis.org

1, 6, 23, 92, 349, 1394, 5491, 21944, 87497, 349902, 1398479, 5593892, 22371109, 89484074, 357919803, 1431678080, 5726645377, 22906581142, 91626057879, 366504227292, 1466015859181, 5864063418866, 23456249463283, 93824997852744, 375299974563657, 1501199898183502
Offset: 1

Views

Author

Seiichi Manyama, May 29 2021

Keywords

Comments

Partial sums of A339684.

Links

Crossrefs

Column k=4 of A344821.
Sums of the form Sum_{k=1..n} q^(k-1)*floor(n/k): A344820 (q=-n), A344819 (q=-4), A344818 (q=-3), A344817 (q=-2), A059851 (q=-1), A006218 (q=1), A268235 (q=2), A344814 (q=3), this sequence (q=4), A344816 (q=5), A332533 (q=n).

Programs

  • Magma
    A344815:= func< n | (&+[Floor(n/j)*4^(j-1): j in [1..n]]) >;
[A344815(n): n in [1..40]]; // G. C. Greubel, Jun 27 2024
  • Maple
    seq(add(4^(k-1)*floor(n/k), k=1..n), n=1..60); # Ridouane Oudra, Feb 16 2023
  • 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 A344815(n): return sum((n//j)*4^(j-1) for j in range(1,n+1))
[A344815(n) for n in range(1,41)] # G. C. Greubel, Jun 27 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) ~ 4^n / 3. - Vaclav Kotesovec, Jun 05 2021
a(n) = (1/3) * Sum_{k=1..n} (4^floor(n/k) - 1). - Ridouane Oudra, Feb 16 2023

A344816 a(n) = Sum_{k=1..n} floor(n/k) * 5^(k-1).

Original entry on oeis.org

1, 7, 33, 164, 790, 3946, 19572, 97828, 488479, 2442235, 12207861, 61039267, 305179893, 1525898649, 7629414925, 38147071306, 190734961932, 953674808838, 4768372074464, 23841860356470, 119209292012746, 596046459981502, 2980232250997128, 14901161254984784
Offset: 1

Views

Author

Seiichi Manyama, May 29 2021

Keywords

Comments

Partial sums of A339685.

Links

Crossrefs

Column k=5 of A344821.
Sums of the form Sum_{k=1..n} q^(k-1)*floor(n/k): A344820 (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), this sequence (q=5), A332533 (q=n).

Programs

  • Magma
    A344816:= func< n | (&+[Floor(n/k)*5^(k-1): k in [1..n]]) >;
[A344816(n): n in [1..40]]; // G. C. Greubel, Jun 26 2024
  • Maple
    seq(add(5^(k-1)*floor(n/k), k=1..n), n=1..60); # Ridouane Oudra, Mar 05 2023
  • Mathematica
    a[n_] := Sum[5^(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*5^(k-1));
  • PARI
    a(n) = sum(k=1, n, sumdiv(k, d, 5^(d-1)));
  • PARI
    my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, x^k/(1-5*x^k))/(1-x))
  • PARI
    my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, 5^(k-1)*x^k/(1-x^k))/(1-x))
  • SageMath
    def A344816(n): return sum((n//k)*5^(k-1) for k in range(1,n+1))
[A344816(n) for n in range(1,41)] # G. C. Greubel, Jun 26 2024

Formula

a(n) = Sum_{k=1..n} Sum_{d|k} 5^(d-1).
G.f.: (1/(1 - x)) * Sum_{k>=1} x^k/(1 - 5*x^k).
G.f.: (1/(1 - x)) * Sum_{k>=1} 5^(k-1) * x^k/(1 - x^k).
a(n) ~ 5^n / 4. - Vaclav Kotesovec, Jun 05 2021
a(n) = (1/4) * Sum_{k=1..n} (5^floor(n/k) - 1). - Ridouane Oudra, Mar 05 2023

A344824 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, 1, 3, 1, 0, 3, 4, 1, -1, 5, 2, 5, 1, -2, 9, -4, 4, 6, 1, -3, 15, -20, 13, 4, 7, 1, -4, 23, -52, 62, -16, 6, 8, 1, -5, 33, -106, 205, -174, 49, 4, 9, 1, -6, 45, -188, 520, -806, 556, -88, 7, 10, 1, -7, 59, -304, 1109, -2584, 3291, -1660, 173, 7, 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, 1,   0,   -1,   -2,    -3,    -4, ...
  3, 3,   5,    9,   15,    23,    33, ...
  4, 2,  -4,  -20,  -52,  -106,  -188, ...
  5, 4,  13,   62,  205,   520,  1109, ...
  6, 4, -16, -174, -806, -2584, -6636, ...
Antidiagonal triangle, T(n, k), begins:
  1;
  1,  2;
  1,  1,   3;
  1,  0,   3,    4;
  1, -1,   5,    2,   5;
  1, -2,   9,   -4,   4,    6;
  1, -3,  15,  -20,  13,    4,   7;
  1, -4,  23,  -52,  62,  -16,   6,   8;
  1, -5,  33, -106, 205, -174,  49,   4,  9;
  1, -6,  45, -188, 520, -806, 556, -88,  7,  10;

Links

Crossrefs

Columns k=0..4 give A000027, A059851, A344817, A344818, A344819.
A(n,n) gives A344820.
Cf. A344821.

Programs

  • Magma
    A:= func< n,k | k eq n select n else (&+[Floor(n/j)*(-k)^(j-1): j in [1..n]]) >;
A344824:= func< n,k | A(k+1, n-k-1) >;
[A344824(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 A344824(n,k): return A(k+1, n-k-1)
flatten([[A344824(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) * (k-n+1)^(j-1), for n >= 1, 0 <= k <= n-1 (antidiagonal triangle). - G. C. Greubel, Jun 27 2024

A345032 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, 2, 2, 1, 3, 3, 2, 1, 4, 6, 4, 3, 1, 5, 11, 12, 5, 3, 1, 6, 18, 32, 21, 6, 4, 1, 7, 27, 70, 87, 41, 7, 4, 1, 8, 38, 132, 263, 258, 74, 8, 5, 1, 9, 51, 224, 633, 1047, 745, 144, 9, 5, 1, 10, 66, 352, 1305, 3158, 4120, 2224, 275, 10, 6, 1, 11, 83, 522, 2411, 7821, 15659, 16460, 6605, 541, 11, 6
Offset: 1

Views

Author

Seiichi Manyama, Jun 06 2021

Keywords

Examples

			Square array begins:
  1, 1,  1,   1,    1,    1,    1, ...
  1, 2,  3,   4,    5,    6,    7, ...
  2, 3,  6,  11,   18,   27,   38, ...
  2, 4, 12,  32,   70,  132,  224, ...
  3, 5, 21,  87,  263,  633, 1305, ...
  3, 6, 41, 258, 1047, 3158, 7821, ...

Crossrefs

Columns k=0..3 give A110654, A000027, A345028, A345029.
T(n,n) gives A345030.
Cf. A344821, A345033.

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 1-5 of 5 results.

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