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 1-8 of 8 results.

A163744 Smallest positive m such that A050410(m) = 0 (mod n!).

Original entry on oeis.org

1, 1, 3, 4, 23, 23, 608, 3703, 59063, 65975, 65975, 65975, 3227648, 83180983
Offset: 1

Views

Author

Zak Seidov, Aug 03 2009

Keywords

Comments

Smallest positive m such that m-th truncated square pyramid number tsp(m)=m*(7*m-1)*(2*m-1)/6 is divisible by n!, n=0,1,.. .

Examples

			a(3)=3 because tsp(3)=126 is divisible by 3!: 126/3!=21
a(5)=23 because tsp(23)=27600 is divisible by 5!: 27600/5!=230
a(6)=608 because tsp(608)=523875600 is divisible by 6!: 523875600/6!=727605.

Crossrefs

Cf. A050410 Truncated square pyramid numbers:a(n)=sum(k^2, k=n..2*n-1)=(1/6)n(7n-1)(2n-1).

A072474 Sum of next n squares.

Original entry on oeis.org

1, 13, 77, 294, 855, 2071, 4403, 8492, 15189, 25585, 41041, 63218, 94107, 136059, 191815, 264536, 357833, 475797, 623029, 804670, 1026431, 1294623, 1616187, 1998724, 2450525, 2980601, 3598713, 4315402, 5142019, 6090755, 7174671, 8407728, 9804817, 11381789, 13155485
Offset: 1

Views

Author

Amarnath Murthy, Jun 20 2002

Keywords

Examples

			a(1) = 1^2 = 1;
a(2) = 2^2 + 3^2 = 13;
a(3) = 4^2 + 5^2 + 6^2 = 77.

Links

Crossrefs

Cf. A000290, A000330, A072475, A050410.
Cf. A006003 (for natural numbers), A260513 (for triangular numbers), A372583 (for pentagonal numbers), A372751 (for hexagonal numbers), A075664 (for cubes).

Programs

  • Magma
    [n*(3*n^2+1)*(n^2+2)/12: n in [1..35]]; // Vincenzo Librandi, Dec 31 2024
  • Mathematica
    Table[Sum[ i^2, {i, n(n - 1)/2 + 1, n(n + 1)/2}], {n, 1, 35}]
  • PARI
    a(n) = n*(3*n^2+1)*(n^2+2)/12

Formula

a(n) = k*(k+1)*(2*k+1)/6 - r*(r+1)*(2*r+1)/6, where k = n*(n+1)/2 and r = n*(n-1)/2.
a(n) = A000330(n*(n+1)/2) - A000330(n*(n-1)/2).
a(n) = (n/12)*(3*n^2 + 1)*(n^2 + 2). - Benoit Cloitre, Jun 26 2002
G.f.: x*(1+3*x+x^2)*(1+4*x+x^2)/(1-x)^6. - Colin Barker, Mar 23 2012
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6) for n > 6. - Jinyuan Wang, May 25 2020
E.g.f.: exp(x)*x*(12 + 66*x + 82*x^2 + 30*x^3 + 3*x^4)/12. - Stefano Spezia, May 14 2024

Extensions

Edited by Robert G. Wilson v, Jun 21 2002

A094415 Triangle T read by rows: dot product * .

Original entry on oeis.org

1, 4, 5, 10, 13, 13, 20, 26, 28, 26, 35, 45, 50, 50, 45, 56, 71, 80, 83, 80, 71, 84, 105, 119, 126, 126, 119, 105, 120, 148, 168, 180, 184, 180, 168, 148, 165, 201, 228, 246, 255, 255, 246, 228, 201, 220, 265, 300, 325, 340, 345, 340, 325, 300, 265, 286, 341
Offset: 0

Views

Author

Ralf Stephan, May 02 2004

Keywords

Examples

			Triangle begins as:
   1;
   4,  5;
  10, 13, 13;
  20, 26, 28, 26;
  35, 45, 50, 50, 45;
  56, 71, 80, 83, 80, 71;

Links

Crossrefs

Columns 0-6 are A000292, A008778, A026054, A026057, A026060, A026063, A026066.
Half-diagonal is A050410.
Row sums are A000537.
See also A094414, A088003.

Programs

  • GAP
    Flat(List([0..12], n-> List([0..n], k-> (n+1)*((n+2)*(n+3) + 3*k*(n-k+1))/6 ))); # G. C. Greubel, Oct 30 2019
  • Magma
    [(n+1)*((n+2)*(n+3) + 3*k*(n-k+1))/6: k in [0..n], n in [0..12]]; // G. C. Greubel, Oct 30 2019
  • Maple
    seq(seq( (n+1)*((n+2)*(n+3) + 3*k*(n-k+1))/6 , k=0..n), n=0..12); # G. C. Greubel, Oct 30 2019
  • Mathematica
    Table[(n+1)*((n+2)*(n+3) + 3*k*(n-k+1))/6, {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Oct 30 2019 *)
  • PARI
    T(n,k) = (n+1)*((n+2)*(n+3) + 3*k*(n-k+1))/6;
for(n=0,12, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Oct 30 2019
  • Sage
    [[(n+1)*((n+2)*(n+3) + 3*k*(n-k+1))/6 for k in (0..n)] for n in (0..12)] # G. C. Greubel, Oct 30 2019

Formula

T(n, k) = n*(n^2 + 3*n*(1+k) + 2 - 3*k^2)/6 for n >= 0, 0 <= k <= n.

A240137 Sum of n consecutive cubes starting from n^3.

Original entry on oeis.org

0, 1, 35, 216, 748, 1925, 4131, 7840, 13616, 22113, 34075, 50336, 71820, 99541, 134603, 178200, 231616, 296225, 373491, 464968, 572300, 697221, 841555, 1007216, 1196208, 1410625, 1652651, 1924560, 2228716, 2567573, 2943675, 3359656, 3818240, 4322241, 4874563
Offset: 0

Views

Author

Bruno Berselli, Apr 02 2014

Keywords

Comments

Sum_{i>=1} 1/a(i) = 1.0356568858420883122567711052556541...
Consider the partitions of 2n into two parts (p,q) where p <= q. Then a(n) is the total volume of the family of cubes with side length q. - Wesley Ivan Hurt, Apr 15 2018
A180920 lists the numbers k such that a(k) is a square. - Jon E. Schoenfield, Mar 13 2022

Examples

			a(3) = 216 because 216 = 3^3 + 4^3 + 5^3.

Links

Crossrefs

Subsequence of A217843.
Cf. A116149: sum of n consecutive cubes after n^3.
Cf. A050410: sum of n consecutive squares starting from n^2.
Cf. A000326 (pentagonal numbers): sum of n consecutive integers starting from n.
Cf. A126274: n-th triangular number (A000217) * n-th pentagonal number (A000326).

Programs

  • Magma
    [n^2*(3*n-1)*(5*n-3)/4: n in [0..40]];
  • Maple
    A240137:=n->n^2*(3*n-1)*(5*n-3)/4; seq(A240137(n), n=0..40); # Wesley Ivan Hurt, May 09 2014
  • Mathematica
    Table[n^2 (3 n - 1) (5 n - 3)/4, {n, 0, 40}]
CoefficientList[Series[x (1 + 30 x + 51 x^2 + 8 x^3)/(1 - x)^5, {x, 0, 40}], x] (* Vincenzo Librandi, May 09 2014 *)
  • PARI
    a(n)=n^2*(3*n-1)*(5*n-3)/4 \\ Charles R Greathouse IV, Oct 07 2015
  • Sage
    [n^2*(3*n-1)*(5*n-3)/4 for n in [0..40]]

Formula

G.f.: x*(1 + 30*x + 51*x^2 + 8*x^3)/(1 - x)^5.
a(n) = n^2*(3*n - 1)*(5*n - 3)/4 = A000326(n)*A000566(n).
a(n) = A116149(-n), with A116149(0)=0.
a(n) = Sum_{j=n..2n-1} j^3. - Jon E. Schoenfield, Mar 13 2022

A262925 Sum of n consecutive fourth powers starting with n^4.

Original entry on oeis.org

0, 1, 97, 962, 4578, 14979, 38995, 86996, 173636, 318597, 547333, 891814, 1391270, 2092935, 3052791, 4336312, 6019208, 8188169, 10941609, 14390410, 18658666, 23884427, 30220443, 37834908, 46912204, 57653645, 70278221, 85023342, 102145582, 121921423
Offset: 0

Views

Author

Colin Barker, Oct 04 2015

Keywords

Examples

			a(3) = 3^4 + 4^4 + 5^4 = 962.

Links

Crossrefs

Cf. A050410, A240137, A262926.

Programs

  • Mathematica
    LinearRecurrence[{6,-15,20,-15,6,-1},{0,1,97,962,4578,14979},30] (* Harvey P. Dale, Jul 30 2017 *)
  • PARI
    vector(40, n, n--; sum(k=n, 2*n-1, k^4))
  • PARI
    concat(0, Vec(x*(16*x^4+241*x^3+395*x^2+91*x+1)/(x-1)^6 + O(x^40)))

Formula

a(n) = n*(-1+70*n^2-225*n^3+186*n^4)/30.
G.f.: x*(16*x^4+241*x^3+395*x^2+91*x+1) / (x-1)^6.

A262926 Sum of n consecutive n-th powers starting with n^n.

Original entry on oeis.org

0, 1, 31, 3408, 873580, 405071029, 295716738515, 312086923732368, 449317984129326216, 846136323944158864793, 2018612200059553898143707, 5949463230586042065279268128, 21227845340442717633531647231668, 90172805592203250075964230466892813
Offset: 0

Views

Author

Colin Barker, Oct 04 2015

Keywords

Examples

			a(3) = 3^3 + 4^4 + 5^5 = 3408.

Crossrefs

Cf. A050410, A240137, A262925.

Programs

  • Mathematica
    Table[Sum[k^k, {k, n, 2 n - 1}], {n, 0, 13}] (* Michael De Vlieger, Oct 05 2015 *)
  • PARI
    vector(20, n, n--; sum(k=n, 2*n-1, k^k))
  • Sage
    [sum(k^k for k in (n..2*n-1)) for n in (0..20)] # Bruno Berselli, Oct 05 2015

A299646 a(n) = Sum_{k = n..2*n+1} k^2.

Original entry on oeis.org

1, 14, 54, 135, 271, 476, 764, 1149, 1645, 2266, 3026, 3939, 5019, 6280, 7736, 9401, 11289, 13414, 15790, 18431, 21351, 24564, 28084, 31925, 36101, 40626, 45514, 50779, 56435, 62496, 68976, 75889, 83249, 91070, 99366, 108151, 117439, 127244, 137580, 148461, 159901
Offset: 0

Views

Author

Bruno Berselli, Feb 20 2018

Keywords

Comments

Inverse binomial transform is 1, 13, 27, 14, 0, 0, 0, ... (0 continued).

Links

Crossrefs

Subsequence of A008854, A047388, A174070 (after 1).
Cf. A050409: Sum_{k = n..2*n} k^2; A050410: Sum_{k = n..2*n-1} k^2.

Programs

  • GAP
    List([0..50], n -> (n+2)*(14*n^2+11*n+3)/6);
  • Magma
    [(n+2)*(14*n^2+11*n+3)/6: n in [0..50]];
  • Maple
    seq((n + 2)*(14*n^2 + 11*n + 3)/6, n=0..50); # Peter Luschny, Feb 21 2018
  • Mathematica
    Table[(n + 2) (14 n^2 + 11 n + 3)/6, {n, 0, 50}]
(* Second program: *)
LinearRecurrence[{4, -6, 4, -1}, {1, 14, 54, 135}, 41] (* Jean-François Alcover, Feb 21 2018 *)
  • Maxima
    makelist((n+2)*(14*n^2+11*n+3)/6, n, 0, 50);
  • PARI
    a(n)=(n+2)*(14*n^2+11*n+3)/6 \\ Charles R Greathouse IV, Feb 21 2018
  • PARI
    Vec((1 + 10*x + 4*x^2 - x^3)/(1 - x)^4 + O(x^60)) \\ Colin Barker, Feb 22 2018
  • Sage
    [(n+2)*(14*n^2+11*n+3)/6 for n in (0..50)]

Formula

O.g.f.: (1 + 10*x + 4*x^2 - x^3)/(1 - x)^4.
E.g.f.: (6 + 78*x + 81*x^2 + 14*x^3)*exp(x)/6.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
a(n) = (n + 2)*(14*n^2 + 11*n + 3)/6. Therefore:
a(6*k + r) = 504*k^3 + 18*(14*r + 13)*k^2 + (42*r^2 + 78*r + 25)*k + a(r), with 0 <= r <= 5. Example: for r=5, a(6*k + 5) = (6*k + 7)*(84*k^2 + 151*k + 68).

A322135 Table of truncated square pyramid numbers, read by antidiagonals.

Original entry on oeis.org

1, 4, 5, 9, 13, 14, 16, 25, 29, 30, 25, 41, 50, 54, 55, 36, 61, 77, 86, 90, 91, 49, 85, 110, 126, 135, 139, 140, 64, 113, 149, 174, 190, 199, 203, 204, 81, 145, 194, 230, 255, 271, 280, 284, 285, 100, 181, 245, 294, 330, 355, 371, 380, 384, 385, 121, 221, 302
Offset: 1

Views

Author

Allan C. Wechsler, Nov 27 2018

Keywords

Comments

The n-th row contains n numbers: n^2, n^2 + (n-1)^2, ..., n^2 + (n-1)^2 + ... + 1^2.
All numbers that appear in the table are listed in ascending order at A034705.
All numbers that appear twice or more are listed at A130052.
The left column is A000290 (the squares).
The top row is A000330 (the square pyramidal numbers).
The columns are A000290, A099776 (or a tail of A001844), a tail of A005918 or A120328, a tail of A027575, a tail of A027578, a tail of A027865, ...
The first two rows are A000330 and a tail of A168599, but subsequent rows are not currently in the OEIS, and are all tails of A000330 minus various constants.
The main diagonal is A050410.

Examples

			The 17th term is entry 2 on antidiagonal 6, so we sum two terms: 6^2 + 5^2 = 61.
Table begins:
   1   5  14  30  55  91 140 204 ...
   4  13  29  54  90 139 203 ...
   9  25  50  86 135 199 ...
  16  41  77 126 190 ...
  25  61 110 174 ...
  36  85 149 ...
  49 113 ...
  64 ...
  ...

Crossrefs

See comments; also cf. A000330, A059255.

Programs

  • Mathematica
    T[n_,k_] = Sum[(n+i)^2, {i,0,k-1}]; Table[T[n-k+1, k], {n,1,10},  {k,1,n}] // Flatten (* Amiram Eldar, Nov 28 2018 *)
f[n_] := Table[SeriesCoefficient[-((y (y (1 + y) + x (1 - 2 y - 3 y^2) + x^2 (1 - 3 y + 4 y^2)))/((-1 + x)^3 (-1 + y)^4)) , {x, 0,
i + 1 - j}, {y, 0, j}], {i, n, n}, {j, 1, n}]; Flatten[Array[f, 10]] (* Stefano Spezia, Nov 28 2018 *)

Formula

T(n,k) = n^2 + (n+1)^2 + ... + (n+k-1)^2 = A000330(n + k - 1) - A000330(n - 1) = T(n, k) = k*n^2 + (k^2 - k)*n + (1/3*k^3 - 1/2*k^2 + 1/6*k)
G.f.: -y*(y*(1 + y) + x*(1 - 2*y - 3*y^2) + x^2*(1 - 3*y + 4*y^2))/((- 1 + x)^3*(- 1 + y)^4). - Stefano Spezia, Nov 28 2018
Showing 1-8 of 8 results.

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