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 1-7 of 7 results.

A261835 Number A(n,k) of compositions of n into distinct parts where each part i is marked with a word of length i over a k-ary alphabet whose letters appear in alphabetical order; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 3, 3, 0, 1, 4, 6, 16, 3, 0, 1, 5, 10, 46, 21, 5, 0, 1, 6, 15, 100, 75, 50, 11, 0, 1, 7, 21, 185, 195, 231, 205, 13, 0, 1, 8, 28, 308, 420, 736, 1414, 292, 19, 0, 1, 9, 36, 476, 798, 1876, 6032, 2376, 587, 27, 0
Offset: 0

Views

Author

Alois P. Heinz, Sep 02 2015

Keywords

Comments

Also matrices with k rows of nonnegative integers with distinct positive column sums and total element sum n.
A(2,2) = 3: (matrices and corresponding marked compositions are given)
[1] [2] [0]
[1] [0] [2]
2ab, 2aa, 2bb.

Examples

			A(3,2) = 16: 3aaa, 3aab, 3abb, 3bbb, 2aa1a, 2aa1b, 2ab1a, 2ab1b, 2bb1a, 2bb1b, 1a2aa, 1a2ab, 1a2bb, 1b2aa, 1b2ab, 1b2bb.
Square array A(n,k) begins:
  1,  1,   1,    1,     1,     1,      1,      1, ...
  0,  1,   2,    3,     4,     5,      6,      7, ...
  0,  1,   3,    6,    10,    15,     21,     28, ...
  0,  3,  16,   46,   100,   185,    308,    476, ...
  0,  3,  21,   75,   195,   420,    798,   1386, ...
  0,  5,  50,  231,   736,  1876,   4116,   8106, ...
  0, 11, 205, 1414,  6032, 19320,  51114, 117936, ...
  0, 13, 292, 2376, 11712, 42610, 126288, 322764, ...

Links

Crossrefs

Columns k=0-10 give: A000007, A032020, A261840, A261841, A261842, A261843, A261844, A261845, A261846, A261847, A261848.
Rows n=0-4 give: A000012, A001477, A000217, A255211, A228317(n+2).
Main diagonal gives A261837.
Cf. A261780, A261836.

Programs

  • Maple
    b:= proc(n, i, p, k) option remember;
      `if`(i*(i+1)/2n, 0, b(n-i, i-1, p+1, k)*binomial(i+k-1, k-1))))
    end:
A:= (n, k)-> b(n$2, 0, k):
seq(seq(A(n, d-n), n=0..d), d=0..12);
  • Mathematica
    b[n_, i_, p_, k_] := b[n, i, p, k] = If[i*(i+1)/2 < n, 0, If[n == 0, p!, b[n, i-1, p, k] + If[i>n, 0, b[n-i, i-1, p+1, k]*Binomial[i+k-1, k-1]]]]; A[n_, k_] := b[n, n, 0, k]; Table[A[n, d-n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, Jan 16 2017, translated from Maple *)

Formula

A(n,k) = Sum_{i=0..k} C(k,i) * A261836(n,k-i).

A304993 a(n) = n*(n + 1)*(7*n + 5)/6.

Original entry on oeis.org

0, 4, 19, 52, 110, 200, 329, 504, 732, 1020, 1375, 1804, 2314, 2912, 3605, 4400, 5304, 6324, 7467, 8740, 10150, 11704, 13409, 15272, 17300, 19500, 21879, 24444, 27202, 30160, 33325, 36704, 40304, 44132, 48195, 52500, 57054, 61864, 66937, 72280, 77900, 83804, 89999, 96492
Offset: 0

Views

Author

Bruno Berselli, May 23 2018

Keywords

Comments

The sequence provides the sums of the triangular numbers from A000217(n) to A000217(2*n).

Links

Crossrefs

Cf. A000217, A255211.
Partial sums of A022265.
Cf. A045943: Sum_{k = n..2*n} k.
Cf. A050409: Sum_{k = n..2*n} k^2.
Row sums of the triangle in A141433.

Programs

  • Mathematica
    Table[n (n + 1) (7 n + 5)/6, {n, 0, 50}]
LinearRecurrence[{4,-6,4,-1},{0,4,19,52},50] (* Harvey P. Dale, May 03 2023 *)
  • PARI
    concat(0, Vec(x*(4 + 3*x)/(1 - x)^4 + O(x^40))) \\ Colin Barker, May 25 2018

Formula

O.g.f.: x*(4 + 3*x)/(1 - x)^4.
E.g.f.: x*(24 + 33*x + 7*x^2)*exp(x)/6.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
a(n) = -A255211(-n-1).
a(n) + a(-n) = A016742(n).
a(n) = Sum_{k = n..2*n} k*(k+1)/2.

A178238 Triangle read by rows: partial column sums of the triangle of natural numbers (written sequentially by rows).

Original entry on oeis.org

1, 3, 3, 7, 8, 6, 14, 16, 15, 10, 25, 28, 28, 24, 15, 41, 45, 46, 43, 35, 21, 63, 68, 70, 68, 61, 48, 28, 92, 98, 101, 100, 94, 82, 63, 36, 129, 136, 140, 140, 135, 124, 106, 80, 45, 175, 183, 188, 189, 185, 175, 158, 133, 99, 55, 231, 240, 246, 248, 245, 236, 220, 196, 163, 120, 66
Offset: 1

Views

Author

Gary W. Adamson, May 23 2010

Keywords

Comments

T(n,k) is the n-th partial sum of the k-th column of the triangle of natural numbers.

Examples

			First few rows of the triangle:
    1;
    3,   3;
    7,   8,   6;
   14,  16,  15,  10;
   25,  28,  28,  24,  15;
   41,  45,  46,  43,  35,  21;
   63,  68,  70,  68,  61,  48,  28;
   92,  98, 101, 100,  94,  82,  63,  36;
  129, 136, 140, 140, 135, 124, 106,  80,  45;
  175, 183, 188, 189, 185, 175, 158, 133,  99,  55;
  231, 240, 246, 248, 245, 236, 220, 196, 163, 120,  66;
  298, 308, 314, 318, 316, 308, 293, 270, 238, 196, 143, 78;
  ...
These are the partial sums of the columns of the triangle:
  1;
  2, 3;
  4, 5, 6;
  7, 8, 9, 10;
  ...
For example, T(4,2) = 3 + 5 + 8 = 16.

Links

Crossrefs

Column 1 is A004006.
Main diagonal is A000217.
Row sums are A002817.
Cf. A000012, A000027.

Programs

  • PARI
    T(n,k) = {binomial(n+1, 3) - binomial(k, 3) + k*(n-k+1)}
{ for(n=1, 10, for(k=1, n, print1(T(n,k), ", ")); print) } \\ Andrew Howroyd, Apr 18 2021

Formula

As infinite lower triangular matrices, A000012 * A000027.
From Andrew Howroyd, Apr 18 2021: (Start)
T(n,k) = Sum_{j=k..n} (k + j*(j-1)/2).
T(n,k) = binomial(n+1, 3) - binomial(k, 3) + k*(n-k+1).
T(2*n, n) = A255211(n).
(End)

Extensions

Name changed and terms a(56) and beyond from Andrew Howroyd, Apr 18 2021

A141434 Triangle T(n, k) = (k-1)*(3*n-k-1), read by rows.

Original entry on oeis.org

0, 0, 3, 0, 6, 10, 0, 9, 16, 21, 0, 12, 22, 30, 36, 0, 15, 28, 39, 48, 55, 0, 18, 34, 48, 60, 70, 78, 0, 21, 40, 57, 72, 85, 96, 105, 0, 24, 46, 66, 84, 100, 114, 126, 136, 0, 27, 52, 75, 96, 115, 132, 147, 160, 171
Offset: 1

Views

Author

Roger L. Bagula and Gary W. Adamson, Aug 06 2008

Keywords

Examples

			Triangle begins as:
  0;
  0,  3;
  0,  6, 10;
  0,  9, 16, 21;
  0, 12, 22, 30, 36;
  0, 15, 28, 39, 48,  55;
  0, 18, 34, 48, 60,  70,  78;
  0, 21, 40, 57, 72,  85,  96, 105;
  0, 24, 46, 66, 84, 100, 114, 126, 136;
  0, 27, 52, 75, 96, 115, 132, 147, 160, 171;

Links

Crossrefs

Cf. A255211 (row sums).

Programs

  • Magma
    [(k-1)*(3*n-k-1): k in [1..n], n in [1..12]]; // G. C. Greubel, Apr 01 2021
  • Maple
    A141434:= (n,k) -> (k-1)*(3*n-k-1); seq(seq(A141434(n,k), k=1..n), n=1..12); # G. C. Greubel, Apr 01 2021
  • Mathematica
    Table[(k-1)*(3*n-k-1), {n, 12}, {k, n}]//Flatten (* modified by G. C. Greubel, Apr 01 2021 *)
  • Sage
    flatten([[(k-1)*(3*n-k-1) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Apr 01 2021

Formula

Sum_{k=1..n} T(n,k) = (n-1)*n*(7*n-5)/6. - R. J. Mathar, Sep 07 2011

A255687 a(n) = n*(n + 1)*(7*n + 11)/6.

Original entry on oeis.org

0, 6, 25, 64, 130, 230, 371, 560, 804, 1110, 1485, 1936, 2470, 3094, 3815, 4640, 5576, 6630, 7809, 9120, 10570, 12166, 13915, 15824, 17900, 20150, 22581, 25200, 28014, 31030, 34255, 37696, 41360, 45254, 49385, 53760, 58386, 63270, 68419, 73840, 79540, 85526
Offset: 0

Views

Author

Luce ETIENNE, Mar 02 2015

Keywords

Comments

This sequence gives the number of triangles of all sizes in (3*n^2+2*n)-polyiamonds in a pentagonal or heptagonal configuration.
Also sum of 2*n*(n+1)*(n+2)/3 triangles oriented in one direction and n*(n+1)^2/2 oriented in the opposite direction.

Links

Crossrefs

First bisection of A212977.
Partial sums of A179986.
Cf. A000292, A006002, A007584, A255211.

Programs

  • Magma
    [n*(n+1)*(7*n+11)/6: n in [0..50]]; // Bruno Berselli, Mar 02 2015
  • Maple
    A255687:=n->n*(n+1)*(7*n+11)/6: seq(A255687(n), n=0..50); # Wesley Ivan Hurt, Mar 03 2015
  • Mathematica
    Table[n (n + 1) (7 n + 11)/6, {n, 0, 50}] (* Bruno Berselli, Mar 02 2015 *)
LinearRecurrence[{4,-6,4,-1},{0,6,25,64},50] (* Harvey P. Dale, Jul 17 2015 *)
  • PARI
    vector(50, n, n--; n*(n+1)*(7*n+11)/6)
  • PARI
    concat(0, Vec(x*(x+6)/(x-1)^4 + O(x^100))) \\ Colin Barker, Mar 02 2015
  • Sage
    [n*(n+1)*(7*n+11)/6 for n in (0..50)] # Bruno Berselli, Mar 02 2015

Formula

a(n) = (1/2)*(Sum_{j=0..n} (n+1-j)*(3*n-j) + Sum_{j=0..n-1} (n-j)*(3*n+1-3*j)).
From Colin Barker, Mar 02 2015: (Start)
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
G.f.: x*(x + 6)/(x - 1)^4. (End)
a(n) = -A007584(-n-1). - Bruno Berselli, Mar 02 2015
From Elmo R. Oliveira, Aug 18 2025: (Start)
E.g.f.: exp(x)*x*(36 + 39*x + 7*x^2)/6.
a(n) = A212977(2*n). (End)

A257093 a(n) = n*(n+1)*(13*n+2)/6.

Original entry on oeis.org

0, 5, 28, 82, 180, 335, 560, 868, 1272, 1785, 2420, 3190, 4108, 5187, 6440, 7880, 9520, 11373, 13452, 15770, 18340, 21175, 24288, 27692, 31400, 35425, 39780, 44478, 49532, 54955, 60760, 66960, 73568, 80597, 88060, 95970, 104340, 113183, 122512, 132340
Offset: 0

Views

Author

Luce ETIENNE, Apr 16 2015

Keywords

Comments

This sequence gives the number of triangles of all sizes in (5*n^2)-polyiamonds in a tetragonal or hexagonal or heptagonal configuration.
It is the sum of (1/2)*Sum_{j=0..n-1} (n-j)*(5*n+1-j) triangles oriented in one direction and (1/2)*Sum_{j-0..n-1} (n-j)*(5*n-1-3*j) oriented in the opposite direction.
Shäfli's notation: 3.3.3.3.3 for a(1).
The difference between this sequence and A050409(n) equals A000292(n-1).
Also, (1/3)*(A002717(2*n) + A255211(n) - 2*A000330(n)) gives A033994(n): a (5*n^2)-polyiamond in pentagonal configuration that does not belong to this sequence because a(1)=6.
a(n) is odd only when n mod 4 = 1.

Examples

			Second comment a(0) = 0; a(1) = 3 + 2; a(2) = 16 + 12; a(3) = 46 + 36; a(4) = 100 + 80; a(5) = 185 + 150; a(6) = 308 + 252.

Links

Crossrefs

Cf. A002411, A011379, A033429, A050409, A000292, A002717, A000330, A033994, A255211.

Programs

  • Magma
    [n*(n+1)*(13*n+2)/6: n in [0..40]]; // Vincenzo Librandi, Apr 16 2015
  • Mathematica
    Table[n (n + 1) (13 n + 2)/6, {n, 0, 40}] (* Vincenzo Librandi, Apr 16 2015 *)
CoefficientList[Series[x (5+8x)/(1-x)^4,{x,0,50}],x] (* or *) LinearRecurrence[{4,-6,4,-1},{0,5,28,82},60] (* Harvey P. Dale, Feb 12 2023 *)

Formula

a(n) = Sum_{j=0..n-1} (n-j)*(5*n-2*j).
From Vincenzo Librandi, Apr 16 2015: (Start)
G.f.: x*(5+8*x)/(1-x)^4.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). (End)
E.g.f.: exp(x)*x*(30 + 54*x + 13*x^2)/6. - Stefano Spezia, Mar 02 2025

Extensions

Corrected by Harvey P. Dale, Feb 12 2023

A296636 Sequences n*(n+1)*(6*n+1)/2 and n*(n+1)*(7*n+1)/2 interleaved.

Original entry on oeis.org

0, 7, 8, 39, 45, 114, 132, 250, 290, 465, 540, 777, 903, 1204, 1400, 1764, 2052, 2475, 2880, 3355, 3905, 4422, 5148, 5694, 6630, 7189, 8372, 8925, 10395, 10920, 12720, 13192, 15368, 15759, 18360, 18639, 21717, 21850, 25460, 25410, 29610, 29337, 34188, 33649, 39215, 38364, 44712
Offset: 0

Views

Author

Luce ETIENNE, Dec 17 2017

Keywords

Comments

Difference between these subsequences is A002411.
This sequence gives numbers of triangles all sizes in every n-th stage [of what? - N. J. A. Sloane, Feb 09 2018].

Links

Crossrefs

Cf. A002411, A002717, A006331, A033581, A033582, A255211.

Programs

  • GAP
    List([0..50], n -> (2*n+1-(-1)^n)*(2*n+5-(-1)^n)*(13*n+10+(n-6)*(-1)^n)/128); # Bruno Berselli, Feb 12 2018
  • Magma
    [(2*n+1-(-1)^n)*(2*n+5-(-1)^n)*(13*n+10+(n-6)*(-1)^n)/128: n in [0..50]]; // Bruno Berselli, Feb 12 2018
  • Mathematica
    CoefficientList[Series[x (7 + 8 x + 11 x^2 + 13 x^3)/((1 - x)^4*(1 + x)^4), {x, 0, 46}], x] (* Michael De Vlieger, Dec 18 2017 *)
LinearRecurrence[{0,4,0,-6,0,4,0,-1},{0,7,8,39,45,114,132,250},50] (* Harvey P. Dale, May 01 2018 *)
Rest[Flatten[Table[With[{c=(n(n+1))/2},{c*(6n+1),c*(7n+1)}],{n,0,30}]]] (* Harvey P. Dale, Oct 11 2020 *)
  • PARI
    concat(0, Vec(x*(7 + 8*x + 11*x^2 + 13*x^3) / ((1 - x)^4*(1 + x)^4) + O(x^80))) \\ Colin Barker, Dec 18 2017

Formula

a(n) = a(n-1)+4*a(n-2)-4*a(n-3)-6*a(n-4)+6*a(n-5)+4*a(n-6)-4*a(n-7)-a(n-8)+a(n-9).
a(n) = (2*n+1-(-1)^n)*(2*n+5-(-1)^n)*(13*n+10+(n-6)*(-1)^n)/128.
From Colin Barker, Dec 18 2017: (Start)
G.f.: x*(7 + 8*x + 11*x^2 + 13*x^3) / ((1 - x)^4*(1 + x)^4).
a(n) = 4*a(n-2) - 6*a(n-4) + 4*a(n-6) - a(n-8) for n>7.
(End)
Showing 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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