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.

A083487 Triangle read by rows: T(n,k) = 2*n*k + n + k (1 <= k <= n).

Original entry on oeis.org

4, 7, 12, 10, 17, 24, 13, 22, 31, 40, 16, 27, 38, 49, 60, 19, 32, 45, 58, 71, 84, 22, 37, 52, 67, 82, 97, 112, 25, 42, 59, 76, 93, 110, 127, 144, 28, 47, 66, 85, 104, 123, 142, 161, 180, 31, 52, 73, 94, 115, 136, 157, 178, 199, 220, 34, 57, 80, 103, 126, 149, 172, 195, 218, 241, 264
Offset: 1

Views

Author

Artemario Tadeu Medeiros da Silva (artemario(AT)uol.com.br), Jun 09 2003

Keywords

Comments

T(n,k) gives number of edges (of unit length) in a k X n grid.
The values 2*T(n,k)+1 = (2*n+1)*(2*k+1) are nonprime and therefore in A047845.

Examples

			Triangle begins:
   4;
   7, 12;
  10, 17, 24;
  13, 22, 31, 40;
  16, 27, 38, 49,  60;
  19, 32, 45, 58,  71,  84;
  22, 37, 52, 67,  82,  97, 112;
  25, 42, 59, 76,  93, 110, 127, 144;
  28, 47, 66, 85, 104, 123, 142, 161, 180;

Links

Crossrefs

Cf. A000217, A016777, A016873, A017017, A017209, A017449, A033954.
Cf. A056220, A082652, A083003, A090288, A182868, A186113, A271625.
See A151890 for another version.

Programs

  • Magma
    [(2*n*k + n + k): k in [1..n], n in [1..11]]; // Vincenzo Librandi, Jun 01 2014
  • Mathematica
    T[n_,k_]:= 2 n k + n + k; Table[T[n, k], {n, 10}, {k, n}]//Flatten (* Vincenzo Librandi, Jun 01 2014 *)
  • Python
    def T(r, c): return 2*r*c + r + c
a = [T(r, c) for r in range(12) for c in range(1, r+1)]
print(a) # Michael S. Branicky, Sep 07 2022
  • SageMath
    flatten([[2*n*k +n +k for k in range(1,n+1)] for n in range(1,14)]) # G. C. Greubel, Oct 17 2023

Formula

From G. C. Greubel, Oct 17 2023: (Start)
T(n, 1) = A016777(n).
T(n, 2) = A016873(n).
T(n, 3) = A017017(n).
T(n, 4) = A017209(n).
T(n, 5) = A017449(n).
T(n, 6) = A186113(n).
T(n, n-1) = A056220(n).
T(n, n-2) = A090288(n-2).
T(n, n-3) = A271625(n-2).
T(n, n) = 4*A000217(n).
T(2*n, n) = A033954(n).
Sum_{k=1..n} T(n, k) = A162254(n).
Sum_{k=1..n} (-1)^(k-1)*T(n, k) = A182868((n+1)/2) if n is odd otherwise A182868(n/2) + 1. (End)

Extensions

Edited by N. J. A. Sloane, Jul 23 2009
Name edited by Michael S. Branicky, Sep 07 2022

A213831 Rectangular array: (row n) = b**c, where b(h) = 2*h-1, c(h) = 3*n-5+3*h, n>=1, h>=1, and ** = convolution.

Original entry on oeis.org

1, 7, 4, 24, 19, 7, 58, 51, 31, 10, 115, 106, 78, 43, 13, 201, 190, 154, 105, 55, 16, 322, 309, 265, 202, 132, 67, 19, 484, 469, 417, 340, 250, 159, 79, 22, 693, 676, 616, 525, 415, 298, 186, 91, 25, 955, 936, 868, 763, 633
Offset: 1

Views

Author

Clark Kimberling, Jul 04 2012

Keywords

Comments

Principal diagonal: A213832.
Antidiagonal sums: A212560.
row 1, (1,3,5,7,...)**(1,4,7,10,...): A081436.
Row 2, (1,3,5,7,...)**(4,7,10,13,...): A162254.
Row 3, (1,3,5,7,...)**(7,10,13,16,...): (2*k^3 + 11*k^2 + k)/2.
For a guide to related arrays, see A212500.

Examples

			1....7....24....58....115
4....19...51....106...190
7....31...78....154...265
10...43...105...202...340
13...55...132...250...415

Crossrefs

Cf. A212500

Programs

  • Mathematica
    b[n_]:=2n-1;c[n_]:=3n-2;
t[n_,k_]:=Sum[b[k-i]c[n+i],{i,0,k-1}]
TableForm[Table[t[n,k],{n,1,10},{k,1,10}]]
Flatten[Table[t[n-k+1,k],{n,12},{k,n,1,-1}]]
r[n_]:=Table[t[n,k],{k,1,60}] (* A213831 *)
Table[t[n,n],{n,1,40}] (* A213832 *)
s[n_]:=Sum[t[i,n+1-i],{i,1,n}]
Table[s[n],{n,1,50}] (* A212560 *)

Formula

T(n,k) = 4*T(n,k-1)-6*T(n,k-2)+4*T(n,k-3)-T(n,k-4).
G.f. for row n: f(x)/g(x), where f(x) = x*((3*n-2) + 3*x - (3*n-5)*x^2) and g(x) = (1-x)^4.
Northwest corner (the array is read by falling antidiagonals):

A155156 Triangle T(n, k) = 4*n*k + 2*n + 2*k, read by rows.

Original entry on oeis.org

8, 14, 24, 20, 34, 48, 26, 44, 62, 80, 32, 54, 76, 98, 120, 38, 64, 90, 116, 142, 168, 44, 74, 104, 134, 164, 194, 224, 50, 84, 118, 152, 186, 220, 254, 288, 56, 94, 132, 170, 208, 246, 284, 322, 360, 62, 104, 146, 188, 230, 272, 314, 356, 398, 440, 68, 114, 160, 206, 252, 298, 344, 390, 436, 482, 528
Offset: 1

Views

Author

Vincenzo Librandi, Jan 21 2009

Keywords

Comments

First column: A016933, second column: A017317, third column: A063151, fourth column: 2*A017209. - Vincenzo Librandi, Nov 21 2012

Examples

			Triangle begins:
   8;
  14,  24;
  20,  34,  48;
  26,  44,  62,  80;
  32,  54,  76,  98, 120;
  38,  64,  90, 116, 142, 168;
  44,  74, 104, 134, 164, 194, 224;
  50,  84, 118, 152, 186, 220, 254, 288;
  56,  94, 132, 170, 208, 246, 284, 322, 360;
  62, 104, 146, 188, 230, 272, 314, 356, 398, 440;

Links

Crossrefs

Cf. A016933, A017317, A063151, A017209. A083487, A162254, A163832 (row sums).

Programs

  • Magma
    [4*n*k + 2*n + 2*k : k in [1..n], n in [1..11]]; // Vincenzo Librandi, Nov 21 2012
  • Maple
    seq(seq( 2*(2*n*k +n+k), k=1..n), n=1..15); # G. C. Greubel, Mar 20 2021
  • Mathematica
    T[n_,k_]:=4*n*k +2*n +2*k; Table[T[n, k], {n, 15}, {k, n}]//Flatten (* Vincenzo Librandi, Nov 21 2012 *)
  • Sage
    flatten([[2*(2*n*k +n+k) for k in (1..n)] for n in (1..15)]) # G. C. Greubel, Mar 20 2021

Formula

T(n, k) = 2*A083487(n, k). - R. J. Mathar, Jan 05 2011
Sum_{k=0..n} T(n,k) = n*(2*n^2 + 5*n + 1) = 2*A162254(n) = A163832(n). - G. C. Greubel, Mar 20 2021

Extensions

Edited by Robert Hochberg, Jun 21 2010

A163832 a(n) = n*(2*n^2 + 5*n + 1).

Original entry on oeis.org

0, 8, 38, 102, 212, 380, 618, 938, 1352, 1872, 2510, 3278, 4188, 5252, 6482, 7890, 9488, 11288, 13302, 15542, 18020, 20748, 23738, 27002, 30552, 34400, 38558, 43038, 47852, 53012, 58530, 64418, 70688, 77352, 84422, 91910, 99828, 108188, 117002
Offset: 0

Views

Author

Vincenzo Librandi, Aug 05 2009

Keywords

Comments

Row sums of triangle A155156.

Links

Crossrefs

Cf. A155156.

Programs

  • Mathematica
    Table[n(2n^2+5n+1),{n,0,40}] (* or *) LinearRecurrence[{4,-6,4,-1},{0,8,38,102},40] (* Harvey P. Dale, Feb 02 2012 *)
  • PARI
    for(n=0, 40, print1(n*(2*n^2+5*n+1)", ")); \\ Vincenzo Librandi, Feb 22 2012

Formula

G.f.: -2*x*(1+x)*(x-4)/(x-1)^4.
a(n) = 4*a(n-1) -6*a(n-2) +4*a(n-3) -a(n-4).
a(n) = A163683(n) + n = A163815(n) - 2*n = 2*A162254(n).
a(n) = -n*A168244(n+2). - Bruno Berselli, Feb 02 2012
E.g.f.: x*(8 + 11*x + 2*x^2)*exp(x). - G. C. Greubel, Aug 05 2017

Extensions

Edited by R. J. Mathar, Aug 05 2009

A360665 Square array T(n, k) = k*((2*n-1)*k+1)/2 read by rising antidiagonals.

Original entry on oeis.org

0, 0, 0, 0, 1, -1, 0, 2, 3, -3, 0, 3, 7, 6, -6, 0, 4, 11, 15, 10, -10, 0, 5, 15, 24, 26, 15, -15, 0, 6, 19, 33, 42, 40, 21, -21, 0, 7, 23, 42, 58, 65, 57, 28, -28, 0, 8, 27, 51, 74, 90, 93, 77, 36, -36, 0, 9, 31, 60, 90, 115, 129, 126, 100, 45, -45
Offset: 0

Views

Author

Paul Curtz, Mar 17 2023

Keywords

Examples

			By rows:
   0,   0,  -1,  -3,  -6,  -10,  -15,  -21,  -28, ...   = -A161680
   0,   1,   3,   6,  10,   15,   21,   28,   36, ...   =  A000217
   0,   2,   7,  15,  26,   40,   57,   77,  100, ...   =  A005449
   0,   3,  11,  24,  42,   65,   93,  126,  164, ...   =  A005475
   0,   4,  15,  33,  58,   90,  129,  175,  228, ...   =  A022265
   0,   5,  19,  42,  74,  115,  165,  224,  292, ...   =  A022267
   0,   6,  23,  51,  90,  140,  201,  273,  356, ...   =  A022269
   ... .

Crossrefs

Cf. Antidiagonal sums: A034827(n+1).
Cf. A000217, A000290, A005449, A005475, A022265.
Cf. A022267, A022269, A059270, A081436, A101986.
Cf. A161680, A162254, A179297, A360962, A361226.

Programs

  • Mathematica
    T[n_, k_] := ((2*n - 1)*k^2 + k)/2; Table[T[n - k, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, Mar 31 2023 *)
  • PARI
    T(n, k) = ((2*n-1)*k^2+k)/2 \\ Thomas Scheuerle, Mar 17 2023

Formula

T(n,k) = T(n,k-1)+k^2.
T(n,n) = A081436(n-1).
T(n,n+1) = A059270(n).
T(n,n+4) = -3*A179297(n+4).
T(n+3,n) = A162254(n).
T(n+5,n) = 3*A101986(n).
From Stefano Spezia, Mar 31 2023: (Start)
O.g.f.: (x*y - y^2 + 2*x*y^2)/((1 - x)^2*(1 - y)^3).
E.g.f.: exp(x+y)*y*(2*x - y + 2*x*y)/2. (End)
Showing 1-5 of 5 results.

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

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