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 1-4 of 4 results.

A185872 Accumulation array of the (odd,odd)-polka dot array A185868, by antidiagonals.

Original entry on oeis.org

1, 5, 7, 16, 24, 22, 38, 59, 65, 50, 75, 120, 141, 136, 95, 131, 215, 262, 274, 245, 161, 210, 352, 440, 480, 470, 400, 252, 316, 539, 687, 770, 790, 741, 609, 372, 453, 784, 1015, 1160, 1225, 1208, 1099, 880, 525, 625, 1095, 1436, 1666, 1795, 1825, 1750, 1556, 1221, 715, 836, 1480, 1962, 2304, 2520, 2616, 2590, 2432, 2124, 1640, 946, 1090, 1947, 2605, 3090, 3420, 3605, 3647, 3540
Offset: 1

Views

Author

Clark Kimberling, Feb 05 2011

Keywords

Comments

See A144112 for the definition of accumulation array.

Examples

			Northwest corner:
   1,   5,  16,  38,  75
   7,  24,  59, 120, 215
  22,  54, 141, 262, 440
  50, 136, 174, 480, 770

Links

Crossrefs

Cf. A185868.
Row 1: A174723; column 1: A002412.

Programs

  • Mathematica
    f[n_,k_]:=2n-1+(2n+2k-4)(2n+2k-3)/2;
TableForm[Table[f[n,k],{n,1,10},{k,1,15}]] (* A185868 *)
Table[f[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten
s[n_,k_]:=Sum[f[i,j],{i,1,n},{j,1,k}]; (* accumulation array of {f(n,k)} *)
FullSimplify[s[n,k]] (*formula for A185872 *)
g[n_]:=Sum[f[n+1-k,k],{k,1,n}];
Table[g[n],{n,50}] (* A185872 *)
TableForm[Table[s[n,k],{n,1,10},{k,1,15}]]

Formula

T(n,k) = (k*n/6)*(4*n^2 + 6*n*k + 4*k^2 - 3*n - 9*k + 4), k>=1, n>=1.

A185869 (Odd,even)-polka dot array in the natural number array A000027; read by antidiagonals.

Original entry on oeis.org

2, 7, 9, 16, 18, 20, 29, 31, 33, 35, 46, 48, 50, 52, 54, 67, 69, 71, 73, 75, 77, 92, 94, 96, 98, 100, 102, 104, 121, 123, 125, 127, 129, 131, 133, 135, 154, 156, 158, 160, 162, 164, 166, 168, 170, 191, 193, 195, 197, 199, 201, 203, 205, 207, 209, 232, 234, 236, 238, 240, 242, 244, 246, 248, 250, 252, 277, 279, 281, 283, 285, 287, 289, 291, 293, 295, 297, 299, 326, 328, 330, 332, 334, 336, 338, 340, 342, 344, 346, 348, 350, 379, 381, 383, 385, 387, 389, 391, 393, 395, 397, 399, 401, 403, 405
Offset: 1

Views

Author

Clark Kimberling, Feb 05 2011

Keywords

Comments

This is the second of four polka dot arrays; see A185868.
row 1: A130883;
row 2: A100037;
row 3: A100038;
row 4: A100039;
col 1: A014107;
col 2: A033537;
col 3: A100040;
col 4: A100041;
diag (2,18,...): A077591;
diag (7,31,...): A157914;
diag (16,48,...): A035008;
diag (29,69,...): A108928;
antidiagonal sums: A033431;
antidiagonal sums: 2*(1^3, 2^3, 3^3, 4^3,...) = 2*A000578.
A060432(n) + n is odd if and only if n is in this sequence. - Peter Kagey, Feb 03 2016

Examples

			Northwest corner:
  2....7....16...29...46
  9....18...31...48...69
  20...33...50...71...96
  35...52...73...98...127

Links

Crossrefs

Cf. A000027 (as an array), A060432, A185868, A185870, A185871.

Programs

  • Haskell
    a185869 n = a185869_list !! (n - 1)
a185869_list = scanl (+) 2 $ a' 1
  where  a' n = 2 * n + 3 : replicate n 2 ++ a' (n + 1)
-- Peter Kagey, Sep 02 2016
  • Mathematica
    f[n_,k_]:=2n-1+(2n+2k-3)(n+k-1);
TableForm[Table[f[n,k],{n,1,10},{k,1,15}]]
Table[f[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten
  • Python
    from math import isqrt, comb
def A185869(n):
    a = (m:=isqrt(k:=n<<1))+(k>m*(m+1))
    x = n-comb(a,2)
    y = a-x+1
    return y*((y+(c:=x<<1)<<1)-5)+x*(c-3)+2 # Chai Wah Wu, Jun 18 2025

Formula

T(n,k) = 2n-1+(n+k-1)*(2n+2k-3), k>=1, n>=1.

A185870 (Even,odd)-polka dot array in the natural number array A000027, by antidiagonals.

Original entry on oeis.org

3, 8, 10, 17, 19, 21, 30, 32, 34, 36, 47, 49, 51, 53, 55, 68, 70, 72, 74, 76, 78, 93, 95, 97, 99, 101, 103, 105, 122, 124, 126, 128, 130, 132, 134, 136, 155, 157, 159, 161, 163, 165, 167, 169, 171, 192, 194, 196, 198, 200, 202, 204, 206, 208, 210, 233, 235, 237, 239, 241, 243, 245, 247, 249, 251, 253, 278, 280, 282, 284, 286, 288, 290, 292, 294, 296, 298, 300, 327, 329, 331, 333, 335, 337, 339, 341, 343, 345, 347, 349, 351, 380, 382, 384, 386, 388, 390, 392, 394, 396, 398, 400, 402, 404, 406
Offset: 1

Views

Author

Clark Kimberling, Feb 05 2011

Keywords

Comments

This is the third of four polka dot arrays in the array A000027. See A185868.
row 1: A033816
col 1: A014105
col 2: -A168244
antidiagonal sums: A061317
antidiagonal sums: 3*(octahedral numbers) = 3*A005900.

Examples

			Northwest corner:
  3....8....17...30...47
  10...19...32...49...70
  21...34...51...72...97
  36...53...74...99...128

Links

Crossrefs

Cf. A000027 (as an array), A185868, A185869, A185871.

Programs

  • Mathematica
    f[n_,k_]:=2n+(2n+2k-3)(n+k-1);
TableForm[Table[f[n,k],{n,1,10},{k,1,15}]]
Table[f[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten
  • Python
    from math import comb, isqrt
def A185870(n):
    a = (m:=isqrt(k:=n<<1))+(k>m*(m+1))
    x = n-comb(a,2)
    y = a-x+1
    return y*((y+(c:=x<<1)<<1)-5)+x*(c-3)+3 # Chai Wah Wu, Jun 18 2025

Formula

T(n,k) = 2*n + (n+k-1)*(2*n+2*k-3), k>=1, n>=1.

A185871 (Even,even)-polka dot array in the natural number array A000027, by antidiagonals.

Original entry on oeis.org

5, 12, 14, 23, 25, 27, 38, 40, 42, 44, 57, 59, 61, 63, 65, 80, 82, 84, 86, 88, 90, 107, 109, 111, 113, 115, 117, 119, 138, 140, 142, 144, 146, 148, 150, 152, 173, 175, 177, 179, 181, 183, 185, 187, 189, 212, 214, 216, 218, 220, 222, 224, 226, 228, 230, 255, 257, 259, 261, 263, 265, 267, 269, 271, 273, 275, 302, 304, 306, 308, 310, 312, 314, 316, 318, 320, 322, 324, 353, 355, 357, 359, 361, 363, 365, 367, 369, 371, 373, 375, 377, 408, 410, 412, 414, 416, 418, 420, 422, 424, 426, 428, 430, 432, 434
Offset: 1

Views

Author

Clark Kimberling, Feb 05 2011

Keywords

Comments

This is the fourth of four polka dot arrays in the natural number array A000027. See A185868.
row 1: A096376
col 1: A014106
col 2: A071355
diag (5,25,...): A080856
diag (12,40,...): A033586
antidiagonal sums: A048395 (sums of consecutive squares)

Examples

			Northwest corner:
  5....12...23...38...57
  14...25...40...59...82
  27...42...61...84...111
  44...63...86...113..144

Links

Crossrefs

Cf. A000027 (as an array), A185868, A185869, A185870.

Programs

  • Mathematica
    f[n_,k_]:=2n+(n+k-1)(2n+2k-1);
TableForm[Table[f[n,k],{n,1,10},{k,1,15}]]
Table[f[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten
  • Python
    from math import comb, isqrt
def A185871(n):
    a = (m:=isqrt(k:=n<<1))+(k>m*(m+1))
    x = n-comb(a,2)
    y = a-x+1
    return y*((y+(c:=x<<1)<<1)-3)+x*(c-1)+1 # Chai Wah Wu, Jun 18 2025

Formula

T(n,k) = 2*n + (n+k-1)*(2*n+2*k-1), k>=1, n>=1.
Showing 1-4 of 4 results.

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