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.

A185868 (Odd,odd)-polka dot array in the natural number array A000027, by antidiagonals.

Original entry on oeis.org

1, 4, 6, 11, 13, 15, 22, 24, 26, 28, 37, 39, 41, 43, 45, 56, 58, 60, 62, 64, 66, 79, 81, 83, 85, 87, 89, 91, 106, 108, 110, 112, 114, 116, 118, 120, 137, 139, 141, 143, 145, 147, 149, 151, 153, 172, 174, 176, 178, 180, 182, 184, 186, 188, 190, 211, 213, 215, 217, 219, 221, 223, 225, 227, 229, 231, 254, 256, 258, 260, 262, 264, 266, 268, 270, 272, 274, 276, 301, 303, 305, 307, 309, 311, 313, 315, 317, 319, 321, 323, 325, 352, 354, 356, 358, 360, 362, 364, 366, 368, 370, 372, 374, 376, 378
Offset: 1

Views

Author

Clark Kimberling, Feb 05 2011

Keywords

Comments

This is one of four polka dot arrays in the natural number array A000027:
(odd,odd): A185868
(odd,even): A185869
(even,odd): A185870
(even,even): A185871
row 1: A084849
col 1: A000384
col 2: A091823
diag (1,13,...): A102083
diag (4,24,...): A085250
antidiagonal sums: A059722

Examples

			The natural number array A000027 has northwest corner
  1...2...4...7...11
  3...5...8...12..17
  6...9...13..18..24
  10..14..19..25..32
  15..20..26..33..41
The numbers in (odd,odd) positions comprise A185868:
  1....4....11...22...37
  6....13...24...39...58
  15...26...41...60...83
  28...43...62...85...112

Links

Crossrefs

Cf. A000027 (as an array), A185872, A185869, A185870, A185871.

Programs

  • Mathematica
    f[n_,k_]:=2n-1+(n+k-2)(2n+2k-3);
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 A185868(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)-7)+x*(c-5)+5 # Chai Wah Wu, Jun 18 2025

Formula

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

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.

A274701 First differences of A259280.

Original entry on oeis.org

1, 2, 1, 2, 2, 2, 3, 2, 3, 2, 3, 3, 3, 3, 3, 4, 3, 4, 3, 4, 3, 4, 4, 4, 4, 4, 4, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 5, 6, 5, 6, 5, 6, 5, 6, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 7, 8, 7, 8, 7, 8, 7, 8, 7, 8, 7, 8
Offset: 1

Views

Author

Braxton Carrigan, Peter Kagey, Joseph O'Brien, Jul 06 2016

Keywords

Links

Crossrefs

Cf. A259280, A185869, A002024.

Formula

If n is in A185869 then a(n) = ceiling(k) else a(n) = floor(k) where k is (A002024(n) + 1)/2.
Showing 1-4 of 4 results.

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