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.

A081298 Main diagonal of the square array A081297.

Original entry on oeis.org

1, 1, 7, 25, 461, 2821, 84883, 734161, 30684601, 342800821, 18348174791, 251203133545, 16394732478853, 265727053328101, 20464206411678331, 383172119935376161, 34011762638354230001, 722380674949394645269
Offset: 0

Views

Author

Paul Barry, Mar 17 2003

Keywords

Links

Crossrefs

Cf. A081299, A081300, A081301, A081302.

Programs

  • Magma
    [((n+1)^(n+1)-(-n)^(n+1))/(2*n+1): n in [0..20]]; // Vincenzo Librandi, Aug 07 2013
  • Mathematica
    Table[((n + 1)^(n + 1) - (-n)^(n + 1)) / (2 n + 1),{n, 0, 20}] (* Vincenzo Librandi, Aug 07 2013 *)

Formula

a(n) = ((n+1)^(n+1)-(-n)^(n+1))/(2n+1).

A081299 Diagonal of square array A081297.

Original entry on oeis.org

1, 3, 13, 181, 1281, 32551, 314245, 11638089, 141943681, 6914792611, 101829922701, 6152865979261, 106138316846017, 7657555132292703, 151395000617362741, 12699162274678909201, 283052059672669084161
Offset: 0

Views

Author

Paul Barry, Mar 17 2003

Keywords

Links

Crossrefs

Cf. A081298, A081300, A081301, A081302.

Programs

  • Magma
    [((n+1)^(n+2)-(-n)^(n+2))/(2*n+1): n in [0..20]]; // Vincenzo Librandi, Aug 08 2013
  • Mathematica
    Table[((n + 1)^(n + 2) - (-n)^(n + 2)) / (2 n + 1), {n, 0, 20}] (* Vincenzo Librandi, Aug 08 2013 *)

Formula

a(n) = ((n+1)^(n+2)-(-n)^(n+2))/(2*n+1).

A081300 Diagonal of square array A081297.

Original entry on oeis.org

1, 5, 55, 481, 10501, 117181, 3879331, 52751105, 2351234953, 37766866501, 2120129149711, 39311679607201, 2663716583547085, 56019878838007085, 4448878347069812251, 104660471059169187841, 9534251497305019644433
Offset: 0

Views

Author

Paul Barry, Mar 17 2003

Keywords

Links

Crossrefs

Cf. A081298, A081299, A081301, A081302.

Programs

  • Magma
    [((n+1)^(n+3)-(-n)^(n+3))/(2*n+1): n in [0..20]]; // Vincenzo Librandi, Aug 07 2013
  • Mathematica
    Table[((n + 1)^(n + 3) - (-n)^(n + 3)) / (2 n + 1), {n, 0, 20}] (* Vincenzo Librandi, Aug 07 2013 *)

Formula

a(n) = ((n+1)^(n+3)-(-n)^(n+3))/(2*n+1).

A081301 Subdiagonal of square array A081297.

Original entry on oeis.org

1, 1, 13, 41, 991, 5461, 194713, 1545265, 73022131, 758924981, 44709567013, 575279386969, 40614439994311, 623479972408021, 51316625644764721, 915589327332039905, 86090052046429522747, 1750836276286883890741
Offset: 0

Views

Author

Paul Barry, Mar 17 2003

Keywords

Links

Crossrefs

Cf. A081298, A081299, A081300, A081302.

Programs

  • Magma
    [((n+2)^(n+1)-(-(n+1))^(n+1))/(2*n+3):n in [0..20]]; // Vincenzo Librandi, Aug 08 2013
  • Mathematica
    Table[((n + 2)^(n + 1) - (-(n + 1))^(n + 1)) / (2 n + 3), {n, 0, 20}] (* Vincenzo Librandi, Aug 08 2013 *)

Formula

a(n) = ((n+2)^(n+1)-(-(n+1))^(n+1))/(2*n+3).

A081302 Subdiagonal of square array A081297.

Original entry on oeis.org

1, 1, 21, 61, 1891, 9633, 404713, 2997541, 159902271, 1564345201, 101406750589, 1236882490845, 94479649710811, 1382731226210881, 121677107761110993, 2079381120597925237, 207197254527662127511, 4051708966720224576081
Offset: 0

Views

Author

Paul Barry, Mar 17 2003

Keywords

Links

Crossrefs

Cf. A081298, A081299, A081300, A081301.

Programs

  • Magma
    [((n+3)^(n+1)-(-(n+2))^(n+1))/(2*n+5): n in [0..20]]; // Vincenzo Librandi, Aug 08 2013
  • Mathematica
    Table[((n + 3)^(n + 1) - (-(n + 2))^(n + 1)) / (2 n + 5), {n, 0, 20}] (* Vincenzo Librandi, Aug 08 2013 *)

Formula

a(n) = ((n+3)^(n+1)-(-(n+2))^(n+1))/(2*n+5).

A072024 Table by antidiagonals of T(n,k) = ((n+1)^k - (-n)^k)/(2*n+1).

Original entry on oeis.org

0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 3, 1, 1, 0, 1, 5, 7, 1, 1, 0, 1, 11, 13, 13, 1, 1, 0, 1, 21, 55, 25, 21, 1, 1, 0, 1, 43, 133, 181, 41, 31, 1, 1, 0, 1, 85, 463, 481, 461, 61, 43, 1, 1, 0, 1, 171, 1261, 2653, 1281, 991, 85, 57, 1, 1, 0, 1, 341, 4039, 8425, 10501, 2821, 1891, 113, 73, 1, 1, 0
Offset: 0

Views

Author

Henry Bottomley, Jun 06 2002

Keywords

Comments

Rows of the array have g.f. x/((1+k*x)*(1-(k+1)*x)). - Philippe Deléham, Nov 24 2013

Examples

			Rows start:
0 1 1  1   1    1     1      1       1        1 ...
0 1 1  3   5   11    21     43      85      171 ...
0 1 1  7  13   55   133    463    1261     4039 ...
0 1 1 13  25  181   481   2653    8425    40261 ...
0 1 1 21  41  461  1281  10501   36121   246141 ...
0 1 1 31  61  991  2821  32551  117181  1093711 ...
0 1 1 43  85 1891  5461  84883  314245  3879331 ...
0 1 1 57 113 3305  9633 194713  734161 11638089 ...
...

Links

Crossrefs

Rows include A057427, A001045, A015441, A053404, A053428, A053430, A065874, etc. Columns include A000004, A000012, A000012, A002061, A001844, A072025, etc.
Cf. A081297.

Programs

  • Magma
    [((k+1)^(n-k) - (-k)^(n-k))/(2*k+1): k in [0..n], n in [0..12]]; // G. C. Greubel, Jan 27 2020
  • Maple
    seq(seq( ((k+1)^(n-k) - (-k)^(n-k))/(2*k+1), k=0..n), n=0..12); # G. C. Greubel, Jan 27 2020
  • Mathematica
    T[n_, k_]:= ((n + 1)^k - (-n)^k)/(2n + 1); Flatten[Join[{0}, Table[T[k, n- k], {n, 1, 15}, {k, 0, n}]]] (* Indranil Ghosh, Mar 27 2017 *)
  • PARI
    for(n=0, 10, for(k=0, 9, print1(((n+1)^k-(-n)^k)/(2*n+1), ", "); ); print(); ) \\ Andrew Howroyd, Mar 26 2017
  • Sage
    def T(n, k): return ((n+1)^k - (-n)^k)/(2*n+1)
[[T(k,n-k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Jan 27 2020

Formula

T(n, k) = T(n, k-1) + n*(n+1)*T(n, k-2) = A060959(A002378(n), k).
T(k, 2n) = (2n+1)*A047969(n, k+1).

A243201 Odd octagonal numbers indexed by triangular numbers.

Original entry on oeis.org

1, 21, 133, 481, 1281, 2821, 5461, 9633, 15841, 24661, 36741, 52801, 73633, 100101, 133141, 173761, 223041, 282133, 352261, 434721, 530881, 642181, 770133, 916321, 1082401, 1270101, 1481221, 1717633, 1981281, 2274181, 2598421, 2956161, 3349633, 3781141, 4253061, 4767841, 5328001
Offset: 0

Views

Author

Mathew Englander, Jun 01 2014

Keywords

Examples

			a(2) = 133 because the second triangular number is 3 and third odd octagonal number is 133.
a(3) = 481 because the third triangular number is 6 and the sixth odd octagonal number is 481.
a(4) = 1281 because the fourth triangular number is 10 and the tenth odd octagonal number is 1281.

Links

Crossrefs

Row 5 of A059259 (coefficients of 1 + 4*n + 7*n^2 + 6*n^3 + 3*n^4 + 0*n^5 which is a formula for the within sequence).
Column 5 of A081297.
Column 6 of A072024.
Diagonal T(n + 1, n) of A219069, n > 0.
Cf. A000217, A000567, A002061, A003215, A005408, A014641, A022522, A140676, A187156.

Programs

  • Magma
    [3*n^4+6*n^3+7*n^2+4*n+1: n in [0..40]]; // Bruno Berselli, Jun 03 2014
  • Mathematica
    Table[((3 n^2 + 3 n + 2)^2 - 1)/3, {n, 0, 39}] (* Alonso del Arte, Jun 01 2014 *)
  • Sage
    [3*n^4+6*n^3+7*n^2+4*n+1 for n in (0..40)] # Bruno Berselli, Jun 03 2014

Formula

a(n) = 3*n^4 + 6*n^3 + 7*n^2 + 4*n + 1.
a(n) = (n^2 + n + 1)*(3*n^2 + 3*n + 1).
a(n) = ((3*n^2 + 3*n + 2)^2 - 1)/3.
a(n) = A003215(n) * A002061(n + 1).
a(n) = A022522(n) / A005408(n).
a(n) = A000567(n^2 + n + 1).
a(n) = A014641((n^2 + n)/2).
a(n) = 1 + A140676(n^2 + n).
a(n) = 1 + A187156((n^2 + n + 4)/2) (empirical).
G.f.: (1 + 16*x + 38*x^2 + 16*x^3 + x^4)/(1 - x)^5. - Bruno Berselli, Jun 03 2014
E.g.f.: exp(x)*(1 + 20*x + 46*x^2 + 24*x^3 + 3*x^4). - Stefano Spezia, Apr 16 2022
Showing 1-7 of 7 results.

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