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.

Previous Showing 11-17 of 17 results.

A082736 LCM of n-th group of terms in A074147.

Original entry on oeis.org

1, 4, 105, 120, 109395, 55440, 1856277675, 42325920, 966710699955, 7210803600, 303646176781042095, 43790142876480, 2432266195067253069525, 6338767304469600, 12793596869123737224933375, 659267412349963697280
Offset: 1

Views

Author

Amarnath Murthy, Apr 14 2003

Keywords

Crossrefs

Cf. A074147, A074148, A074149, A082735.

Programs

Formula

a(1) = 1, a(2n) = LCM of next 2n even numbers. a(2n+1) = LCM of next 2n+1 odd numbers.

Extensions

More terms from R. J. Mathar, Jul 17 2007

A261243 Row lengths of the irregular triangles A258643 and A261242: maximal number of 0-islands (holes) of certain bisymmetric n X n matrices with 0 or 1 entries only.

Original entry on oeis.org

1, 1, 2, 3, 6, 9, 14, 19, 26, 33, 42, 51, 62, 73, 86, 99, 114, 129, 146, 163, 182, 201, 222, 243, 266, 289, 314, 339, 366, 393, 422, 451, 482, 513, 546, 579, 614, 649, 686, 723, 762, 801, 842, 883, 926
Offset: 1

Views

Author

Wolfdieter Lang, Aug 18 2015

Keywords

Comments

A shifted version of A061925. - R. J. Mathar, Aug 23 2015

Crossrefs

Cf. A000982, A258643, A261242.

Formula

a(n) = ceiling(((n-2)^2)/2) + 1, n >= 2, a(1) = 1.
a(n) = (1/2)*(n-2)^2+1 if n is even, a(n) = (ceiling((n-2)/2))^2 + (floor((n-2)/2))^2 + 1 if n is odd >= 3, and a(1) = 1.
O.g.f.: x*(1 - x + x^3 + x^4)/((1-x^2)*(1-x)^2) (from the o.g.f. of A000982).

A109857 Next 2*n - 1 odd numbers in decreasing order followed by next 2*n even numbers in decreasing order.

Original entry on oeis.org

1, 4, 2, 7, 5, 3, 12, 10, 8, 6, 17, 15, 13, 11, 9, 24, 22, 20, 18, 16, 14, 31, 29, 27, 25, 23, 21, 19, 40, 38, 36, 34, 32, 30, 28, 26, 49, 47, 45, 43, 41, 39, 37, 35, 33, 60, 58, 56, 54, 52, 50, 48, 46, 44, 42, 71, 69, 67, 65, 63, 61, 59, 57, 55, 53, 51, 84, 82, 80, 78, 76, 74
Offset: 1

Views

Author

Amarnath Murthy, Jul 08 2005

Keywords

Comments

This sequence is a permutation of the positive integers. - Werner Schulte, Jul 29 2023

Examples

			 1;
 4,  2;
 7,  5,  3;
12, 10,  8,  6;
17, 15, 13, 11,  9;
24, 22, 20, 18, 16, 14;
31, 29, 27, 25, 23, 21, 19;
40, 38, 36, 34, 32, 30, 28, 26;

Crossrefs

Cf. A074147 (row reversed), A074149 (row sums), A074148 (column 1), A001844, A061925 (main diagonal).

Programs

  • PARI
    T(n,k)=n*(n+1)/2+floor(n/2)-2*(k-1) \\ Werner Schulte, Jul 29 2023

Formula

From Werner Schulte, Jul 29 2023: (Start)
T(n, k) = n*(n+1)/2 + floor(n/2) - 2*(k-1) for 1 <= k <= n.
T(n, n) = (n^2-3*n+4)/2 + floor(n/2) for n > 0.
T(2*n-1, n) = n^2 + (n-1)^2 = A001844(n-1) for n > 0. (End)

Extensions

More terms from Joshua Zucker, May 05 2006

A173154 a(n) = n^3/6 + 3*n^2/4 + 7*n/3 + 7/8 + (-1)^n/8.

Original entry on oeis.org

1, 4, 10, 19, 33, 52, 78, 111, 153, 204, 266, 339, 425, 524, 638, 767, 913, 1076, 1258, 1459, 1681, 1924, 2190, 2479, 2793, 3132, 3498, 3891, 4313, 4764, 5246, 5759, 6305, 6884, 7498, 8147, 8833, 9556, 10318, 11119, 11961, 12844, 13770, 14739, 15753, 16812, 17918, 19071, 20273, 21524
Offset: 0

Views

Author

Paul Curtz, Feb 11 2010

Keywords

Comments

Generated by reading the table shown in A172002 down the diagonal starting at 1.
The inverse binomial transform yields 1, 3, 3, 0, 2, -4, 8, -16, 32, -64, 128, -256, 512, -1024, ... with a pattern of powers of 2.

Links

Programs

  • Magma
    [n^3/6 + 3*n^2/4 + 7*n/3 + 7/8 + (-1)^n/8: n in [0..50]]; // Vincenzo Librandi, Aug 05 2011
  • Mathematica
    Table[n^3/6+(3n^2)/4+(7n)/3+7/8+(-1)^n/8,{n,0,50}] (* or *) LinearRecurrence[{3,-2,-2,3,-1},{1,4,10,19,33},50] (* Harvey P. Dale, Jan 04 2012 *)

Formula

G.f.: ( 1 + x - x^3 + x^4 ) / ( (1+x)*(x-1)^4 ).
a(n) = 3*a(n-1) - 2*a(n-2) - 2*a(n-3) + 3*a(n-4) - a(n-5).
a(n+4) - a(n) = 4*A152948(n+5) = 4*A089071(n+5).
First differences: a(n+1) - a(n) = A061925(n+2).
Second differences: a(n+2) - 2*a(n+1) + a(n) = n + 5/2 + (-1)^n/2 = 3, 3, 5, 5, 7, 7, 9, 9, ... , duplicated A144396.

A219527 a(n) = (6*n^2 + 7*n - 9 + 2*n^3)/12 - (-1)^n*(n+1)/4.

Original entry on oeis.org

1, 3, 11, 19, 37, 55, 87, 119, 169, 219, 291, 363, 461, 559, 687, 815, 977, 1139, 1339, 1539, 1781, 2023, 2311, 2599, 2937, 3275, 3667, 4059, 4509, 4959, 5471, 5983, 6561, 7139, 7787, 8435, 9157, 9879, 10679, 11479, 12361, 13243
Offset: 1

Views

Author

Paul Curtz, Nov 21 2012

Keywords

Comments

First column of the Mendeleyev-Moseley-Seaborg table (with alkali metals) or 31st column of the Janet table. See A138726.
(a(n+10) - a(n))/10 = 29, 36, 45, 54, ... = A061925(n+7) + 3.
b(n) = a(n+1) - 2*a(n) = 1, 5, -3, -1, -19, -23, -55, -69, -119, -147, -219, -265, -363, -431, ... contains -a(2*n).
b(2*n-1) - b(2*n-2) = 4, 2, -4, -14, -28, -46, -68, ... = A147973(n+3).

Links

Crossrefs

Cf. A147973.

Programs

  • Mathematica
    a[n_] := (6*n^2 + 7*n - 9 + 2*n^3)/12 - (-1)^n*(n + 1)/4; Table[ a[n], {n, 1, 42}] (* Jean-François Alcover, Apr 05 2013 *)
LinearRecurrence[{2,1,-4,1,2,-1},{1,3,11,19,37,55},50] (* Harvey P. Dale, Apr 01 2018 *)

Formula

a(n) = A168380(n+1) - 1.
a(n+2) - a(n+1) = A093907(n) = A137583(n+1).
a(n+3) - a(n+1) = 10,16,26,36,... = A137928(n+3).
G.f. x*(1 + x + 4*x^2 - 2*x^3 + x^5 - x^4) / ( (1+x)^2*(x-1)^4 ). - R. J. Mathar, Mar 27 2013

A294070 a(n) = (1/4)*(n^2 - 2*n)^2 + (9/4)*(n^2 - 2*n) + 6.

Original entry on oeis.org

4, 6, 15, 40, 96, 204, 391, 690, 1140, 1786, 2679, 3876, 5440, 7440, 9951, 13054, 16836, 21390, 26815, 33216, 40704, 49396, 59415, 70890, 83956, 98754, 115431, 134140, 155040, 178296, 204079, 232566, 263940, 298390, 336111, 377304, 422176, 470940, 523815
Offset: 1

Views

Author

Jan Lakota Nono, Aug 14 2018

Keywords

Examples

			2*2, 2*3, 3*5, 5*8, 8*12, 12*17, 17*23, 23*30, 30*38, ...

Links

Crossrefs

Cf. A027689, A152948.

Programs

  • GAP
    List([1..40],n->(n^2-3*n+6)*(n^2-n+4)/4); # Muniru A Asiru, Aug 16 2018
  • Magma
    [(n^2-3*n+6)*(n^2-n+4)/4: n in [1..40]]; // Vincenzo Librandi, Aug 30 2018
  • Maple
    b:=n->(n^2-3*n+6)/2: seq(b(n)*b(n+1),n=1..40); # Muniru A Asiru, Aug 16 2018
  • Mathematica
    Times@@@Partition[Array[(#^2 -3# +6)/2 &, 40], 2, 1] (* Michael De Vlieger, Sep 24 2018 *)
LinearRecurrence[{5,-10,10,-5,1}, {4,6,15,40,96}, 40] (* G. C. Greubel, Feb 10 2019 *)
  • PARI
    Vec(x*(4 - 14*x + 25*x^2 - 15*x^3 + 6*x^4)/(1-x)^5 + O(x^40)) \\ Colin Barker, Nov 26 2018
  • Sage
    [(n^2-3*n+6)*(n^2-n+4)/4 for n in (1..40)] # G. C. Greubel, Feb 10 2019

Formula

a(n) = A152948(n) * A152948(n+1).
From Muniru A Asiru, Aug 16 2018: (Start)
a(n) = (n^2 - 3*n + 6)*(n^2 - n + 4)/4.
a(n) = A152948(n)*A027689(n-1)/2. (End)
a(n) = A266883(A061925(n-1)). - Bruno Berselli, Aug 30 2018
From Colin Barker, Nov 26 2018: (Start)
G.f.: x*(4 - 14*x + 25*x^2 - 15*x^3 + 6*x^4)/(1 - x)^5.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n > 5.
a(n) = (24 - 18*n + 13*n^2 - 4*n^3 + n^4)/4. (End)
E.g.f.: (1/4)*exp(x)*(16 + 8*x + 14*x^2 + 6*x^3 + x^4). - Stefano Spezia, Nov 30 2018

A376133 Triangle T read by rows: T(n, 1) = (2*n*n - 4*n + 7 + (-1)^n) / 4 and T(n, k) = T(n, k-1) + (-1)^k * 2 * (n+1-k) for k >= 2.

Original entry on oeis.org

1, 2, 4, 3, 7, 5, 6, 12, 8, 10, 9, 17, 11, 15, 13, 14, 24, 16, 22, 18, 20, 19, 31, 21, 29, 23, 27, 25, 26, 40, 28, 38, 30, 36, 32, 34, 33, 49, 35, 47, 37, 45, 39, 43, 41, 42, 60, 44, 58, 46, 56, 48, 54, 50, 52, 51, 71, 53, 69, 55, 67, 57, 65, 59, 63, 61, 62, 84, 64, 82, 66, 80, 68, 78, 70, 76, 72, 74
Offset: 1

Views

Author

Werner Schulte, Sep 11 2024

Keywords

Comments

Row n consists of the next n odd/even natural numbers if n is odd/even. So the sequence yields a permutation of the natural numbers.

Examples

			Row n=5: Next (1,3,5,7 see rows 1 and 3) five odd numbers are 9,11,13,15 and 17; with "9+8-6+4-2" we get 9,17,11,15,13 for row 5.
Row n=8: Next (2,4,..,24 see rows 2, 4 and 6) eight even numbers are 26,28,..,40; with "26+14-12+10-8+6-4+2" we get 26,40,28,38,30,36,32,34 for row 8.
Triangle T(n, k) for 1 <= k <= n starts:
n\ k :   1   2   3   4   5   6   7   8   9  10  11  12
======================================================
   1 :   1
   2 :   2   4
   3 :   3   7   5
   4 :   6  12   8  10
   5 :   9  17  11  15  13
   6 :  14  24  16  22  18  20
   7 :  19  31  21  29  23  27  25
   8 :  26  40  28  38  30  36  32  34
   9 :  33  49  35  47  37  45  39  43  41
  10 :  42  60  44  58  46  56  48  54  50  52
  11 :  51  71  53  69  55  67  57  65  59  63  61
  12 :  62  84  64  82  66  80  68  78  70  76  72  74
  etc.

Crossrefs

Cf. A061925 (column 1), A074148 (column 2), A074149 (row sums), A236283 (main diagonal).

Programs

  • Maple
    T := (n, k) -> ((-1)^k*(2 + 4*(n - k)) + 2*n^2 + (-1)^n + 5)/4:
seq(seq(T(n, k), k = 1..n), n = 1..12);  # Peter Luschny, Sep 13 2024
  • PARI
    T(n,k)=(2*n*n+(-1)^k*4*(n-k)+5+2*(-1)^k+(-1)^n)/4

Formula

T(n, k) = (2*n*n + (-1)^k * 4 * (n - k) + 5 + 2 * (-1)^k + (-1)^n) / 4.
T(n, 1) = (2*n*n - 4*n + 7 + (-1)^n) / 4 = A061925(n-1).
T(n, 2) = (2*n*n + 4*n - 1 + (-1)^n) / 4 = A074148(n) for n > 1.
T(n, k) = T(n, k-2) - (-1)^k * 2 for 3 <= k <= n.
G.f.: x*y*(1 + 2*x*y + 2*x^5*y^2 + x^6*y^3 - x^4*y*(3 + y + y^2) - x^2*(1 + y + 3*y^2) + 2*x^3*(1 + y^3))/((1 - x)^3*(1 + x)*(1 - x*y)^3*(1 + x*y)). - Stefano Spezia, Sep 12 2024
Previous Showing 11-17 of 17 results.

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