A074149 -id:A074149 - cp's OEIS frontend

A074148 a(n) = n + floor(n^2/2).

1, 4, 7, 12, 17, 24, 31, 40, 49, 60, 71, 84, 97, 112, 127, 144, 161, 180, 199, 220, 241, 264, 287, 312, 337, 364, 391, 420, 449, 480, 511, 544, 577, 612, 647, 684, 721, 760, 799, 840, 881, 924, 967, 1012, 1057, 1104, 1151, 1200, 1249, 1300, 1351, 1404, 1457

Offset: 1

Amarnath Murthy, Aug 28 2002

Last term in each group in A074147.
Index of the last occurrence of n in A100795.
Equals row sums of an infinite lower triangular matrix with alternate columns of (1, 3, 5, 7, ...) and (1, 1, 1, ...). - Gary W. Adamson, May 16 2010
a(n) = A214075(n+2,2). - Reinhard Zumkeller, Jul 03 2012
The heart pattern appears in (n+1) X (n+1) coins. Abnormal orientation heart is A065423. Normal heart is A093005 (A074148 - A065423). Void is A007590. See illustration in links. - Kival Ngaokrajang, Sep 11 2013
a(n+1) is the smallest size of an n-prolific permutation; a permutation of s letters is n-prolific if each (s - n)-subset of the letters in its one-line notation forms a unique pattern. - David Bevan, Nov 30 2016
For n > 2, a(n-1) is the smallest size of a nontrivial permuted packing of diamond tiles with diagonal length n; a permuted packing is a translational packing for which the set of translations is the plot of a permutation. - David Bevan, Nov 30 2016
Also the length of a longest path in the (n+1) X (n+1) bishop and black bishop graphs. - Eric W. Weisstein, Mar 27 2018
Row sums of A143182 triangle - Nikita Sadkov, Oct 10 2018

			Equals row sums of the generating triangle:
   1;
   3,  1;
   5,  1,  1;
   7,  1,  3,  1;
   9,  1,  5,  1,  1;
  11,  1,  7,  1,  3,  1;
  13,  1,  9,  1,  5,  1,  1;
  15,  1, 11,  1,  7,  1,  3,  1;
  ...
Example: a(5) = 17 = (9 + 1 + 5 + 1 + 1). - _Gary W. Adamson_, May 16 2010
The smallest 1-prolific permutations are 3142 and its symmetries; a(2) = 4. The smallest 2-prolific permutations are 3614725 and its symmetries; a(3) = 7. - _David Bevan_, Nov 30 2016

Vincenzo Librandi, Table of n, a(n) for n = 1..2000
D. Bevan, C. Homberger, and B. E. Tenner, Prolific permutations and permuted packings: downsets containing many large patterns, arXiv preprint arXiv:1608.06931 [math.CO], 2016.
Peter M. Chema, Illustration of initial terms, n > 1.
A. Edelman and M. La Croix, The Singular Values of the GUE (Less is More), arXiv preprint arXiv:1410.7065 [math.PR], 2014-2015. See Section 7.
Kival Ngaokrajang, Illustration of initial terms.
Eric Weisstein's World of Mathematics, Bishop Graph.
Eric Weisstein's World of Mathematics, Black Bishop Graph.
Eric Weisstein's World of Mathematics, Longest Path Problem.
Wikipedia, Cartan decomposition.
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A074147, A074149, A100795, A061925.
a(n) = A000982(n+1) - 1.
Antidiagonal sums of A237447 & A237448.

List([1..60],n->n+Int(n^2/2)); # Muniru A Asiru, Jan 04 2019

[(2*n^2+4*n+(-1)^n-1)/4: n in [1..60]]; // Vincenzo Librandi, Jun 16 2011

seq(floor(n^4/(2*n^2+1)),n=2..25); # Gary Detlefs, Feb 11 2010

f[x_, y_] := Floor[Abs[y/x - x/y]]; Table[Floor[f[1, n^2 + 2 n + 1]/2], {n, 60}] (* Robert G. Wilson v, Aug 11 2010 *)
Table[n + Floor[n^2/2], {n, 20}] (* Eric W. Weisstein, Mar 27 2018 *)
Table[((-1)^n + 2 n (n + 2) - 1)/4, {n, 10}] (* Eric W. Weisstein, Mar 27 2018 *)
LinearRecurrence[{2, 0, -2, 1}, {1, 4, 7, 12}, 20] (* Eric W. Weisstein, Mar 27 2018 *)
CoefficientList[Series[(-1 - 2 x + x^2)/((-1 + x)^3 (1 + x)), {x, 0, 20}], x] (* Eric W. Weisstein, Mar 27 2018 *)

a(n)=(2*n^2+4*n-1)\/4 \\ Charles R Greathouse IV, Apr 17 2012

def A074148(n): return n + n**2//2 # Chai Wah Wu, Jun 07 2022

a(n) = (2*n^2 + 4*n + (-1)^n - 1)/4. - Vladeta Jovovic, Apr 06 2003
a(n) = A109225(n,2) for n > 1. - Reinhard Zumkeller, Jun 23 2005
a(n) = +2*a(n-1) - 2*a(n-3) + 1*a(n-4). - Joerg Arndt, Apr 02 2011
a(n) = a(n-2) + 2*n, a(0) = 0, a(1) = 1. - Paul Barry, Jul 17 2004
From R. J. Mathar, Aug 30 2008: (Start)
G.f.: x*(1 + 2*x - x^2)/((1 - x)^3*(1 + x)).
a(n) + a(n+1) = A028387(n).
a(n+1) - a(n) = A109613(n+1). (End)
a(n) = floor(n^4/(2n^2 + 1)) with offset 2..a(2) = 1. - Gary Detlefs, Feb 11 2010
a(n) = n + floor(n^2/2). - Wesley Ivan Hurt, Jun 14 2013
From Franck Maminirina Ramaharo, Jan 04 2019: (Start)
a(n) = n*(n + 1)/2 + floor(n/2) = A000217(n) + A004526(n).
E.g.f.: (exp(-x) - (1 - 6*x - 2*x^2)*exp(x))/4. (End)
Sum_{n>=1} 1/a(n) = 1 - cot(Pi/sqrt(2))*Pi/(2*sqrt(2)). - Amiram Eldar, Sep 16 2022

More terms from Vladeta Jovovic, Apr 06 2003
Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, May 31 2007
Further edited by N. J. A. Sloane, Sep 06 2008 at the suggestion of R. J. Mathar
Description simplified by Eric W. Weisstein, Mar 27 2018

A061925 a(n) = ceiling(n^2/2) + 1.

Original entry on oeis.org

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, 969, 1014, 1059, 1106, 1153, 1202, 1251, 1302, 1353, 1406

Offset: 0

Henry Bottomley, May 17 2001

a(n+1) gives index of the first occurrence of n in A100795. - Amarnath Murthy, Dec 05 2004
First term in each group in A074148. - Amarnath Murthy, Aug 28 2002
From Christian Barrientos, Jan 01 2021: (Start)
For n >= 3, a(n) is the number of square polyominoes with at least 2n - 2 cells whose bounding box has size 2 X n.
For n = 3, there are 6 square polyominoes with a bounding box of size 2 X 3:
      _       _       _       _       _      
   |||_|   |||_|   |||_|   |||_|   |||_|   |||_
   |||_|   |||     || ||       ||     ||       |||
(End)
Except for a(2), a(n) agrees with the lower matching number of the (n+1) X (n+1) bishop graph up to at least n = 13. - Eric W. Weisstein, Dec 23 2024

Harry J. Smith, Table of n, a(n) for n = 0..1000
Kassie Archer, Ethan Borsh, Jensen Bridges, Christina Graves, and Millie Jeske, Cyclic permutations avoiding patterns in both one-line and cycle forms, arXiv:2312.05145 [math.CO], 2023. See p. 2.
Eric Weisstein's World of Mathematics, Bishop Graph.
Eric Weisstein's World of Mathematics, Lower Matching Number.
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A100795, A074147-A074149.

seq(floor((n^2+3)/2),n=0..25); # Gary Detlefs, Feb 13 2010

Table[Ceiling[n^2/2]+1,{n,0,60}] (* Vladimir Joseph Stephan Orlovsky, Apr 02 2011 *)
LinearRecurrence[{2,0,-2,1},{1,2,3,6},60] (* Harvey P. Dale, Jan 03 2024 *)

a(n) = { ceil(n^2/2) + 1 } \\ Harry J. Smith, Jul 29 2009

a(n) = a(n-1) + 2*floor((n-1)/2) + 1 = A061926(3, k) = 2*A002620(n+1) - (n-1) = A000982(n) + 1.
a(2*n) = a(2*n-1) + 2*n - 1 = 2*n^2 + 1 = A058331(n).
a(2*n+1) = a(2*n) + 2*n + 1 = 2*(n^2 + n + 1) = A051890(n+1).
a(n) = floor((n^2+3)/2). - Gary Detlefs, Feb 13 2010
From R. J. Mathar, Feb 19 2010: (Start)
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4).
G.f.: (1-x^2+2*x^3)/((1+x) * (1-x)^3). (End)
a(n) = (2*n^2 - (-1)^n + 5)/4. - Bruno Berselli, Sep 29 2011
a(n) = A007590(n+1) - n + 1. - Wesley Ivan Hurt, Jul 15 2013
a(n) + a(n+1) = A027688(n). a(n+1) - a(n) = A109613(n). - R. J. Mathar, Jul 20 2013
E.g.f.: ((2 + x + x^2)*cosh(x) + (3 + x + x^2)*sinh(x))/2. - Stefano Spezia, May 07 2021

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Jun 09 2007

A074147 (2n-1) odd numbers followed by 2n even numbers.

Original entry on oeis.org

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

Offset: 1

Amarnath Murthy, Aug 28 2002

			As a triangular array:
1,
2, 4,
3, 5, 7,
6, 8, 10, 12,
9, 11, 13, 15, 17,
14, 16, 18, 20, 22, 24,
...

Ivan Neretin, Table of n, a(n) for n = 1..5050

Cf. A074148, A074149 (row sums), A001614.

A074147 := proc(n,k)
    ceil((n-1)^2/2)+1+2*k ;
end proc:
seq(seq( A074147(n,k),k=0..n-1) ,n=1..12) ; # R. J. Mathar, Nov 02 2023

Flatten@Table[Ceiling[(n - 1)^2/2] + 2 k - 1, {n, 12}, {k, n}] (* Ivan Neretin, Dec 15 2016 *)

T(n,k) = A061925(n-1)+2k, 0<=kR. J. Mathar, Jul 17 2007, corrected Nov 02 2023

a(n) = A116941(n-1)+1. - Robert G. Wilson v, Mar 09 2017

A082735 Product of n-th group of terms in A074147.

Original entry on oeis.org

1, 8, 105, 5760, 328185, 42577920, 5568833025, 1300252262400, 304513870485825, 111644006842368000, 40992233865440682825, 21695920874860629196800, 11492457771692770753505625, 8291067715225260172247040000

Offset: 1

Amarnath Murthy, Apr 14 2003

nonn

Cf. A074147, A074148, A074149, A082736.

A074148 := proc(n) (2*n^2+4*n+(-1)^n-1)/4 ; end: A082735 := proc(n) doublefactorial(A074148(n))/doublefactorial(A074148(n-2)) ; end: seq(A082735(n),n=1..20) ; # R. J. Mathar, Jul 17 2007

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

a(n)=A006882[A074148(n)]/A006882[A074148(n-2)]. a(2n-1)=A062031(n). a(2n)=A062030(n). # R. J. Mathar, Jul 17 2007

Corrected and extended by R. J. Mathar, Jul 17 2007

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

Amarnath Murthy, Apr 14 2003

nonn

Cf. A074147, A074148, A074149, A082735.

A061925 := proc(n) ceil(n^2/2)+1 ; end: A082736 := proc(n) seq(A061925(n-1)+2*k,k=0..n-1) ; ilcm(%) ; end: seq(A082736(n),n=1..20) ; # R. J. Mathar, Jul 17 2007

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

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

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

Amarnath Murthy, Jul 08 2005

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

			 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;

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

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

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)

More terms from Joshua Zucker, May 05 2006

A376133 Triangle T read by rows: T(n, 1) = (2nn - 4n + 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

Werner Schulte, Sep 11 2024

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.

			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.

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

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

T(n,k)=(2*n*n+(-1)^k*4*(n-k)+5+2*(-1)^k+(-1)^n)/4

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

cp's OEIS Frontend

A074148 a(n) = n + floor(n^2/2).

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A061925 a(n) = ceiling(n^2/2) + 1.

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A074147 (2n-1) odd numbers followed by 2n even numbers.

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A082735 Product of n-th group of terms in A074147.

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

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

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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.

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

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