A237447 -id:A237447 - 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

A237265 Irregular table: n X n matrices (n=1,2,3,...), read by rows filled with numbers 1..n, with k moved to the front in the k-th row.

Original entry on oeis.org

1, 1, 2, 2, 1, 1, 2, 3, 2, 1, 3, 3, 1, 2, 1, 2, 3, 4, 2, 1, 3, 4, 3, 1, 2, 4, 4, 1, 2, 3, 1, 2, 3, 4, 5, 2, 1, 3, 4, 5, 3, 1, 2, 4, 5, 4, 1, 2, 3, 5, 5, 1, 2, 3, 4, 1, 2, 3, 4, 5, 6, 2, 1, 3, 4, 5, 6, 3, 1, 2, 4, 5, 6, 4, 1, 2, 3, 5, 6, 5, 1, 2, 3, 4, 6, 6, 1, 2, 3, 4, 5

Offset: 1

R. J. Cano, Feb 09 2014

Cases of enumeration in ascending order for the first m positive integers when one of them, j, is previously excluded (m=1,2,3,..., 1 <= j <= m).
Table of the k initial permutations, one per block, when all the k! permutations in lexicographic ascending order are split uniformly into k blocks. Such table read by rows for k=1,2,3,... .
These permutations might be considered the initial inputs for a parallel/distributed variant of the Narayana Pandita's algorithm. Such variant would deliver to each thread/core/host one or more of the mentioned inputs, then the remaining permutations can be obtained with (k-1)!-1 executions of the classic Narayana Pandita's algorithm for the next permutation in lexical order.
The terms of A237450 give the positions of rows of this table among the rows of A030298. The finite n X n square matrices converge towards the infinite square array A237447. Please see further comments there. - Antti Karttunen, Feb 10 2014
Alternative way to express this is that each row k=1..n of each n X n matrix contains the lexicographically earliest n-letter permutation beginning with number k, or equally, that each of the n X n square matrices contain in their n rows those n-letter permutations of the symmetric group S_n that correspond to the inverses of cycles (1), (1 2), (1 2 3), ..., (1 2 ... n). Please see the Example section. - Antti Karttunen, Feb 12 2014

			By excluding 2, the natural numbers between 1 and 4 are 1,3,4, then the second row of the corresponding matrix must be [2,1,3,4] and a(22)=4; that is, when reading by rows, a(22) must be placed at the 4th matrix since 22 is greater than the sum of elements there in the preceding matrices and it is smaller than the next of such sums: 14 = (1 + 2^2 + 3^2) <= (22) <= (1 + 2^2 + 3^2 + 4^2) = 30. Therefore 14 is subtracted from 22 leaving 8. This means that a(22) is the 8th element in the fourth matrix read by rows, so a(22) = A(4)[2,4] (see formula).
The irregular table starts consists of successively larger squares (beginning with a 1 X 1 square {1}), where each larger (n+1) X (n+1) square contains the previous n X n square in its upper left corner, with the first n rows followed by n+1, and the last row consisting of (n+1) followed by the first row of the previous n X n square (i.e., terms 1, 2, ..., n):
Permutation  In cycle notation.  Inverse in cycle notation
1;           ( )                 ( )    [Note: ( ) stands for identity]
1,2;         ( )                 ( )
2,1;         (1 2)               (1 2)
1,2,3;       ( )                 ( )
2,1,3;       (1 2)               (1 2)
3,1,2;       (1 3 2)             (1 2 3)
1,2,3,4;     ( )                 ( )
2,1,3,4;     (1 2)               (1 2)
3,1,2,4;     (1 3 2)             (1 2 3)
4,1,2,3;     (1 4 3 2)           (1 2 3 4)
1,2,3,4,5;   ( )                 ( )
2,1,3,4,5;   (1 2)               (1 2)
3,1,2,4,5;   (1 3 2)             (1 2 3)
4,1,2,3,5;   (1 4 3 2)           (1 2 3 4)
5,1,2,3,4;   (1 5 4 3 2)         (1 2 3 4 5)
...
The table starts with 1 since the definition must be read in the mathematical sense of its statement. If we have N elements and one of them must be excluded, there are no elements available to exclude when N=1.

Donald Knuth, The Art of Computer Programming, Volume 4: "Generating All Tuples and Permutations" Fascicle 2, first printing. Addison-Wesley, 2005. ISBN 0-201-85393-0.

R. J. Cano, Table of n, a(n) for n = 1..10000
R. J. Cano, Additional information.
R. J. Cano, Illustration of the recursive algorithm defining this sequence.
R. J. Cano, Recursive and iterative algorithms for A237265, OEIS wiki.

Cf. A237447, A030298, A000330, A074279, A237450, A237451, A237452.

a(n,k=0)=if(k,if(k>1,k-(k<=n),n),a(A238013(n),A121997(n))) \\ M. F. Hasler, Feb 16 2014

A237265_mth_matrix(m,zeroless=1)={my(c=Vec(numtoperm(m,0))-!zeroless*vector(m,i,1),M=matrix(m,m,i,j,0));for(j=1,m,M[j,]=concat([j-!zeroless],concat(c[1..j-1],c[j+1..m])));M}
a(n)=my(p,q,r,s); while(sA237265_mth_matrix(p,1)[q[1],q[2]] \\ R. J. Cano, May 08 2017

;; Implemented as a recurrence: (uses memoization macro definec from Antti Karttunen's IntSeq-library)
(definec (A237265 n) (cond ((zero? (A237452 (+ n (A074279 n)))) (+ (A237451 n) (if (zero? (A237451 n)) (A074279 n) 0))) ((zero? (A237451 (+ n 1))) (A074279 n)) (else (A237265 (+ 1 (A000330 (- (A074279 n) 2)) (* (- (A074279 n) 1) (A237452 n)) (A237451 n))))))
;; Version which uses the array A237447:
(define (A237265 n) (let ((col (A237451 n)) (row (A237452 n))) (A237447 (+ 1 (/ (+ (expt (+ col row) 2) col (* 3 row)) 2)))))
;; Antti Karttunen, Feb 08-10 2014

a(n) = A237447(1 + ((1/2) * ((col+row)^2 + col + 3*row)))[where col = A237451(n) and row = A237452(n)] = A237447bi(A237452(n),A237451(n)) [where A237447bi(row,col) is square array A237447 considered as a bivariate function]. - Antti Karttunen, Feb 10-12 2014

Name changed and more terms added by Antti Karttunen, Feb 10 2014

Further edits by M. F. Hasler, Mar 09 2014

A237448 Square array T(row >= 1, col >= 1): The first row, row=1, T(1,col) = col = A000027. When row > col, T(row,col) = row, otherwise (when 1 < row <= col), T(row,col) = row-1.

Original entry on oeis.org

1, 2, 2, 3, 1, 3, 4, 1, 3, 4, 5, 1, 2, 4, 5, 6, 1, 2, 4, 5, 6, 7, 1, 2, 3, 5, 6, 7, 8, 1, 2, 3, 5, 6, 7, 8, 9, 1, 2, 3, 4, 6, 7, 8, 9, 10, 1, 2, 3, 4, 6, 7, 8, 9, 10, 11, 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 13, 1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13

Offset: 1

Antti Karttunen, Feb 10 2014

This is transpose of A237447, please see comments there.

			The top left 9 X 9 corner of this infinite square array:
  1 2 3 4 5 6 7 8 9
  2 1 1 1 1 1 1 1 1
  3 3 2 2 2 2 2 2 2
  4 4 4 3 3 3 3 3 3
  5 5 5 5 4 4 4 4 4
  6 6 6 6 6 5 5 5 5
  7 7 7 7 7 7 6 6 6
  8 8 8 8 8 8 8 7 7
  9 9 9 9 9 9 9 9 8

Antti Karttunen, Table of first 144 antidiagonals of array, flattened

Transpose: A237447.
The leftmost column and the topmost row: A000027. Second row: A054977. Central diagonal: A028310 (note the different starting offsets).
Antidiagonal sums: A074148.
Cf. A237265, A000330, A002024, A002260, A004736, A000124, A010054.

(define (A237448 n) (cond ((= 1 (A010054 (- n 1))) (A002024 n)) ((< (A004736 n) (A002260 n)) (A002260 n)) (else (- (A002260 n) 1))))

As a one-dimensional sequence:
If A010054(n-1) = 1 [that is, if n is in A000124], then a(n) = A002024(n), otherwise, if A004736(n) < A002260(n), a(n) = A002260(n), and if A004736(n) >= A002260(n), a(n) = A002260(n)-1.
Equivalently, as a square array T:
When col < row, T(row,col) = row, for 1 < row <= col, T(row,col) = row-1, and for the first row T(1,col) = col = A000027(col).
Can be computed also as a transposed version of the infinite limit of the finite square arrays in sequence A237265: T(row,col) = A237265((A000330(max(row,col)-1)+1) + (max(row,col)*(col-1)) + (row-1)).

cp's OEIS Frontend

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

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A237265 Irregular table: n X n matrices (n=1,2,3,...), read by rows filled with numbers 1..n, with k moved to the front in the k-th row.

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A237448 Square array T(row >= 1, col >= 1): The first row, row=1, T(1,col) = col = A000027. When row > col, T(row,col) = row, otherwise (when 1 < row <= col), T(row,col) = row-1.

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