A127366 -id:A127366 - cp's OEIS frontend

A127367 Inverse permutation to A127366.

0, 2, 1, 5, 3, 7, 4, 10, 6, 12, 8, 14, 9, 17, 11, 19, 13, 21, 15, 23, 16, 26, 18, 28, 20, 30, 22, 32, 24, 34, 25, 37, 27, 39, 29, 41, 31, 43, 33, 45, 35, 47, 36, 50, 38, 52, 40, 54, 42, 56, 44, 58, 46, 60, 48, 62, 49, 65, 51, 67, 53, 69, 55, 71, 57, 73, 59, 75, 61, 77, 63, 79, 64

Offset: 0

Franklin T. Adams-Watters, Jan 11 2007

nonn

Let m be the largest number such that n >= m(m+1). If n is even, a(n) = n - m; otherwise a(n) = n + m + 1.

a(A005408(n)) > 0; a(A005843(n)) <= 0. [Reinhard Zumkeller, Oct 12 2011]

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Index entries for sequences that are permutations of the natural numbers

Cf. A127366, A002378, A000194.

Cf. A002378 (oblong numbers).

a127367 n | even n    = n - m + 1
          | otherwise = n + m
          where m = length $ takeWhile (<= n) a002378_list
-- Reinhard Zumkeller, Oct 12 2011

A195437 Triangle formed by: 1 even, 2 odd, 3 even, 4 odd... starting with 2.

Original entry on oeis.org

2, 5, 7, 10, 12, 14, 17, 19, 21, 23, 26, 28, 30, 32, 34, 37, 39, 41, 43, 45, 47, 50, 52, 54, 56, 58, 60, 62, 65, 67, 69, 71, 73, 75, 77, 79, 82, 84, 86, 88, 90, 92, 94, 96, 98, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 122, 124, 126, 128, 130, 132

Offset: 0

Reinhard Zumkeller, Oct 12 2011

A127366(T(n,k)) mod 2 = 1;
T(n,k) mod 2 + A000196(T(n,k)) mod 2 = 1;
complement of A133280.

Reinhard Zumkeller, Rows n=0..100 of triangle, flattened

Cf. A002522 (left edge), A008865 (right edge), A011379 (row sums), A002939 (central terms).

a195437 n k = a195437_tabl !! n !! k
a195437_tabl = tail $ g 1 1 [0..] where
   g m j xs = (filter ((== m) . (`mod` 2)) ys) : g (1 - m) (j + 2) xs'
     where (ys,xs') = splitAt j xs
b195437 = bFile' "A195437" (concat $ take 101 a195437_tabl) 0
-- Reinhard Zumkeller, Nov 23 2011 (fixed), Oct 12 2011

p[n_,k_]:=NestList[#+2&,n,k-1]; Module[{nn=6,ev,od},ev=p@@@Partition[Riffle[Table[ 4n^2-4n+2,{n,nn}],Range[ 1,2nn,2]],2];od=p@@@Partition[Riffle[Table[4n^2+1,{n,nn}],Range[ 2,2nn+1,2]],2];Sort[Flatten[Join[ev,od]]]] (* Harvey P. Dale, Aug 27 2023 *)

A133280 Triangle formed by: 1 even, 2 odd, 3 even, 4 odd, ... starting with zero.

Original entry on oeis.org

0, 1, 3, 4, 6, 8, 9, 11, 13, 15, 16, 18, 20, 22, 24, 25, 27, 29, 31, 33, 35, 36, 38, 40, 42, 44, 46, 48, 49, 51, 53, 55, 57, 59, 61, 63, 64, 66, 68, 70, 72, 74, 76, 78, 80, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 100, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120

Offset: 0

Omar E. Pol, Aug 27 2008

This sequence is related to the Connell sequence (A001614).
First member of every row is a square (A000290).
A127366(T(n,k)) mod 2 = 0 or equal parity of T(n,k) and A000196(T(n,k)); complement of A195437. - Reinhard Zumkeller, Oct 12 2011
Written as a square array the main diagonal gives A002943. - Omar E. Pol, Aug 13 2013
Last member of every row is one less than a square (A005563). - Harvey P. Dale, Oct 02 2013

			Written as a triangle the sequence begins:
    0;
    1,   3;
    4,   6,   8;
    9,  11,  13,  15;
   16,  18,  20,  22,  24;
   25,  27,  29,  31,  33,  35;
   36,  38,  40,  42,  44,  46,  48;
   49,  51,  53,  55,  57,  59,  61,  63;
   64,  66,  68,  70,  72,  74,  76,  78,  80;
   81,  83,  85,  87,  89,  91,  93,  95,  97,  99;
  100, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120;

Reinhard Zumkeller, Rows n=0..100 of triangle, flattened

Column 1 is A000290. Right border gives A005563.
Cf. A001614.
Cf. A045991 (row sums). - R. J. Mathar, Jul 20 2009

a133280 n k = a133280_tabl !! n !! k
a133280_tabl = f 0 1 [0..] where
   f m j xs = (filter ((== m) . (`mod` 2)) ys) : f (1 - m) (j + 2) xs'
     where (ys,xs') = splitAt j xs
b133280 = bFile' "A133280" (concat $ take 101 a133280_tabl) 0
-- Reinhard Zumkeller, Oct 12 2011

Flatten[Table[Range[(n-1)^2,n^2-1,2],{n,20}]] (* Harvey P. Dale, Oct 02 2013 *)

T(n,k) = n^2 + 2*k;
for(n=0,10,for(k=0,n,print1(T(n,k),", "))); \\ Joerg Arndt, Aug 13 2013

from math import isqrt
def A133280(n): return (m:=(n<<1)+1)-((isqrt(m+1<<2)+1)>>1) # Chai Wah Wu, Aug 01 2022

a(n) = A005408(n) - A002024(n+1). - Ivan N. Ianakiev, Aug 13 2013

T(n,k) = n^2 + 2*k. - Joerg Arndt, Aug 13 2013

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A127367 Inverse permutation to A127366.

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A195437 Triangle formed by: 1 even, 2 odd, 3 even, 4 odd... starting with 2.

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A133280 Triangle formed by: 1 even, 2 odd, 3 even, 4 odd, ... starting with zero.

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