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

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

Author

Omar E. Pol, Aug 27 2008

Keywords

• easy
• nonn
• tabl

Comments

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

Examples

			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;
		

Links

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

Crossrefs

Column 1 is A000290. Right border gives A005563.

Cf. A001614.

Cf. A045991 (row sums). - R. J. Mathar, Jul 20 2009

Programs

Haskell

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

Mathematica

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

PARI

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

Python

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

Formula

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