A014255 -id:A014255 - cp's OEIS frontend

A114752 a(2n)=2n, a(2n+1)=4n+1.

1, 2, 5, 4, 9, 6, 13, 8, 17, 10, 21, 12, 25, 14, 29, 16, 33, 18, 37, 20, 41, 22, 45, 24, 49, 26, 53, 28, 57, 30, 61, 32, 65, 34, 69, 36, 73, 38, 77, 40, 81, 42, 85, 44, 89, 46, 93, 48, 97, 50, 101, 52, 105, 54, 109, 56, 113, 58, 117, 60, 121, 62, 125, 64, 129, 66, 133, 68, 137

Offset: 1

Amarnath Murthy, Nov 15 2005

Original definition (typos corrected):  The following triangle contains n consecutive numbers beginning from n in ascending order if n is odd else in descending order. 1 3 2 3 4 5 7 6 5 4 5 6 7 8 9 11 10 9 8 7 6 ... Sequence contains the leading diagonal.
Equals A133566 * [1,2,3,...]. - Gary W. Adamson, Sep 16 2007
The sequence satisfies a divisibility property described by E. Angelini on the SeqFan list, cf. link. - M. F. Hasler, Mar 22 2013
First difference of A014255 (shown easily from the Nurikabe property of that sequence, or by manipulating the linear recurrence representations). - Allan C. Wechsler, Oct 20 2022

			Contribution by _M. F. Hasler_, Mar 22 2013: (Start)
The triangle described in the original definition starts
   1
   3  2
   3  4  5
   7  6  5  4
   5  6  7  8  9
  11 10  9  8  7  6. (End)

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Eric Angelini, k-chunks sum and division by k, post to the SeqFan list, Mar 22 2013.
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Cf. A114751, A114753, A133566, A014255

With[{nn=40},Riffle[Range[1,4nn,4],Range[2,2nn,2]]] (* Harvey P. Dale, Apr 14 2017 *)

a(n)=n+if(n%2,n\2*2) \\ Charles R Greathouse IV, Oct 16 2015

a(2n) = 2n, a(2n+1) = 4n+1. - Joshua Zucker, May 05 2006
G.f.: x*(1+2*x+3*x^2)/(1-x^2)^2. - Philippe Deléham, Mar 02 2012
a(n) = (3n-(n-1)*(-1)^n-1)/2. - Bruno Berselli, Mar 02 2012

More terms from Joshua Zucker, May 05 2006

Simpler definition from M. F. Hasler, Mar 22 2013

A032438 a(n) = n^2 - floor((n+1)/2)^2.

Original entry on oeis.org

0, 0, 3, 5, 12, 16, 27, 33, 48, 56, 75, 85, 108, 120, 147, 161, 192, 208, 243, 261, 300, 320, 363, 385, 432, 456, 507, 533, 588, 616, 675, 705, 768, 800, 867, 901, 972, 1008, 1083, 1121, 1200, 1240, 1323, 1365, 1452, 1496, 1587, 1633, 1728, 1776, 1875, 1925

Offset: 0

N. J. A. Sloane

The answer to a question from Mike and Laurie Crain (2crains(AT)concentric.net): how many even numbers are there in an n X n multiplication table starting at 1 X 1?
a(n+1) is the number of pairs (x,y) with x and y in {0,...,n}, x and y of the same parity, and x+y >= n. - Clark Kimberling, Jul 02 2012
From J. M. Bergot, Aug 08 2013: (Start)
Define a triangle to have T(1,1)=0 and T(n,c) = n^2 - c^2. The difference of the sum of the terms in antidiagonal(n+1) and those in antidiagonal(n)=a(n).
Column 0 is vertical and T(n,n)=0. The first few rows are 0; 3,0; 8,5,0; 15,12,7,0; 24,21,16,9,0; 35,32,27,20,11,0; the first few antidiagonals are 0; 3; 8,0; 15,5; 24,12,0; 35,21,7; 48,32,16,0; the first few sum of terms in the antidiagonals are 0, 3, 8, 20, 36, 63, 96, 144, 200, 275, 360, 468, 588, 735, 896, 1088, 1296, 1539. (End)
Sum of the largest parts in the partitions of 2n into two distinct odd parts. For example, a(5) = 16; the partitions of 2(5) = 10 into two distinct odd parts are (9,1) and (7,3). The sum of the largest parts is then 9+7 = 16. - Wesley Ivan Hurt, Nov 27 2017

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

First differences are in A059029, partial sums in A143785.

Cf. A008794, A014255.

[n^2-Floor( (n+1)/2 )^2 : n in [0..60]]; // Vincenzo Librandi, Sep 27 2011

A032438:=n->n^2-floor((n+1)/2)^2; seq(A032438(n), n=0..100) # Wesley Ivan Hurt, Nov 25 2013

Table[n^2-Floor[((n+1)/2)]^2,{n,0,50}] (* or *) LinearRecurrence[ {1,2,-2,-1,1},{0,0,3,5,12},51]

a(n)=n^2 - ((n+1)\2)^2 \\ Charles R Greathouse IV, Feb 19 2017

a(n) = n^2 - A008794(n+1).
G.f.: x^2*(x^2 + 2*x + 3)/(1-x^2)^2/(1-x). - Ralf Stephan, Jun 10 2003
a(n) = (1/8)*(2*n*(3*n-1)+(2*n+1)*(-1)^n-1). a(-n-1) = A014255(n). - Bruno Berselli, Sep 27 2011
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5), n > 4. - Harvey P. Dale, Nov 24 2011
E.g.f.: (x*(1 + 3*x)*cosh(x) + (3*x^2 + 3*x - 1)*sinh(x))/4. - Stefano Spezia, Aug 01 2022

A130266 A051340 * A128174.

Original entry on oeis.org

1, 1, 2, 4, 1, 3, 2, 5, 1, 4, 7, 2, 6, 1, 5, 3, 8, 2, 7, 1, 6, 10, 3, 9, 2, 8, 1, 7, 4, 11, 3, 10, 2, 9, 1, 8, 13, 4, 12, 3, 11, 2, 10, 1, 9, 5, 14, 4, 13, 3, 12, 2, 11, 1, 10, 16, 5, 15, 4, 14, 3, 13, 2, 12, 1, 11, 6, 17, 5, 16, 4, 15, 3, 14, 2, 13

Offset: 0

Gary W. Adamson, May 18 2007

Row sums = A014255: (1, 3, 8, 12, 21, 27, 40, ...).

Left border = A123684: (1, 1, 4, 2, 7, 3, 10, 4, ...).

			First few rows of the triangle:
   1;
   1, 2;
   4, 1, 3;
   2, 5, 1, 4;
   7, 2, 6, 1, 5;
   3, 8, 2, 7, 1, 6;
  10, 3, 9, 2, 8, 1, 7;
  ...

Cf. A051340, A128174, A014255, A123684.

A128174 := proc(n,k)
    if k > n or k < 1 then
        0;
    else
        modp(k+n+1,2) ;
    end if;
end proc:
A051340 := proc(n,k)
    if k = n then
        n ;
    elif k <= n then
        1;
    else
        0;
    end if;
end proc:
A130266 := proc(n,k)
    add( A051340(n,j)*A128174(j,k),j=k..n) ;
end proc:
seq(seq(A130266(n,k),k=1..n),n=1..15) ; # R. J. Mathar, Aug 06 2016

A051340 * A128174 as infinite lower triangular matrices.

A264798 Irregular triangle read by rows: odd-valued terms of A094728(n+1).

Original entry on oeis.org

1, 3, 9, 5, 15, 7, 25, 21, 9, 35, 27, 11, 49, 45, 33, 13, 63, 55, 39, 15, 81, 77, 65, 45, 17, 99, 91, 75, 51, 19, 121, 117, 105, 85, 57, 21, 143, 135, 119, 95, 63, 23, 169, 165, 153, 133, 105, 69, 25, 195, 187, 171, 147, 115, 75, 27, 225, 221, 209, 189, 161, 125, 81, 29, 255, 247

Offset: 0

Paul Curtz, Nov 25 2015

A094728(n+1) comes from A120070(n+2). a(n) approximates frequencies of the spectral lines of the hydrogen atom.
Row sums: 1, 3, 14, 22, ... = A024598(n+1).
First column: A085046(n+1).
Row sums of A261046(n) = 1, 3, 8, 12, ... = A014255(n). See the formula.

			Irregular triangle begins:
1,
3,
9,  5,
15, 7,
25, 21,  9,
35, 27, 11,
49, 45, 33, 13,
63, 55, 39, 15,
...

Cf. A014255, A024598, A085046, A094728, A120070, A167268, A249947, A261046.

Table[n^2 - k^2, {n, 14}, {k, 0, n - 1}] /. n_ /; EvenQ@ n -> Nothing // Flatten (* Michael De Vlieger, Nov 25 2015 *)

for(n=1,20,for(k=0,n-1,s=n^2-k^2;if(s%2,print1(s,", ")))) \\ Derek Orr, Dec 24 2015

a(n) = A261046(n)*A167268(n+1)/2, where A167268 is Janet's sequence.

cp's OEIS Frontend

A114752 a(2n)=2n, a(2n+1)=4n+1.

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A032438 a(n) = n^2 - floor((n+1)/2)^2.

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A130266 A051340 * A128174.

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A264798 Irregular triangle read by rows: odd-valued terms of A094728(n+1).

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