A065423 -id:A065423 - cp's OEIS frontend

A131355 Partial sums of A065423 plus one.

1, 1, 1, 3, 4, 8, 10, 16, 19, 27, 31, 41, 46, 58, 64, 78, 85, 101, 109, 127, 136, 156, 166, 188, 199, 223, 235, 261, 274, 302, 316, 346, 361, 393, 409, 443, 460, 496, 514, 552, 571, 611, 631, 673, 694, 738, 760, 806, 829, 877, 901, 951, 976, 1028, 1054, 1108

Offset: 0

Paul Curtz, Sep 30 2007

Number of 132-avoiding even Grassmannian permutations of size n. - Juan B. Gil, Mar 10 2023

Juan B. Gil and Jessica A. Tomasko, Pattern-avoiding even and odd Grassmannian permutations, arXiv:2207.12617 [math.CO], 2022.
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A065423.

[(6*n^2-10*n+17-(1+2*n)*(-1)^n)/16: n in [0..70]]; // Vincenzo Librandi, Jul 29 2015

A065423 := proc(n) if n mod 2 <> 0 then n-1 ; else n/2-1 ; fi ; end: A131355 := proc(n) 1+add(A065423(i), i=1..n) ; end: seq(A131355(n),n=0..80) ; # R. J. Mathar, Oct 04 2007

Table[(6 n^2 - 10 n + 17 - (1 + 2 n) (-1)^n)/16, {n, 0, 100}] (* Wesley Ivan Hurt, Jul 28 2015 *)
LinearRecurrence[{1, 2, -2, -1, 1}, {1, 1, 1, 3, 4}, 70] (* Vincenzo Librandi, Jul 29 2015 *)
Join[{1},Accumulate[LinearRecurrence[{0,2,0,-1},{0,0,2,1},100]]+1] (* Harvey P. Dale, May 28 2025 *)

From R. J. Mathar, Jul 17 2009: (Start)
G.f.: (1 - 2*x^2 + 2*x^3 + 2*x^4)/((1+x)^2*(1-x)^3).
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5), n > 5. (End)
a(n) = (6*n^2 - 10*n + 17 - (1+2n)*(-1)^n)/16. - Wesley Ivan Hurt, Jul 28 2015
a(n) = 1 + binomial(n,2) - binomial(floor(n/2)+1,2). - Juan B. Gil, Mar 10 2023

More terms from R. J. Mathar, Oct 04 2007

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

Original entry on oeis.org

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

A029578 The natural numbers interleaved with the even numbers.

Original entry on oeis.org

0, 0, 1, 2, 2, 4, 3, 6, 4, 8, 5, 10, 6, 12, 7, 14, 8, 16, 9, 18, 10, 20, 11, 22, 12, 24, 13, 26, 14, 28, 15, 30, 16, 32, 17, 34, 18, 36, 19, 38, 20, 40, 21, 42, 22, 44, 23, 46, 24, 48, 25, 50, 26, 52, 27, 54, 28, 56, 29, 58, 30, 60, 31, 62, 32, 64, 33, 66, 34, 68, 35, 70, 36, 72

Offset: 0

N. J. A. Sloane

a(n) = number of ordered, length two, compositions of n with at least one odd summand - Len Smiley, Nov 25 2001
Also number of 0's in n-th row of triangle in A071037. - Hans Havermann, May 26 2002
For n > 2: a(n) = number of odd terms in row n-2 of triangle A265705. - Reinhard Zumkeller, Dec 15 2015

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Index entries for two-way infinite sequences
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Cf. A065423 (at least one even summand).

Cf. A001477, A005843, A009531, A071037, A211538 (partial sums), A265705.

import Data.List (transpose)
a029578 n =  (n - n `mod` 2) `div` (2 - n `mod` 2)
a029578_list = concat $ transpose [a001477_list, a005843_list]
-- Reinhard Zumkeller, Nov 27 2012

A029578:= func< n | (n + (n-2)*(n mod 2))/2 >;
[A029578(n): n in [0..80]]; // G. C. Greubel, Jan 22 2025

With[{nn=40},Riffle[Range[0,nn],Range[0,2nn,2]]] (* or *) LinearRecurrence[ {0,2,0,-1},{0,0,1,2},80] (* Harvey P. Dale, Aug 23 2015 *)

PARI
```
a(n)=if(n%2,n-1,n/2)
```

def A029578(n): return (n + (n-2)*(n%2))//2
print([A029578(n) for n in range(81)]) # G. C. Greubel, Jan 22 2025

a(n) = (3*n - 2 - (-1)^n*(n - 2))/4.
a(n+4) = 2*a(n+2) - a(n).
G.f.: x^2*(1 + 2*x)/(1-x^2)^2.
a(n) = floor((n+1)/2) + (n is odd)*floor((n+1)/2).
a(n) = (n - n mod 2)/(2 - n mod 2). - Reinhard Zumkeller, Jul 30 2002
a(n) = floor(n/2)*binomial(2, mod(n, 2)) - Paul Barry, May 25 2003
a(2*n) = n.
a(2*n-1) = 2*n-2.
a(-n) = -A065423(n+2).
a(n) = Sum_{k=0..floor((n-2)/2)} (C(n-k-2, k) mod 2)((1+(-1)^k)/2)*2^A000120(n-2k-2). - Paul Barry, Jan 06 2005
a(n) = Sum_{k=0..n-2} gcd(n-k-1, k+1). - Paul Barry, May 03 2005
For n>6: a(n) = floor(a(n-1)*a(n-2)/a(n-3)). - Reinhard Zumkeller, Mar 06 2011
E.g.f.: (1/4)*((x+2)*exp(-x) + (3*x-2)*exp(x)). - G. C. Greubel, Jan 22 2025

Explicated definition by Reinhard Zumkeller, Nov 27 2012

Title simplified by Sean A. Irvine, Feb 29 2020

A153733 Remove all trailing 1's in the binary representation of n.

Original entry on oeis.org

0, 0, 2, 0, 4, 2, 6, 0, 8, 4, 10, 2, 12, 6, 14, 0, 16, 8, 18, 4, 20, 10, 22, 2, 24, 12, 26, 6, 28, 14, 30, 0, 32, 16, 34, 8, 36, 18, 38, 4, 40, 20, 42, 10, 44, 22, 46, 2, 48, 24, 50, 12, 52, 26, 54, 6, 56, 28, 58, 14, 60, 30, 62, 0, 64, 32, 66, 16, 68, 34, 70, 8, 72, 36, 74, 18, 76, 38

Offset: 0

Reinhard Zumkeller, Dec 31 2008

a(n) is also the map n -> A065423(n+1) applied A007814(n+1) times. - Federico Provvedi, Dec 14 2021

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for sequences related to binary expansion of n

Cf. A000265, A007814, A007088.
Cf. A163575.
Cf. A065423, A331739.

a153733 n = if b == 0 then n else a153733 n'  where (n', b) = divMod n 2
-- Reinhard Zumkeller, Jul 22 2014

f:= n -> (n+1)/2^padic:-ordp(n+1,2)-1:
map(f, [$0..100]); # Robert Israel, Mar 18 2018

Table[If[EvenQ[n],n,FromDigits[Flatten[Most[Split[IntegerDigits[n,2]]]],2]],{n,0,100}] (* Harvey P. Dale, Feb 15 2014 *)
a[n_] := BitShiftRight[n + 1, IntegerExponent[n+1, 2]] - 1; a[Range[0,100]] (* Federico Provvedi, Dec 21 2021 *)

A153733(n)=(n+=1)>>valuation(n,2)-1 \\ most efficient variant: use this. - M. F. Hasler, Mar 16 2018

{a(n)=while(bittest(n,0),n>>=1);n} \\ for illustration: as long as there's a trailing bit 1, remove it. - M. F. Hasler, Mar 16 2018

a(n)=for(i=0,n,bittest(n,i)||return(n>>i)) \\ scan the trailing 1's, then remove all of them at once. - M. F. Hasler, Mar 16 2018

def a(n):
    while n&1: n >>= 1
    return n
print([a(n) for n in range(100)]) # Michael S. Branicky, Dec 18 2021

def A153733(n): return n>>(~(n+1)&n).bit_length() # Chai Wah Wu, Jul 08 2022

a(n) = n if n is even, a((n-1)/2) if odd.
a(n)/2 = A025480(n).
a(n) = A000265(n+1) - 1. - M. F. Hasler, Mar 16 2018
a(n) = n - A331739(n+1). - Federico Provvedi, Dec 21 2021

A211539 Number of ordered triples (w,x,y) with all terms in {1,...,n} and 2w = 2n - 2x + y.

Original entry on oeis.org

0, 0, 2, 3, 7, 9, 15, 18, 26, 30, 40, 45, 57, 63, 77, 84, 100, 108, 126, 135, 155, 165, 187, 198, 222, 234, 260, 273, 301, 315, 345, 360, 392, 408, 442, 459, 495, 513, 551, 570, 610, 630, 672, 693, 737, 759, 805, 828, 876, 900, 950, 975, 1027, 1053

Offset: 0

Clark Kimberling, Apr 15 2012

For a guide to related sequences, see A211422.
a(n) = sum of natural numbers in interval (floor((n+1)/2),n]. - Jaroslav Krizek, Mar 05 2014
For n > 0, 2*a(n-1) is the sum of the largest parts of the partitions of 2n into two distinct even parts. - Wesley Ivan Hurt, Dec 19 2017
From Paul Curtz, Oct 23 2018: (Start)
Consider the 51 first nonnegative numbers in the following boustrophedon distribution:
     35--36--37--38--39--40--41--42--43--44--45
         34--33--32--31--30--29--28--27--26--46
             12--13--14--15--16--17--18--25--47
                 11--10---9---8---7--19--24--48
                      1---2---3---6--20--23--49
                          0---4---5--21--22--50
a(n+1) is the union of the main vertical (0,2,  9,15, 30,40, ... ) and of the shifted main antidiagonal (3,7, 18,26,  45,57, ... ). (End)
Sum of the shortest side lengths of all integer-sided triangles with perimeter 3(n+1) whose sides lengths are in arithmetic progression (For example, when n=4 there are two triangles with perimeter 3(4+1) = 15 whose side lengths are in arithmetic progression: [3,5,7] and [4,5,6]; thus a(4) = 3+4 = 7). - Wesley Ivan Hurt, Nov 01 2020

			G.f. = 2*x^2 + 3*x^3 + 7*x^4 + 9*x^5 + 15*x^6 + 18*x^7 + ... - _Michael Somos_, Nov 14 2018

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A211422.

Cf. A000217, A001318, A005449, A045943, A045944, A065423, A211538.

a:=[0];; for n in [2..55] do if n mod 2 = 0 then Add(a,a[n-1]+n); else Add(a,a[n-1]+(n-1)/2); fi; od; Concatenation([0],a); # Muniru A Asiru, Oct 26 2018

I:=[0,0,2,3,7]; [n le 5 select I[n] else Self(n-1)+2*Self(n-2)-2*Self(n-3)-Self(n-4)+Self(n-5): n in [1..60]]; // Vincenzo Librandi, Mar 12 2014

a:=n->add(k,k=floor((n+1)/2)+1..n): seq(a(n),n=0..55); # Muniru A Asiru, Oct 26 2018

t[n_] := t[n] = Flatten[Table[2 w + 2 x - y - 2 n, {w, 1, n}, {x, 1, n}, {y, 1, n}]]
c[n_] := Count[t[n], 0]
t = Table[c[n], {n, 0, 70}]  (* A211539 *)
FindLinearRecurrence[t]
CoefficientList[Series[(x^3 + 2 x^2)/((1 + x)^2 (1 - x)^3), {x, 0, 60}], x] (* Vincenzo Librandi, Mar 12 2014 *)

a(n)=(1/16)*(6*n^2+2*n-3+(2*n+3)*(-1)^n) \\ Ralf Stephan, Mar 10 2014

a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5).
From Jaroslav Krizek, Mar 05 2014: (Start)
a(n) = T(n+1) - T(floor((n+1)/2)) - (n+1), where T(k) = A000217(k).
a(n) = Sum_{k=floor((n+1)/2)+1..n} k.
a(n) = a(n-1) + n for even n; a(n) = a(n-1) + (n-1)/2 for odd n. (End)
From Ralf Stephan, Mar 10 2014: (Start)
a(n) = (1/16) * (6n^2 + 2n - 3 + (2n+3)*(-1)^n ).
G.f.: (x^3+2x^2)/((1+x)^2*(1-x)^3). (End)
From Paul Curtz, Oct 22 2018: (Start)
a(2n) = A005449(n), a(2n+1) = A045943(n).
a(2n) + a(2n+1) = A045944(n).
a(3n) = 3*(0, 1, 5, 10, 19, 28, 42, ...).
a(n+1) = a(n) + A065423(n+2).
a(-n) = A211538(n+2). (End)
E.g.f.: (3*x*(1 + x)*cosh(x) + (-3 + 5*x + 3*x^2)*sinh(x))/8. - Stefano Spezia, Nov 02 2020
a(n) = A001318(n+1) - (n+1). - Davide Rotondo, Apr 07 2024

A097362 a(n) = (n+1)/2 if n is odd, n+2 otherwise.

Original entry on oeis.org

1, 4, 2, 6, 3, 8, 4, 10, 5, 12, 6, 14, 7, 16, 8, 18, 9, 20, 10, 22, 11, 24, 12, 26, 13, 28, 14, 30, 15, 32, 16, 34, 17, 36, 18, 38, 19, 40, 20, 42, 21, 44, 22, 46, 23, 48, 24, 50, 25, 52, 26, 54, 27, 56, 28, 58, 29, 60, 30, 62, 31, 64, 32, 66, 33, 68, 34, 70, 35, 72, 36, 74, 37

Offset: 1

Pierre CAMI, Sep 18 2004

May be regarded as table T(n,j) for j=1 to 2, where T(n,1)=n, T(n,2)=2*n+2 T(n,1)=number of carbon atoms in alkane hydrocarbons C_n H_{2n+2}, T(n,2)=number of hydrogen atoms in alkane hydrocarbons C_nH_{2n+2}.

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

Cf. A065423.

a[n_] := If[OddQ[n], (n + 1)/2, n + 2]; Table[a[n], {n, 100}] (* Stefan Steinerberger, May 13 2006 *)
LinearRecurrence[{0,2,0,-1},{1,4,2,6},100] (* Harvey P. Dale, Sep 14 2016 *)

a(n)=if(n%2,(n+1)/2,n+2) \\ Charles R Greathouse IV, Sep 02 2015

G.f.: (2 + x)/(1 - x^2)^2. - Arkadiusz Wesolowski, Dec 28 2011

More terms from Stefan Steinerberger, May 13 2006

A154958 Antidiagonal sums of number triangle A154957 regarded as a lower triangular infinite matrix.

Original entry on oeis.org

1, 1, 2, 1, 2, 1, 3, 2, 4, 2, 4, 2, 5, 3, 6, 3, 6, 3, 7, 4, 8, 4, 8, 4, 9, 5, 10, 5, 10, 5, 11, 6, 12, 6, 12, 6, 13, 7, 14, 7, 14, 7, 15, 8, 16, 8, 16, 8, 17, 9, 18, 9, 18, 9, 19, 10, 20, 10, 20, 10, 21, 11, 22, 11, 22, 11, 23, 12, 24, 12, 24, 12, 25, 13, 26, 13, 26, 13, 27, 14, 28, 14, 28

Offset: 0

Paul Barry, Jan 18 2009

			G.f. = 1 + x + 2*x^2 + x^3 + 2*x^4 + x^5 + 3*x^6 + 2*x^7 + 4*x^8 + 2*x^9 + ...

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

Cf. A002264, A004523, A026741, A029578, A065423.

CoefficientList[Series[1/((x - 1)^2 (x + 1)^2 (x^2 - x + 1)), {x, 0, 100}], x] (* Vincenzo Librandi, Mar 22 2014 *)
LinearRecurrence[{1,1,-2,1,1,-1},{1,1,2,1,2,1},90] (* Harvey P. Dale, Aug 26 2016 *)

{a(n) = if( n<-5, -a(-6-n), if( n<0, 0, polcoeff( 1 / (1 - x - x^2 + 2*x^3 - x^4 - x^5 + x^6) + x * O(x^n), n)))}; /* Michael Somos, Mar 21 2014 */

a(2n) = A004523(n+2); a(2n+1) = floor((n+3)/3) = A002264(n+3).
G.f.: 1/((x-1)^2*(x+1)^2*(x^2-x+1)). - Philippe Deléham, Mar 21 2014
a(n) = a(n-1) + a(n-2) - 2*a(n-3) + a(n-4) + a(n-5) - a(n-6), a(0) = 1, a(1) = 1, a(2) = 2, a(3) = 1, a(4) = 2, a(5) = 1. - Philippe Deléham, Mar 21 2014
Euler transform of length 6 sequence [ 1, 1, -1, 0, 0, 1]. - Michael Somos, Mar 21 2014
a(-6-n) = -a(n). - Michael Somos, Mar 21 2014
a(3*n) = A026741(n+1). a(3*n + 1) = A029578(n+2). a(3*n + 2) = A065423(n+3). - Michael Somos, Mar 21 2014

A066104 a(2n) = 2n, a(2n+1) = 4(n+1).

Original entry on oeis.org

0, 4, 2, 8, 4, 12, 6, 16, 8, 20, 10, 24, 12, 28, 14, 32, 16, 36, 18, 40, 20, 44, 22, 48, 24, 52, 26, 56, 28, 60, 30, 64, 32, 68, 34, 72, 36, 76, 38, 80, 40, 84, 42, 88, 44, 92, 46, 96, 48, 100, 50, 104, 52, 108, 54, 112, 56, 116, 58, 120, 60, 124, 62, 128, 64, 132, 66, 136

Offset: 0

George E. Antoniou, Dec 04 2001

Fourth column of table A210530 for n>0. - Boris Putievskiy, Jan 29 2013

Harry J. Smith, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Cf. A065423, A210530.

Table[(3*n+2-(n+2)*(-1)^n)/2, {n,0,50}] (* or *) LinearRecurrence[{0, 2, 0, -1}, {0, 4, 2, 8}, 50] (* G. C. Greubel, Dec 24 2016 *)

{ for (n=0, 1000, if(n%2, a=2*n + 2, a=n); write("b066104.txt", n, " ", a) ) } \\ Harry J. Smith, Nov 14 2009

concat([0], Vec(2*x*(x+2)/(1-x^2)^2 + O(x^50))) \\ G. C. Greubel, Dec 24 2016

a(n) = 2*A065423(n+1).
O.g.f.: 2*x(2+x)/(1-x^2)^2. - Len Smiley, Dec 06 2001
a(n) = (3*n+2-(n+2)*(-1)^n)/2. - Boris Putievskiy, Jan 29 2013

A107130 Replace each odd digit d of n with (d-1)/2.

Original entry on oeis.org

0, 0, 2, 1, 4, 2, 6, 3, 8, 4, 0, 0, 2, 1, 4, 2, 6, 3, 8, 4, 20, 20, 22, 21, 24, 22, 26, 23, 28, 24, 10, 10, 12, 11, 14, 12, 16, 13, 18, 14, 40, 40, 42, 41, 44, 42, 46, 43, 48, 44, 20, 20, 22, 21, 24, 22, 26, 23, 28, 24, 60, 60, 62, 61, 64, 62, 66, 63, 68, 64, 30, 30, 32, 31, 34, 32

Offset: 0

Zak Seidov, May 12 2005

Differs from A065423 starting with a(10) = 0, A065423(10) = 10.

Vincenzo Librandi, Table of n, a(n) for n = 0..5000

Cf. A065423.

a[n_]:=FromDigits[Map[If[Mod[ #, 2]==1, (#-1)/2, # ]&, IntegerDigits[n]]];Table[a[n], {n, 0, 100}]

a(n)=fromdigits(apply(d->if(d%2,d\2,d),digits(n))) \\ Charles R Greathouse IV, Apr 07 2022

def A107130(n): return int(str(n).translate({49:48,51:49,53:50,55:51,57:52})) # Chai Wah Wu, Apr 07 2022

Edited by Charles R Greathouse IV, Aug 03 2010

A130656 Interlacing n^3/2 and n^2(n + 1)/2.

Original entry on oeis.org

1, 4, 18, 32, 75, 108, 196, 256, 405, 500, 726, 864, 1183, 1372, 1800, 2048, 2601, 2916, 3610, 4000, 4851, 5324, 6348, 6912, 8125, 8788, 10206, 10976, 12615, 13500, 15376, 16384, 18513, 19652, 22050, 23328, 26011, 27436, 30420, 32000, 35301, 37044

Offset: 1

Olivier Gérard, Jun 21 2007

Index entries for linear recurrences with constant coefficients, signature (1,3,-3,-3,3,1,-1).

Cf. A093005 (quadratic equivalent), A065423 (linear equivalent).

A130656:=n->n^2 * floor((n + 1)/2): seq(A130656(n), n=1..100); # Wesley Ivan Hurt, Jan 21 2017

a[n_Integer] := n^2 * Floor[(n + 1)/2]
LinearRecurrence[{1,3,-3,-3,3,1,-1},{1,4,18,32,75,108,196},50] (* Harvey P. Dale, Feb 18 2015 *)

a(n) = n^2 * floor((n + 1)/2).
G.f.: x*(1+3*x+11*x^2+5*x^3+4*x^4)/((1-x)^4*(1+x)^3). - R. J. Mathar, Sep 09 2008
a(n) = a(n-1)+ 3*a(n-2)-3*a(n-3)-3*a(n-4)+3*a(n-5)+a(n-6)-a(n-7), a(1)=1, a(2)=4, a(3)=18, a(4)=32, a(5)=75, a(6)=108, a(7)=196. - Harvey P. Dale, Feb 18 2015
Sum_{n>=1} 1/a(n) = zeta(3)/4 + Pi^2/4 - 2*log(2). - Amiram Eldar, Mar 15 2024

cp's OEIS Frontend

A131355 Partial sums of A065423 plus one.

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

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A029578 The natural numbers interleaved with the even numbers.

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A153733 Remove all trailing 1's in the binary representation of n.

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A211539 Number of ordered triples (w,x,y) with all terms in {1,...,n} and 2w = 2n - 2x + y.

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A097362 a(n) = (n+1)/2 if n is odd, n+2 otherwise.

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