A135370 -id:A135370 - cp's OEIS frontend

A136047 a(1)=1, a(n)=a(n-1)+n if n even, a(n)=a(n-1)+n^2 if n is odd.

1, 3, 12, 16, 41, 47, 96, 104, 185, 195, 316, 328, 497, 511, 736, 752, 1041, 1059, 1420, 1440, 1881, 1903, 2432, 2456, 3081, 3107, 3836, 3864, 4705, 4735, 5696, 5728, 6817, 6851, 8076, 8112, 9481, 9519, 11040, 11080, 12761, 12803, 14652, 14696, 16721

Offset: 1

Zak Seidov, Dec 12 2007

The only prime terms are 3, 41, 47.
The semiprime terms are A136048.
Cf. A001082/A135370: f(1) = 1, then if n even/odd f(n) = n+f(n-1), if n odd/even f(n) = 2*n+f(n-1).

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

Cf. A001082, A135370, A136048.

a[1]=1;a[n_]:=a[n]=a[n-1]+n^(1+Mod[n,2]); Table[a[n],{n,100}]
nxt[{n_,a_}]:={n+1,If[OddQ[n],a+n+1,a+(n+1)^2]}; Transpose[NestList[nxt,{1,1},50]][[2]] (* Harvey P. Dale, Oct 11 2015 *)

a(n) = (1/12)(1 + n)(2n^2+7n-3) if n is odd, a(n)=(1/12)n(2n^2+3n+4) if n is even.
a(n) = (-3 + 3*(-1)^n + 8*n + 12*n^2 - 6*(-1)^n*n^2 + 4*n^3)/24.
a(1)=1 then a(n) = a(n-1)+n^(if n is even then 1 else 2),
         or a(n) = a(n-1)+n^(1+mod(n,2)),
         or a(n) = a(n-1)+n^((3-(-1)^n)/2).
From R. J. Mathar, Feb 22 2009: (Start)
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).
G.f.: x*(1+2*x+6*x^2-2*x^3+x^4)/((1+x)^3*(x-1)^4). (End)

Edited by Michel Marcus, Mar 02 2022

A144390 a(n) = 3*n^2 - n - 1.

Original entry on oeis.org

1, 9, 23, 43, 69, 101, 139, 183, 233, 289, 351, 419, 493, 573, 659, 751, 849, 953, 1063, 1179, 1301, 1429, 1563, 1703, 1849, 2001, 2159, 2323, 2493, 2669, 2851, 3039, 3233, 3433, 3639, 3851, 4069, 4293, 4523, 4759, 5001, 5249, 5503, 5763, 6029, 6301, 6579

Offset: 1

Paul Curtz, Oct 02 2008

Sequence's original Name was "First bisection of A135370."

The partial sums of this sequence give A081437. - Leo Tavares, Dec 26 2021

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
John Elias, Illustration: Belted Hexagrams
Leo Tavares, Illustration: Bounded Hexagons
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A016933, A049450, A144391.
Cf. A003215, A005408, A016754, A002378.
Cf. A081437 (partial sums).

[3*n^2-n-1: n in [1..60]]; // Vincenzo Librandi, Jun 14 2011

A144390:=n->3*n^2-n-1; seq(A144390(n), n=1..50); # Wesley Ivan Hurt, Mar 26 2014

Table[3*n^2 -n -1 , {n,0,50}] (* G. C. Greubel, Jul 19 2017 *)

a(n)=3*n^2-n-1 \\ Charles R Greathouse IV, Oct 07 2015

a(n+1) = a(n) + 6*n + 2; see A016933.
G.f.: x*(1+6*x-x^2)/(1-x)^3. a(n) = A049450(n)-1. - R. J. Mathar, Oct 24 2008
a(-n) = A144391(n). - Michael Somos, Mar 27 2014
E.g.f.: (3*x^2 + 2*x -1)*exp(x) + 1. - G. C. Greubel, Jul 19 2017
From Leo Tavares, Dec 26 2021: (Start)
a(n) = A003215(n) - 2*A005408(n). See Bounded Hexagons illustration.
a(n) = A016754(n-1) - A002378(n-2). (End)
a(n) = A003154(n) - A049451(n-1). - John Elias, Dec 22 2022

Edited by R. J. Mathar, Oct 24 2008

More terms from Vladimir Joseph Stephan Orlovsky, Oct 25 2008

A144391 a(n) = 3*n^2 + n - 1.

Original entry on oeis.org

3, 13, 29, 51, 79, 113, 153, 199, 251, 309, 373, 443, 519, 601, 689, 783, 883, 989, 1101, 1219, 1343, 1473, 1609, 1751, 1899, 2053, 2213, 2379, 2551, 2729, 2913, 3103, 3299, 3501, 3709, 3923, 4143, 4369, 4601, 4839, 5083, 5333, 5589, 5851, 6119, 6393, 6673

Offset: 1

Paul Curtz, Oct 02 2008

Vincenzo Librandi, Table of n, a(n) for n = 1..3000
Leo Tavares, Illustration: Cropped Hexagons
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A135370, A144390.

[3*n^2+n-1: n in [1..50]]; // Vincenzo Librandi, May 06 2011

Table[3 n^2 + n - 1, {n, 50}] (* or *) LinearRecurrence[{3, -3, 1}, {3, 13, 29}, 50] (* Harvey P. Dale, Sep 18 2016 *)

a(n)=3*n^2+n-1 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = A135370(2*n).
First differences: a(n+1) - a(n) = A016957(n).
a(n) - A144390(n) = 6*n + 4 = A005843(n).
From R. J. Mathar, Oct 24 2008: (Start)
G.f.: x*(3 + 4*x - x^2)/(1 - x)^3.
a(n) = A049451(n) - 1. (End)
E.g.f.: (3*x^2 + 4*x - 1)*exp(x) + 1. - G. C. Greubel, Jul 19 2017
a(n) = 1 + Sum_{i = n-1..2*n-1} 2*i. - Bruno Berselli, Feb 16 2018
a(n) = A003215(n) - (n+1)*2. - Leo Tavares, Jul 04 2021

Edited by R. J. Mathar, Oct 24 2008

More terms from Vladimir Joseph Stephan Orlovsky, Mar 01 2009

A142888 First differences of A142705.

Original entry on oeis.org

3, -1, 13, -9, 29, -23, 51, -43, 79, -69, 113, -101, 153, -139, 199, -183, 251, -233, 309, -289, 373, -351, 443, -419, 519, -493, 601, -573, 689, -659, 783, -751, 883, -849, 989, -953, 1101, -1063, 1219, -1179, 1343, -1301, 1473, -1429, 1609, -1563, 1751

Offset: 1

Paul Curtz, Sep 29 2008

Also obtained from A135370 if adjacent pairs are swapped and if the sequence is then multiplied by (-1)^(n+1).

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

Cf. A005843, A142705.

CoefficientList[Series[(3 + 2 x + 6 x^2 - x^4)/((1 + x)^3 (1 - x)^2), {x, 0, 60}], x] (* Vincenzo Librandi, May 25 2014 *)
LinearRecurrence[{-1,2,2,-1,-1},{3,-1,13,-9,29},50] (* Harvey P. Dale, Apr 02 2018 *)

Vec(x*(3+2*x+6*x^2-x^4)/((1+x)^3*(1-x)^2) + O(x^100)) \\ Colin Barker, Jan 26 2016

a(n) = A142705(n+1)-A142705(n).
a(n) = -a(n-1) +2*a(n-2) +2*a(n-3) -a(n-4) -a(n-5). - R. J. Mathar, Sep 12 2010
a(2n-1)+a(2n) = A005843(n).
G.f.: x(3+2x+6x^2-x^4)/((1+x)^3*(1-x)^2). - R. J. Mathar, Oct 24 2008, parenthesis added Sep 12 2010
From Colin Barker, Jan 26 2016: (Start)
a(n) = (5+3*(-1)^n+(10-6*(-1)^n)*n-6*(-1)^n*n^2)/8.
a(n) = (-3*n^2+2*n+4)/4 for n even.
a(n) = (3*n^2+8*n+1)/4 for n odd.
(End)

Edited by R. J. Mathar, Oct 24 2008

More terms from Vincenzo Librandi, May 25 2014

cp's OEIS Frontend

A136047 a(1)=1, a(n)=a(n-1)+n if n even, a(n)=a(n-1)+n^2 if n is odd.

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A144390 a(n) = 3*n^2 - n - 1.

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A144391 a(n) = 3*n^2 + n - 1.

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A142888 First differences of A142705.

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