A049452 -id:A049452 - cp's OEIS frontend

A036498 Numbers of the form m(6m-1) and m(6m+1), where m is an integer.

0, 5, 7, 22, 26, 51, 57, 92, 100, 145, 155, 210, 222, 287, 301, 376, 392, 477, 495, 590, 610, 715, 737, 852, 876, 1001, 1027, 1162, 1190, 1335, 1365, 1520, 1552, 1717, 1751, 1926, 1962, 2147, 2185, 2380, 2420, 2625, 2667, 2882, 2926, 3151, 3197, 3432, 3480

Offset: 1

Wouter Meeussen

PartitionQ[ p ] is odd and contains an extra even partition; series term z^p in Product_{n>=1}(1-z^n) has coefficient (+1). - Wouter Meeussen
Numbers k such that the number of partitions of k into distinct parts with an even number of parts exceed by 1 the number of partitions of k into distinct parts with an odd number of parts. [See, e.g., the Freitag-Busam reference given under A036499, p. 410. - Wolfdieter Lang, Jan 18 2016]
In formal power series, A010815 = Product_{k>0}(1-x^k), ranks of coefficients 1 (A001318 = ranks of nonzero (1 or -1) in A010815 = ranks of odds terms in A000009).

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).

Union of A049452 and A049453.

Cf. A000009, A001318, A010815, A036499.

[1/8*(-1+(-1)^n+2*n)*(-3+(-1)^n+6*n): n in [1..50]]; // Vincenzo Librandi, Apr 24 2012

/* By definition: */ A036498:=func; [0] cat [A036498(n*m): m in [-1,1], n in [1..25]]; // Bruno Berselli, Nov 13 2012

p1 := n->n*(6*n-1): p2 := n->n*(6*n+1): S:={}: for n from 0 to 100 do S := S union {p1(n), p2(n)} od: S

Table[ 1/8*(-1 + (-1)^k + 2*k)*(-3 + (-1)^k + 6*k), {k, 64} ]
CoefficientList[Series[x*(5+2*x+5*x^2)/((1+x)^2*(1-x)^3),{x,0,50}],x] (* Vincenzo Librandi, Apr 24 2012 *)
Rest[Flatten[{#(6#-1),#(6#+1)}&/@Range[0,30]]] (* or *) LinearRecurrence[ {1,2,-2,-1,1},{0,5,7,22,26},60] (* Harvey P. Dale, Aug 13 2012 *)

\ps 5000; for(n=1,5000,if(polcoeff(eta(x),n,x)==1,print1(n,",")))

concat(0, Vec(x^2*(5+2*x+5*x^2)/((1+x)^2*(1-x)^3) + O(x^100))) \\ Altug Alkan, Jan 19 2016

def A036498(n): return (n*(3*n-5)>>1)+1 if n&1 else n*(3*n-1)>>1 # Chai Wah Wu, Mar 25 2025

a(n) = n(n+1)/6 for n=0 or 5 (mod 6).
a(n) = 1/8*(-1+(-1)^n+2*n)*(-3+(-1)^n+6*n) (see MATHEMATICA code).
G.f.: x^2*(5+2*x+5*x^2)/((1+x)^2*(1-x)^3). - Colin Barker, Apr 02 2012
a(1)=0, a(2)=5, a(3)=7, a(4)=22, a(5)=26, a(n)=a(n-1)+2*a(n-2)- 2*a(n-3)- a(n-4)+a(n-5). - Harvey P. Dale, Aug 13 2012
Bisections: a(2*k+1) = A001318(4*k) = k*(1+6*k) = A049453(k), k >= 0; a(2*k) = A001318(4*k-1) = k*(-1+6*k) = A049452(k), k >= 1. - Wolfdieter Lang, Jan 18 2016
From Amiram Eldar, Feb 13 2024: (Start)
Sum_{n>=2} 1/a(n) = 6 - sqrt(3)*Pi.
Sum_{n>=2} (-1)^n/a(n) = 4*log(2) + 3*log(3) - 6. (End)

Better description from Claude Lenormand (claude.lenormand(AT)free.fr), Feb 12 2001

Additional comments and more terms from James Sellers, Feb 14 2001

A282513 a(n) = floor((3*n + 2)^2/24 + 1/3).

Original entry on oeis.org

0, 1, 3, 5, 8, 12, 17, 22, 28, 35, 43, 51, 60, 70, 81, 92, 104, 117, 131, 145, 160, 176, 193, 210, 228, 247, 267, 287, 308, 330, 353, 376, 400, 425, 451, 477, 504, 532, 561, 590, 620, 651, 683, 715, 748, 782, 817, 852, 888, 925, 963

Offset: 0

Luce ETIENNE, Feb 17 2017

List of quadruples: 2*n*(3*n+1), (2*n+1)*(3*n+1), 6*n^2+8*n+3, (n+1)*(6*n+5). These terms belong to the sequences A033580, A033570, A126587 and A049452, respectively. See links for all the permutations.
After 0, subsequence of A025767.
It seems that a(n) is the smallest number of cells that need to be painted in a (n+1) X (n+1) grid, such that it has no unpainted hexominoes (see link to Kamenetsky and Pratt). - Rob Pratt, Dmitry Kamenetsky, Aug 30 2020

			Rectangular array with four columns:
.   0,   1,   3,   5;
.   8,  12,  17,  22;
.  28,  35,  43,  51;
.  60,  70,  81,  92;
. 104, 117, 131, 145, etc.
From _Rob Pratt_, Aug 30 2020: (Start)
For n = 3, painting only 2 cells would leave an unpainted hexomino, but painting the following 3 cells avoids all unpainted hexominoes:
    . . .
    . . X
    X X .
(End)

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Luce ETIENNE, Permutations
Dmitry Kamenetsky and Rob Pratt, 10x10 grid with no unpainted hexominoes, Puzzling StackExchange, October 2019.
Rob Pratt, Optimal solution for n=11.
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,1,-2,1).

Cf. A033436: floor((3*n)^2/24 + 1/3).
Cf. A000326, A000567, A025767, A033570, A033580, A049452, A064412, A126587, A222017, A269064, A274221.
Cf. A130519.
Minimum number of painted cells in other n-ominoes: A337501, A337502, A337503.

[(3*n^2+4*n+4) div 8: n in [0..50]]; // Bruno Berselli, Feb 17 2017

Table[Floor[(3 n + 2)^2/24 + 1/3], {n, 0, 50}] (* or *) CoefficientList[Series[x (1 + x + x^3)/((1 + x) (1 + x^2) (1 - x)^3), {x, 0, 50}], x] (* or *) Table[(6 n^2 + 8 n + 3 + Cos[n Pi] - 4 Cos[n Pi/2])/16, {n, 0, 50}] (* or *) Table[(3 n + 2)^2/24 + 1/3 + (-6 + (1 + (-1)^n) (1 + 2 I^((n + 1) (n + 2))))/16, {n, 0, 50}] (* Michael De Vlieger, Feb 17 2017 *)
LinearRecurrence[{2,-1,0,1,-2,1},{0,1,3,5,8,12},60] (* Harvey P. Dale, Aug 10 2024 *)

a(n)=(3*n^2 + 4*n + 4)\8 \\ Charles R Greathouse IV, Feb 17 2017

G.f.: x*(1 + x + x^3)/((1 + x)*(1 + x^2)*(1 - x)^3).
a(n) = 2*a(n-1) - a(n-2) + a(n-4) - 2*a(n-5) + a(n-6) for n>5.
a(n) = floor((3*n + 2)^2/24 + 2/3).
a(n) = (6*n^2 + 8*n + 3 + (-1)^n - 2*((-1)^((2*n - 1 + (-1)^n)/4) + (-1)^((2*n + 1 - (-1)^n)/4)))/16. Therefore:
a(2*k)   = (6*k^2 + 4*k + 1 - (-1)^k)/4,
a(2*k+1) = (k + 1)*(3*k + 2)/2.
a(n) = (6*n^2 + 8*n + 3 + cos(n*Pi) - 4*cos(n*Pi/2))/16.
a(n) = (3*n + 2)^2/24 + 1/3 + (-6 + (1 + (-1)^n)*(1 + 2*i^((n+1)*(n+2))))/16, where i=sqrt(-1).
a(n) = A130519(n+3)+A130519(n+2)+A130519(n). - R. J. Mathar, Jun 23 2021

Corrected and extended by Bruno Berselli, Feb 17 2017

A104585 a(n) = (1/2) * ( 3n^2 - n(-1)^n ).

Original entry on oeis.org

0, 2, 5, 15, 22, 40, 51, 77, 92, 126, 145, 187, 210, 260, 287, 345, 376, 442, 477, 551, 590, 672, 715, 805, 852, 950, 1001, 1107, 1162, 1276, 1335, 1457, 1520, 1650, 1717, 1855, 1926, 2072, 2147, 2301, 2380, 2542, 2625, 2795, 2882, 3060, 3151, 3337, 3432, 3626

Offset: 0

Gary W. Adamson, Mar 17 2005

nonn

Previous name was: Pentagonal wave sequence of the second kind.
Even-indexed terms are pentagonal numbers with even index in A000326. Odd-indexed terms are second pentagonal numbers with odd index in A005449.
A104584, pentagonal wave sequence of the first kind; switches odd and even applications and vice versa in A104585. The pentagonal wave triangle, A104586, has A104584 in odd columns and A104585 in even columns.
Integer values of (n+1)(2n+1)/3 in order of appearance. - Wesley Ivan Hurt, Sep 17 2013
Exponents of q in the identity 1 - Sum_{n >= 0} ( q^(3*n+2)*Product_{k = 1..n} (1 - q^(4*k-1)) ) = 1 - q^2 - q^5 + q^15 + q^22 - q^40 - q^51 + + - - .... Compare with Euler's pentagonal number theorem: Product_{n >= 1} (1 - q^n) = 1 - Sum_{n >= 1} ( q^n*Product_{k = 1..n-1} (1 - q^k) ) = 1 - q - q^2 + q^5 + q^7 - q^12 - q^15  + + - - .... - Peter Bala, Dec 03 2020

			a(5) = 40 = A005449(5), a second pentagonal number.
a(6) = 51 = A000326(6), a pentagonal number.

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

Cf. A000326, A005449, A049452, A033570, A049453, A033568, A104586.

I:=[0, 2, 5, 15, 22]; [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, Apr 04 2013

Table[(1/2) (3 n^2 - n (-1)^n), {n, 0, 100}] (* Vincenzo Librandi, Apr 04 2013 *)
LinearRecurrence[{1,2,-2,-1,1},{0,2,5,15,22},50] (* Harvey P. Dale, Sep 14 2015 *)

a(n) = (1/2) * ( 3*n^2 - n*(-1)^n ). - Ralf Stephan, Nov 13 2010
G.f.: x*(2+3*x+6*x^2+x^3)/(1-x)^3/(1+x)^2. - Colin Barker, Feb 13 2012
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5). - Vincenzo Librandi, Apr 04 2013
From Amiram Eldar, Feb 22 2022: (Start)
Sum_{n>=1} 1/a(n) = 4*log(2) - Pi/sqrt(3).
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*Pi/sqrt(3) - 3*log(3). (End)

More terms from Colin Barker, Feb 13 2012

Better name, using formula from Ralf Stephan, Joerg Arndt, Sep 17 2013

A291582 Maximum number of 6 sphinx tile shapes in a sphinx tiled hexagon of order n.

Original entry on oeis.org

30, 132, 306, 552, 870, 1260, 1722, 2256, 2862, 3540, 4290, 5112, 6006, 6972, 8010, 9120, 10302, 11556, 12882, 14280, 15750, 17292, 18906, 20592, 22350, 24180, 26082, 28056, 30102, 32220, 34410, 36672, 39006, 41412, 43890, 46440, 49062, 51756, 54522, 57360, 60270, 63252

Offset: 1

Craig Knecht, Aug 30 2017

The equilateral triangle composed of 144 smaller equilateral triangles is the smallest triangle that can be tiled with the sphinx. This triangle is used to form all orders of the hexagon.
Walter Trump enumerated all 830 sphinx tilings of this triangle and found six symmetrical examples one of which is used to produce this sequence.
Hyper-packing is a term that describes the ability of a shape to contain a greater area of subshapes than its own area by overlapping the subshapes. There are 864 unit triangles in the order 1 hexagon. 30 of the subshapes hyper-packed into this hexagon would contain 30x6x6 or 1080 unit triangles if summed individually.
The prime numbers cannot be described by a formula. Subsets of the primes such as the balanced primes are more formula friendly (see comments to puzzle 920 below). - Craig Knecht, Apr 19 2018

G. C. Greubel, Table of n, a(n) for n = 1..5000
Craig Knecht, Eight sphinx tile frame tessellations.
Craig Knecht, Example for the sequence.
Craig Knecht, Order 4 Hexagon with 246 subshapes.
Craig Knecht, Sphinx Tile Component Triangles.
Craig Knecht, Sphinx tiling of the triangle used to make the hexagon.
Craig Knecht, T12 symmetric sphinx tilings.
Carlos Rivera, Puzzle 920. An enigma related to A291582, Primes puzzles and problems connections.
Wikipedia, Walter Trump.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A016969, A049452.

Cf. A279887, A287999.

List([1..30], n -> 6*n*(6*n-1)); # G. C. Greubel, Dec 04 2018

[6*n*(6*n-1): n in [1..50]]; // Vincenzo Librandi, Sep 20 2017

seq(6*n*(6*n-1),n=1..100); # Robert Israel, Sep 19 2017

Array[6 # (6 # - 1) &, 42] (* Michael De Vlieger, Sep 19 2017 *)
CoefficientList[Series[2(15 + 21 x)/(1-x)^3,{x, 0, 50}], x] (* Vincenzo Librandi, Sep 20 2017 *)
CoefficientList[Series[6 E^x (5 + 17 x + 6 x^2), {x, 0, 50}], x]*
Table[n!, {n, 0, 50}] (* Stefano Spezia, Dec 07 2018 *)

a(n) = 6*n*(6*n-1); \\ Altug Alkan, Apr 08 2018

[6*n*(6*n-1) for n in (1..50)] # G. C. Greubel, Dec 04 2018

a(n) = 6*n*(6*n-1). - Walter Trump
G.f.: 2*x*(15+21*x)/(1-x)^3. - Vincenzo Librandi, Sep 20 2017
a(n) = 6*A049452(n) = 6*n*A016969(n-1). - Torlach Rush, Nov 28 2018
E.g.f.: 6*exp(x)*(5 + 17*x + 6*x^2). - Stefano Spezia, Dec 07 2018
a(n) = A016970(n-1) + A016969(n-1). - Torlach Rush, Dec 10 2018
From Amiram Eldar, Jul 30 2024: (Start)
Sum_{n>=1} 1/a(n) = log(2)/3 + log(3)/4 - sqrt(3)*Pi/12.
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/6 - log(2)/6 - arccoth(sqrt(3))/sqrt(3). (End)

A221912 Partial sums of floor(n/12).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 66, 69, 72, 76, 80, 84, 88, 92, 96, 100, 104, 108, 112, 116, 120, 125, 130, 135, 140, 145, 150, 155

Offset: 0

Philippe Deléham, Mar 27 2013

Apart from the initial zeros, the same as A008730.

			..0....0....0....0....0....0....0....0....0....0....0....0
..1....2....3....4....5....6....7....8....9...10...11...12
.14...16...18...20...22...24...26...28...30...32...34...36
.39...42...45...48...51...54...57...60...63...66...69...72
.76...80...84...88...92...96..100..104..108..112..116..120
125..130..135..140..145..150..155..160..165..170..175..180
186..192..198..204..210..216..222..228..234..240..246..252
259..266..273..280..287..294..301..308..315..322..329..336
344..352..360..368..376..384..392..400..408..416..424..432
441..450..459..468..477..486..495..504..513..522..531..540
...

Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,0,0,0,0,0,0,0,1,-2,1).

Cf. A000217, A002620, A130518, A130519, A130520, A174709, A174738, A118729, A218470, A131242, A218530.

Accumulate[Floor[Range[0,70]/12]] (* or *) LinearRecurrence[{2,-1,0,0,0,0,0,0,0,0,0,1,-2,1},{0,0,0,0,0,0,0,0,0,0,0,0,1,2},70] (* Harvey P. Dale, Mar 23 2015 *)

a(12n) = A051866(n).
a(12n+1) = A139267(n).
a(12n+2) = A094159(n).
a(12n+3) = A033579(n).
a(12n+4) = A049452(n).
a(12n+5) = A033581(n).
a(12n+6) = A049453(n).
a(12n+7) = A033580(n).
a(12n+8) = A195319(n).
a(12n+9) = A202804(n).
a(12n+10) = A211014(n).
a(12n+11) = A049598(n).
G.f.: x^12/((1-x)^2*(1-x^12)).
a(0)=0, a(1)=0, a(2)=0, a(3)=0, a(4)=0, a(5)=0, a(6)=0, a(7)=0, a(8)=0, a(9)=0, a(10)=0, a(11)=0, a(12)=1, a(13)=2, a(n)=2*a(n-1)- a(n-2)+ a(n-12)- 2*a(n-13)+ a(n-14). - Harvey P. Dale, Mar 23 2015

A008730 Molien series 1/((1-x)^2(1-x^12)) for 3-dimensional group [2,n] = 22n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 66, 69, 72, 76, 80, 84, 88, 92, 96, 100, 104, 108, 112, 116, 120, 125, 130, 135, 140, 145, 150, 155, 160, 165, 170, 175, 180, 186, 192, 198, 204

Offset: 0

N. J. A. Sloane

			..1....2....3....4....5....6....7....8....9...10...11...12
.14...16...18...20...22...24...26...28...30...32...34...36
.39...42...45...48...51...54...57...60...63...66...69...72
.76...80...84...88...92...96..100..104..108..112..116..120
125..130..135..140..145..150..155..160..165..170..175..180
186..192..198..204..210..216..222..228..234..240..246..252
259..266..273..280..287..294..301..308..315..322..329..336
344..352..360..368..376..384..392..400..408..416..424..432
441..450..459..468..477..486..495..504..513..522..531..540
550..560..570..580..590..600..610..620..630..640..650..660
...
The columns are: A051866, A139267, A094159, A033579, A049452, A033581, A049453, A033580, A195319, A202804, A211014, A049598
- _Philippe Deléham_, Apr 03 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 195
Index entries for Molien series
Index entries for linear recurrences with constant coefficients, signature (2, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -2, 1).

Cf. A001840, A001972, A008724, A008725, A008726, A008727, A008728, A008729, A008732. - Vladimir Joseph Stephan Orlovsky, Mar 14 2010

Cf. A221912

R:=PowerSeriesRing(Integers(), 70); Coefficients(R!( 1/((1-x)^2*(1-x^12)) )); // G. C. Greubel, Jul 30 2019

seq(coeff(series(1/(1-x)^2/(1-x^12), x, n+1), x, n), n=0..80);

CoefficientList[Series[1/((1-x)^2*(1-x^12)), {x,0,70}], x] (* Vincenzo Librandi, Jun 11 2013 *)
LinearRecurrence[{2,-1,0,0,0,0,0,0,0,0,0,1,-2,1},{1,2,3,4,5,6,7,8,9,10,11,12,14,16},70] (* Harvey P. Dale, Jan 01 2024 *)

my(x='x+O('x^70)); Vec(1/((1-x)^2*(1-x^12))) \\ G. C. Greubel, Jul 30 2019

(1/((1-x)^2*(1-x^12))).series(x, 70).coefficients(x, sparse=False) # G. C. Greubel, Jul 30 2019

G.f. 1/( (1-x)^3 * (1+x) *(1+x+x^2) *(1-x+x^2) * (1+x^2) *(1-x^2+x^4)). - R. J. Mathar, Aug 11 2021
From Mitch Harris, Sep 08 2008: (Start)
a(n) = Sum_{j=0..n+12} floor(j/12).
a(n-12) = (1/2)*floor(n/12)*(2*n - 10 - 12*floor(n/12)). (End)
a(n) = A221912(n+12). - Philippe Deléham, Apr 03 2013

More terms from Vladimir Joseph Stephan Orlovsky, Mar 14 2010

A104586 Pentagonal wave sequence triangle.

Original entry on oeis.org

1, 7, 2, 12, 5, 1, 26, 15, 7, 2, 35, 22, 12, 5, 1, 57, 40, 26, 15, 7, 2, 70, 51, 35, 22, 12, 5, 1, 100, 77, 57, 40, 26, 15, 7, 2

Offset: 1

Gary W. Adamson, Mar 17 2005

Row sums = A086500: 1, 9, 18, 50, 75, 147, 196...

			The first few rows are:
1;
7, 2;
12, 5, 1;
26, 15, 7, 2;
35, 22, 12, 5, 1;
57, 40, 26, 15, 7, 2;
70, 51, 35, 22, 12, 5, 1;
...

Cf. A086500, A104584, A104585, A000326, A005449, A049452, A033570, A049453, A033568.

Odd columns are terms of A104584, pentagonal wave sequence of the first kind, (starting with 1): 1, 7, 12, 26, 35, 57, 70... Even columns are terms of A104585, pentagonal wave sequence of the second kind (starting with 2): 2, 5, 15, 22, 40, 51... Odd rows are pentagonal numbers (A000326) starting with "1" at the right. Even rows are second pentagonal numbers (A005449) starting with 2 at the right. The triangle is extracted from a matrix product A * B, A = [1; 1, 2; 1, 2, 1; 1, 2, 1, 2;...], B = [1; 3, 1; 5, 3, 1; 7, 5, 3, 1;...] (both infinite lower triangular matrices, with the rest zeros).

A110344 a(n) = Sum_{k=0..n-1} (n+k) = n(3n-1)/2 if n is even; a(n) = Sum_{k=0..n-1} (n-k) = n(n+1)/2 if n is odd.

Original entry on oeis.org

1, 5, 6, 22, 15, 51, 28, 92, 45, 145, 66, 210, 91, 287, 120, 376, 153, 477, 190, 590, 231, 715, 276, 852, 325, 1001, 378, 1162, 435, 1335, 496, 1520, 561, 1717, 630, 1926, 703, 2147, 780, 2380, 861, 2625, 946, 2882, 1035, 3151, 1128, 3432, 1225, 3725, 1326

Offset: 1

Amarnath Murthy, Jul 20 2005

			a(3) = 3 + 2 +1 = 6.
a(6) = 6 + 7 + 8 + 9 + 10 + 11 = 51.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,3,0,-3,0,1).

Cf. A000217, A000326, A049452.

a:=proc(n) if n mod 2=0 then n*(3*n-1)/2 else n*(n+1)/2 fi end: seq(a(n),n=1..60); # Emeric Deutsch

a[n_] := n*(2*n + (n - 1)*(-1)^n)/2; Array[a, 50] (* Amiram Eldar, Sep 11 2022 *)

Vec(-x*(7*x^3+3*x^2+5*x+1)/((x-1)^3*(x+1)^3) + O(x^100)) \\ Colin Barker, Feb 17 2015

From Emeric Deutsch, Aug 01 2005: (Start)
a(2n+1) = A000217(2n+1) = (n+1)(2n+1) (triangular numbers with odd index).
a(2n) = A000326(2n) = A049452(n) = n(6n-1) (pentagonal numbers with even index).
(End)
a(n) = n*( 2*n + (n-1)*(-1)^n )/2. - Luce ETIENNE, Jul 08 2014
From Colin Barker, Feb 17 2015: (Start)
a(n) = 3*a(n-2)-3*a(n-4)+a(n-6).
G.f.: -x*(7*x^3+3*x^2+5*x+1) / ((x-1)^3*(x+1)^3). (End)
Sum_{n>=1} 1/a(n) = 4*log(2) + 3*log(3)/2 - sqrt(3)*Pi/2. - Amiram Eldar, Sep 11 2022

More terms from Emeric Deutsch, Aug 01 2005

A272104 Sum of the even numbers among the larger parts of the partitions of n into two parts.

Original entry on oeis.org

0, 0, 0, 2, 2, 4, 4, 10, 10, 14, 14, 24, 24, 30, 30, 44, 44, 52, 52, 70, 70, 80, 80, 102, 102, 114, 114, 140, 140, 154, 154, 184, 184, 200, 200, 234, 234, 252, 252, 290, 290, 310, 310, 352, 352, 374, 374, 420, 420, 444, 444, 494, 494, 520, 520, 574, 574, 602

Offset: 0

Wesley Ivan Hurt, Apr 20 2016

Essentially, repeated values of A152749.

Sum of the lengths of the distinct rectangles with even length and integer width such that L + W = n, W <= L. For example, a(10) = 14; the rectangles are 2 X 8 and 4 X 6, so 8 + 6 = 14. - Wesley Ivan Hurt, Nov 04 2017

			a(5) = 4; the partitions of 5 into 2 parts are (4,1),(3,2) and the sum of the larger even parts is 4.
a(6) = 4; the partitions of 6 into 2 parts are (5,1),(4,2),(3,3) and the sum of the larger even parts is also 4.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for sequences related to partitions
Index entries for linear recurrences with constant coefficients, signature (1,0,0,2,-2,0,0,-1,1).

Cf. A000326, A001318, A026741, A049450, A049452, A109043, A126964, A152749, A268351, A272212.

[(1+3*(2*n-3-(-1)^n)/2+3*(2*n-3-(-1)^n)^2/8+(2*n-1-(-1)^n)*(-1)^((2*n+1-(-1)^n) div 4)/2)/8 : n in [0..50]];

A272104:=n->(1+3*(2*n-3-(-1)^n)/2+3*(2*n-3-(-1)^n)^2/8+(2*n-1-(-1)^n)*(-1)^((2*n+1-(-1)^n)/4)/2)/8: seq(A272104(n), n=0..100);

Table[(1 + 3(2n-3-(-1)^n)/2 + 3(2n-3-(-1)^n)^2/8 + (2n-1-(-1)^n) * (-1)^((2n+1-(-1)^n)/4)/2) / 8, {n, 0, 50}]
Table[Total@ Map[First, IntegerPartitions[n, {2}] /. {k_, } /; OddQ@ k -> Nothing], {n, 0, 57}] (* _Michael De Vlieger, Apr 20 2016, Version 10.2 *)

concat(vector(3), Vec(2*x^3*(1-x+x^2)*(1+x+x^2)/((1-x)^3*(1+x)^2*(1+x^2)^2) + O(x^50))) \\ Colin Barker, Apr 20 2016

a(n) = (1 + 3*(2n-3-(-1)^n)/2 + 3*(2n-3-(-1)^n)^2/8 + (2n-1-(-1)^n) * (-1)^((2n+1-(-1)^n)/4)/2) / 8.
a(n) = Sum_{i=ceiling(n/2)..n-1} i * (i+1 mod 2).
a(n) = Sum_{i=1..floor(n/2)} (n-i) * (n-i+1 mod 2).
a(2n+1) = a(2n+2) = A152749(n) = 2*A001318(n).
G.f.: 2*x^3*(1-x+x^2)*(1+x+x^2) / ((1-x)^3*(1+x)^2*(1+x^2)^2). - Colin Barker, Apr 20 2016
From Wesley Ivan Hurt, Apr 22 2016, Apr 23 2016: (Start)
a(2n+2)-a(2n) = A109043(n) = 2*A026741(n).
a(4n) = A049450(n) = 2*A000326(n),
a(8n) = A126964(n) = 2*A049452(n),
a(12n) = 2*A268351(n).
a(n+1) = A001318(n) - A272212(n+1). (End)
E.g.f.: ((2 + 3*x*(1 + x))*cosh(x) - 2*(cos(x) + x*cos(x) + x*sin(x)) + (-1 + 3*(-1 + x)*x)*sinh(x))/16. - Ilya Gutkovskiy, Apr 29 2016

A330409 Semiprimes of the form p(6p - 1).

Original entry on oeis.org

22, 51, 145, 287, 1717, 2147, 3151, 5017, 11051, 13207, 16801, 20827, 26867, 63551, 68587, 71177, 76501, 96647, 112477, 147737, 159251, 232657, 237407, 308947, 314417, 342487, 433897, 480251, 587501, 602617, 722107, 772927, 834401, 861467, 879751, 907537, 945257, 1155887, 1177051

Offset: 1

M. F. Hasler, Dec 13 2019

nonn

			A158015(1) = 2 is the smallest prime p such that 6p - 1 = 12 - 1 = 11 is also prime, whence a(1) = A049452(2) = 2*(6*2 - 1) = 22.
prime(5) = 11 is the smallest prime not in A024898 or A158015, because 6p - 1 is not a prime, therefore A049452(11) = 11*(6*11 - 1) is not in the sequence, and idem for A049452(13).
But prime(7) = 17 is in A024898 and A158015, so a(5) = A024898(A158015(5)) = A024898(17) = 17*(6*17 - 1).

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A024898 (6n-1 is prime), A158015 (primes), A049452 = {n(6n-1)}.

Complement of A255584 = A033570(A130800) (semiprimes (2n+1)(3n+1)) in A245365 (primes of the form n(3n-1)/2).

Select[Table[p(6p-1),{p,500}],PrimeOmega[#]==2&] (* Harvey P. Dale, Apr 27 2022 *)

[p*(6*p-1) | p<-primes(99), isprime(6*p-1)]

a(n) = A049452(A158015(n)) = p(6p - 1) with p = A158015(n).

cp's OEIS Frontend

A036498 Numbers of the form m*(6*m-1) and m*(6*m+1), where m is an integer.

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A282513 a(n) = floor((3*n + 2)^2/24 + 1/3).

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A104585 a(n) = (1/2) * ( 3*n^2 - n*(-1)^n ).

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A291582 Maximum number of 6 sphinx tile shapes in a sphinx tiled hexagon of order n.

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A221912 Partial sums of floor(n/12).

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A008730 Molien series 1/((1-x)^2*(1-x^12)) for 3-dimensional group [2,n] = *22n.

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A036498 Numbers of the form m(6m-1) and m(6m+1), where m is an integer.

A104585 a(n) = (1/2) * ( 3n^2 - n(-1)^n ).

A008730 Molien series 1/((1-x)^2(1-x^12)) for 3-dimensional group [2,n] = 22n.