A049453 -id:A049453 - 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

A144314 a(n) = 3n(6*n + 1).

Original entry on oeis.org

0, 21, 78, 171, 300, 465, 666, 903, 1176, 1485, 1830, 2211, 2628, 3081, 3570, 4095, 4656, 5253, 5886, 6555, 7260, 8001, 8778, 9591, 10440, 11325, 12246, 13203, 14196, 15225, 16290, 17391, 18528, 19701, 20910, 22155, 23436, 24753, 26106, 27495

Offset: 0

Reinhard Zumkeller, Sep 17 2008

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

Cf. A000217, A014105, A033585, A049453, A081266, A144312.

[18*n^2+3*n: n in [0..50]]; // Vincenzo Librandi, Dec 18 2014

A144314:=n->3*n*(6*n+1): seq(A144314(n), n=0..70); # Wesley Ivan Hurt, Dec 16 2015

Table[3n(6n+1),{n,0,40}] (* or *) LinearRecurrence[{3,-3,1},{0,21,78},40] (* Harvey P. Dale, Dec 17 2014 *)
CoefficientList[Series[x (21 + 15 x) / (1 - x)^3, {x, 0, 50}], x] (* Vincenzo Librandi, Dec 18 2014 *)

a(n)=3*n*(6*n+1) \\ Charles R Greathouse IV, Oct 07 2015

a(n) = A000217(6*n) = A014105(3*n) = A081266(2*n).
G.f.: 3*x*(7+5*x)/(1-x)^3. - Vincenzo Librandi, Dec 18 2014
From Wesley Ivan Hurt, Dec 16 2015: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>2.
a(n) = 3 * A049453(n). (End)
E.g.f.: 3*exp(x)*x*(7 + 6*x). - Stefano Spezia, Jun 29 2021
From Amiram Eldar, May 11 2025: (Start)
Sum_{n>=1} 1/a(n) = 2 - Pi/(2*sqrt(3)) - 2*log(2)/3 - log(3)/2.
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/3 + log(2)/3 + log(2+sqrt(3))/sqrt(3) - 2. (End)

A202804 a(n) = n(6n+4).

Original entry on oeis.org

0, 10, 32, 66, 112, 170, 240, 322, 416, 522, 640, 770, 912, 1066, 1232, 1410, 1600, 1802, 2016, 2242, 2480, 2730, 2992, 3266, 3552, 3850, 4160, 4482, 4816, 5162, 5520, 5890, 6272, 6666, 7072, 7490, 7920, 8362, 8816, 9282, 9760, 10250, 10752, 11266, 11792, 12330

Offset: 0

Jeremy Gardiner, Dec 24 2011

Sequence found by reading the line from 0, in the direction 0, 10, ..., in the square spiral whose vertices are the generalized pentagonal numbers A001318. Opposite numbers to the members of A033579 in the same spiral. - Omar E. Pol, Jul 17 2012

Partial sums give A163815. - Leo Tavares, Feb 25 2022

Jeremy Gardiner, Table of n, a(n) for n = 0..3000
Milan Janjic and Boris Petkovic, A Counting Function, arXiv 1301.4550 [math.CO], 2013.
Leo Tavares, Illustration: Star Halves.
Pavlos Vavolas, Cellular automaton model of cardiac arrhythmias, University of Sheffield Department of Computer Science (2005), (see page 43). [broken link, abstract]
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A003154, A005408, A033580, A033581, A049453, A163815, A195319.

Cf. A001318, A033579, A045944, A080859.

A202804:=n->n*(6*n+4): seq(A202804(n), n=0..100); # Wesley Ivan Hurt, Apr 09 2017

Table[n(6n+4),{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{0,10,32},50] (* Harvey P. Dale, Dec 28 2015 *)

x='x + O('x^50); concat([0], Vec(-2*x*(5 + x)/(x - 1)^3)) \\ Indranil Ghosh, Apr 10 2017

a(n) = 2*n(3*n+2) = 6*n^2 + 4*n = 2*A045944(n).
a(n) = A080859(n) - 1. - Omar E. Pol, Jul 18 2012
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Dec 28 2015
G.f.: 2*x*(5 + x)/(1 - x)^3. - Indranil Ghosh, Apr 10 2017
a(n) = A003154(n+1) - A005408(n). - Leo Tavares, Feb 25 2022
From Amiram Eldar, Mar 01 2022: (Start)
Sum_{n>=1} 1/a(n) = (Pi/sqrt(3) - 3*log(3) + 3)/8.
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/(4*sqrt(3)) - 3/8. (End)
E.g.f.: 2*exp(x)*x*(5 + 3*x). - Elmo R. Oliveira, Dec 12 2024

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

A194454 a(n) = 12n^2 + 2n + 1.

Original entry on oeis.org

1, 15, 53, 115, 201, 311, 445, 603, 785, 991, 1221, 1475, 1753, 2055, 2381, 2731, 3105, 3503, 3925, 4371, 4841, 5335, 5853, 6395, 6961, 7551, 8165, 8803, 9465, 10151, 10861, 11595, 12353, 13135, 13941, 14771, 15625, 16503, 17405, 18331, 19281

Offset: 0

Bruno Berselli, Aug 24 2011

A142241 gives the first differences.
Inverse binomial transform of this sequence: 1, 14, 24, 0, 0 (0 continued).
a(n)*a(n-1)-11 is a square, precisely 4*A051866(n)^2.
Sequence found by reading the line from 1, in the direction 1, 15, ..., in the square spiral whose vertices are the generalized octagonal numbers A001082. - Omar E. Pol, Jul 18 2012

			Using these numbers we can write:
  1, 15, 53, 115, 201, 311, 445,  603,  785,  991, 1221, ...
  0,  0,  1,  15,  53, 115, 201,  311,  445,  603,  785, ...
  0,  0,  0,   0,   1,  15,  53,  115,  201,  311,  445, ...
  0,  0,  0,   0,   0,   0,   1,   15,   53,  115,  201, ...
  0,  0,  0,   0,   0,   0,   0,    0,    1,   15,   53, ...
  0,  0,  0,   0,   0,   0,   0,    0,    0,    0,    1, ...
  ======================================================
  The sums of the columns give the sequence A172073 (after 0):
  1, 15, 54, 130, 255, 441, 700, 1044, 1485, 2035, 2706, ...

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

Cf. A154106, A172073, A049453.

Magma
```
[12*n^2+2*n+1: n in [0..40]];
```

Table[12 n^2 + 2 n + 1, {n, 0, 50}] (* Vincenzo Librandi, Mar 26 2013 *)

PARI
```
for(n=0, 40, print1(12*n^2+2*n+1", "));
```

G.f.: (1+x)*(1+11*x)/(1-x)^3.
a(n) = A154106(-n-1).
a(n) = 2*A049453(n) + 1.
a(n) = A051866(n) + A051866(n+1). - Charlie Marion, Nov 15 2019
E.g.f.: exp(x)*(1 + 14*x + 12*x^2). - Stefano Spezia, Nov 15 2019

A104584 a(n) = (1/2) * ( 3n^2 + n(-1)^n ).

Original entry on oeis.org

0, 1, 7, 12, 26, 35, 57, 70, 100, 117, 155, 176, 222, 247, 301, 330, 392, 425, 495, 532, 610, 651, 737, 782, 876, 925, 1027, 1080, 1190, 1247, 1365, 1426, 1552, 1617, 1751, 1820, 1962, 2035, 2185, 2262, 2420

Offset: 0

Gary W. Adamson, Mar 17 2005

Previous name was: Pentagonal wave sequence of the first kind.
Odd-indexed terms = A033570, pentagonal numbers with odd index (1, 12, 35, 70, ...). Even-indexed terms = A049453, 2nd pentagonal numbers with even index (0, 7, 26, 57, 100, ...).
Companion sequence A104585 (Pentagonal wave sequence of the second kind), switches odd with even applications and vice versa. The pentagonal wave sequence triangle A104586 has A104584 in odd columns and A104585 in even columns.
Exponents of q in the identity Sum_{n >= 0} ( q^n*Product_{k = 1..n} (1 - q^(4*k-3)) ) = 1 + q - q^7 - q^12 + q^26 + q^35 - - + + .... 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) = 35 = A000326(5).
a(6) = 57 = A005449(6).

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, A049453, A033568, A104585, A104586.

I:=[0, 1, 7, 12, 26]; [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..50]]; // 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},{0,1,7,12,26},50] (* Harvey P. Dale, Feb 14 2023 *)

a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5). - Vincenzo Librandi, Apr 04 2013
a(n) = (1/2) * (3*n^2 + n*(-1)^n ). - Ralf Stephan, May 20 2007
G.f. -x*(1+6*x+3*x^2+2*x^3) / ( (1+x)^2*(x-1)^3 ). - R. J. Mathar, Jan 10 2011
From Amiram Eldar, Feb 22 2022: (Start)
Sum_{n>=1} 1/a(n) = 6 - Pi/sqrt(3) - 4*log(2).
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*Pi/sqrt(3) + 3*log(3) - 6. (End)

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

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

A095894 a(2n) = 6n^2 + 7n + 1; a(2n+1) = 6n^2 + 13n + 7.

Original entry on oeis.org

1, 7, 14, 26, 39, 57, 76, 100, 125, 155, 186, 222, 259, 301, 344, 392, 441, 495, 550, 610, 671, 737, 804, 876, 949, 1027, 1106, 1190, 1275, 1365, 1456, 1552, 1649, 1751, 1854, 1962, 2071, 2185, 2300, 2420, 2541, 2667, 2794, 2926, 3059, 3197, 3336, 3480

Offset: 0

Gary W. Adamson, Jun 11 2004

From Omar E. Pol, Jul 18 2012: (Start)
Positive terms of A051866 and positive terms of A049453 interleaved.
Also sequence found by reading the line from 1, in the direction 1, 14, ..., and the line from 7, in the direction 7, 26, ..., in the square spiral whose vertices are the generalized pentagonal numbers A001318. (End)

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

Cf. A047225 (first differences), A049453, A051866.

LinearRecurrence[{2,0,-2,1},{1,7,14,26},60] (* Harvey P. Dale, Oct 13 2016 *)

x='x+O('x^50); Vec((-1-5*x)/((1+x)*(x-1)^3)) \\ G. C. Greubel, Jun 19 2017

G.f.: ( -1-5*x ) / ( (1+x)*(x-1)^3 ). - R. J. Mathar, Oct 26 2011

Edited by Don Reble, Nov 16 2005

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).

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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A144314 a(n) = 3*n*(6*n + 1).

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A202804 a(n) = n*(6*n+4).

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

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A194454 a(n) = 12*n^2 + 2*n + 1.

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A104584 a(n) = (1/2) * ( 3*n^2 + n*(-1)^n ).

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

A144314 a(n) = 3n(6*n + 1).

A202804 a(n) = n(6n+4).

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

A194454 a(n) = 12n^2 + 2n + 1.

A104584 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.

A095894 a(2n) = 6n^2 + 7n + 1; a(2n+1) = 6n^2 + 13n + 7.