A062717 -id:A062717 - cp's OEIS frontend

A152749 a(n) = (n+1)(3n+1)/4 for n odd, a(n) = n(3n+2)/4 for n even.

0, 2, 4, 10, 14, 24, 30, 44, 52, 70, 80, 102, 114, 140, 154, 184, 200, 234, 252, 290, 310, 352, 374, 420, 444, 494, 520, 574, 602, 660, 690, 752, 784, 850, 884, 954, 990, 1064, 1102, 1180, 1220, 1302, 1344, 1430, 1474, 1564, 1610, 1704, 1752, 1850, 1900, 2002

Offset: 0

Vincenzo Librandi, Dec 31 2009

Interleaving of A049450 and A049451 (for n > 0).
Also, integer values of k*(k+1)/3. - Charles R Greathouse IV, Dec 11 2010
The nonzero coefficients of the expansion of f(a) = Product_{k>=1} (1-a^(2k)), see A194159, occur at the terms of the sequence given above, i.e., f(a) = 1 - a^2 - a^4 + a^10 + a^14 - a^24 - a^30 + a^44 + a^52 - a^70 - a^80 + ... = Sum_{n>=0} (-1)^binomial(n+1,2)*a^A152749(n). - Johannes W. Meijer, Aug 21 2011
Partial sums of A109043. - Reinhard Zumkeller, Mar 31 2012
Nonnegative k such that 12*k+1 is a square. - Vicente Izquierdo Gomez, Jul 22 2013
Equivalently, numbers of the form h*(3*h+1), where h = 0, -1, 1, -2, 2, -3, 3, -4, 4, ... (see also the fifth comment of A062717). - Bruno Berselli, Feb 02 2017
For n > 0, a(n-1) is the sum of the largest parts of the partitions of 2n into two even parts. - Wesley Ivan Hurt, Dec 19 2017
The sequence terms occur as exponents in the expansion of Sum_{n >= 0} q^(n*(n+1)/2) * Product_{k >= n+1} 1 - q^k = 1 - q^2 - q^4 + q^10 + q^14 - q^24 - q^30 + + - -  .... - Peter Bala, Dec 15 2024
Sequence terms occur as exponents in the expansions of Sum_{n >= 0} q^(n*(2*n+1)) * Product_{k >= 2*n+2} 1 - q^k  = Sum_{n >= 0} q^(n*(2*n-1)) * Product_{k >= 2*n+1} 1 - q^k = 1 - q^2 - q^4 + q^10 + q^14 - q^24 - q^30 + + - - .... - Peter Bala, Jun 23 2025

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. A049450 (n*(3*n-1)), A049451 (n*(3*n+1)), A153383 (12n+1 is not prime).

a152749 n = a152749_list !! (n-1)
a152749_list = scanl1 (+) a109043_list
-- Reinhard Zumkeller, Mar 31 2012

[IsOdd(n) select (n+1)*(3*n+1)/4 else n*(3*n+2)/4: n in [0..52]];

f:=func; [0] cat [f(n*m): m in [-1,1], n in [1..30]]; // Bruno Berselli, Nov 13 2012

A152749 := proc(n): if type(n,even) then n*(3*n+2)/4  else (n+1)*(3*n+1)/4 fi: end: seq(A152749(n), n=0..51); # Johannes W. Meijer, Aug 21 2011

Table[If[OddQ[n],(n+1)*(3*n+1)/4,n*(3*n+2)/4],{n,0,60}] (* Vladimir Joseph Stephan Orlovsky, Feb 03 2012 *)
LinearRecurrence[{1,2,-2,-1,1}, {0, 2, 4, 10, 14}, 50] (* Vincenzo Librandi, Feb 22 2012 *)
Select[Range[1,1000], IntegerQ[Sqrt[12#+1]]&] (* Vicente Izquierdo Gomez, Jul 22 2013 *)

From R. J. Mathar, Jan 03-06 2009: (Start)
G.f.: 2*x*(1+x+x^2)/((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) = A003154(n+1)/8 - (-1)^n*A005408(n)/8.
a(n) = 2*A001318(n) = ((6*n^2+6*n+1) - (2*n+1)*(-1)^n)/8. (End)
From Amiram Eldar, Mar 15 2022: (Start)
Sum_{n>=1} 1/a(n) = 3 - Pi/sqrt(3).
Sum_{n>=1} (-1)^(n+1)/a(n) = 3*(log(3)-1). (End)

Edited, typo corrected and extended by Klaus Brockhaus, Jan 02 2009

Leading term a(0)=0 added by Johannes W. Meijer, Aug 21 2011

A132355 Numbers of the form 9h^2 + 2h, for h an integer.

Original entry on oeis.org

0, 7, 11, 32, 40, 75, 87, 136, 152, 215, 235, 312, 336, 427, 455, 560, 592, 711, 747, 880, 920, 1067, 1111, 1272, 1320, 1495, 1547, 1736, 1792, 1995, 2055, 2272, 2336, 2567, 2635, 2880, 2952, 3211, 3287, 3560, 3640, 3927, 4011, 4312, 4400, 4715, 4807

Offset: 1

Mohamed Bouhamida, Nov 08 2007

X values of solutions to the equation 9*X^3 + X^2 = Y^2.
The set of all m such that 9*m + 1 is a perfect square. - Gary Detlefs, Feb 22 2010
The concatenation of any term with 11..11 (1 repeated an even number of times, see A099814) belongs to the list. Example: 87 is a term, so also 8711, 871111, 87111111, 871111111111, ... are terms of this sequence. - Bruno Berselli, May 15 2017

Jason Kimberley, Table of n, a(n) for n = 1..2108
S. Cooper and M. D. Hirschhorn, Results of Hurwitz type for three squares. Discrete Math., Vol. 274, No. 1-3 (2004), pp. 9-24. See A(q).
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

A205808 is the characteristic function.
Cf. A000217, A001082, A002378, A005563, A028347, A036666, A046092, A054000, A056220, A062717, A087475, A132209, A010701, A056020.
Numbers of the form 9*n^2+k*n, for integer n: A016766 (k=0), this sequence (k=2), A185039 (k=4), A057780 (k=6), A218864 (k=8). - Jason Kimberley, Nov 09 2012
For similar sequences of numbers m such that 9*m+k is a square, see list in A266956.

a:=func; [0] cat [a(n*m): m in [-1,1], n in [1..25]]; // Jason Kimberley, Nov 08 2012

readlib(issqr); for n from 0 to 3560 do if(issqr(9*n+1)) then print(n) fi od; # Gary Detlefs, Feb 22 2010
seq(n^2+n+5*ceil(n/2)^2,n=0..39); # Gary Detlefs, Feb 23 2010

f[n_]:=IntegerQ[Sqrt[1+9*n]]; Select[Range[0,8! ],f[ # ]&] (* Vladimir Joseph Stephan Orlovsky, Feb 19 2010 *)
Sort[Table[9n^2+2n,{n,-30,30}]] (* Harvey P. Dale, Dec 06 2013 *)

a(n)=n^2-n+5*(n\2)^2 \\ Charles R Greathouse IV, Sep 28 2015

a(2*k) = k*(9*k-2), a(2*k+1) = k*(9*k+2).
a(n) = n^2 - n + 5*floor(n/2)^2. - Gary Detlefs, Feb 23 2010
From R. J. Mathar, Mar 17 2010: (Start)
a(n) = +a(n-1) +2*a(n-2) -2*a(n-3) -a(n-4) +a(n-5).
G.f.: x^2*(7 + 4*x + 7*x^2)/((1 + x)^2*(1 - x)^3). (End)
a(n) = (2*n - 1 + (-1)^n)*(9*(2*n - 1) + (-1)^n)/16. - Luce ETIENNE, Sep 13 2014
Sum_{n>=2} 1/a(n) = 9/4 - cot(2*Pi/9)*Pi/2. - Amiram Eldar, Mar 15 2022

Simpler definition and minor edits from N. J. A. Sloane, Feb 03 2012

Since this is a list, offset changed to 1 and formulas translated by Jason Kimberley, Nov 18 2012

A033580 Four times second pentagonal numbers: a(n) = 2n(3*n+1).

Original entry on oeis.org

0, 8, 28, 60, 104, 160, 228, 308, 400, 504, 620, 748, 888, 1040, 1204, 1380, 1568, 1768, 1980, 2204, 2440, 2688, 2948, 3220, 3504, 3800, 4108, 4428, 4760, 5104, 5460, 5828, 6208, 6600, 7004, 7420, 7848, 8288, 8740, 9204, 9680, 10168, 10668, 11180, 11704, 12240

Offset: 0

N. J. A. Sloane

Subsequence of A062717: A010052(6*a(n)+1) = 1. - Reinhard Zumkeller, Feb 21 2011
Sequence found by reading the line from 0, in the direction 0, 8,..., in the square spiral whose vertices are the generalized pentagonal numbers A001318. Opposite numbers to the members of A139267 in the same spiral - Omar E. Pol, Sep 09 2011
a(n) is the number of edges of the octagonal network O(n,n); O(m,n) is defined by Fig. 1 of the Siddiqui et al. reference. - Emeric Deutsch May 13 2018
The partial sums of this sequence give A035006. - Leo Tavares, Oct 03 2021

Ivan Panchenko, Table of n, a(n) for n = 0..1000
M. K. Siddiqui, M. Naeem, N. A. Rahman, and M. Imran, Computing topological indices of certain networks, J. of Optoelectronics and Advanced Materials, 18, No. 9-10 (2016), pp. 884-892.
Leo Tavares, Illustration: Crossed Stars
Leo Tavares, Illustration: Four Quarter Star Crosses
Leo Tavares, Illustration: Triangulated Star Crosses
Leo Tavares, Illustration: Oblong Star Crosses
Leo Tavares, Illustration: Crossed Diamond Stars
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A033579, A049451, A045945, A186423.
Cf. sequences listed in A254963.
Cf. A003154, A016813.
Cf. A035006.
Cf. A005449, A046092, A033996, A002378, A016742, A000290, A016754, A001105.

List([0..50], n-> 2*n*(3*n+1)); # G. C. Greubel, Oct 09 2019

[2*n*(3*n+1): n in [0..50]]; // G. C. Greubel, Oct 09 2019

seq(2*n*(3*n+1), n=0..50); # G. C. Greubel, Oct 09 2019

4*Binomial[3*Range[50]-2, 2]/3 (* G. C. Greubel, Oct 09 2019 *)

a(n)=2*n*(3*n+1) \\ Charles R Greathouse IV, Sep 28 2015

[2*n*(3*n+1) for n in (0..50)] # G. C. Greubel, Oct 09 2019

a(n) = a(n-1) +12*n -4 (with a(0)=0). - Vincenzo Librandi, Aug 05 2010
G.f.: 4*x*(2+x)/(1-x)^3. - Colin Barker, Feb 13 2012
a(-n) = A033579(n). - Michael Somos, Jun 09 2014
E.g.f.: 2*x*(4 + 3*x)*exp(x). - G. C. Greubel, Oct 09 2019
From Amiram Eldar, Jan 14 2021: (Start)
Sum_{n>=1} 1/a(n) = 3/2 - Pi/(4*sqrt(3)) - 3*log(3)/4.
Sum_{n>=1} (-1)^(n+1)/a(n) =  -3/2 + Pi/(2*sqrt(3)) + log(2). (End)
From Leo Tavares, Oct 12 2021: (Start)
a(n) = A003154(n+1) - A016813(n). See Crossed Stars illustration.
a(n) = 4*A005449(n). See Four Quarter Star Crosses illustration.
a(n) = 2*A049451(n).
a(n) = A046092(n-1) + A033996(n). See Triangulated Star Crosses illustration.
a(n) = 4*A000217(n-1) + 8*A000217(n).
a(n) = 4*A000217(n-1) + 4*A002378. See Oblong Star Crosses illustration.
a(n) = A016754(n) + 4*A000217(n). See Crossed Diamond Stars illustration.
a(n) = 2*A001105(n) + 4*A000217(n).
a(n) = A016742(n) + A046092(n).
a(n) = 4*A000290(n) + 4*A000217(n). (End)

A132356 a(2k) = k(10k+2), a(2k+1) = 10k^2 + 18k + 8, with k >= 0.

Original entry on oeis.org

0, 8, 12, 36, 44, 84, 96, 152, 168, 240, 260, 348, 372, 476, 504, 624, 656, 792, 828, 980, 1020, 1188, 1232, 1416, 1464, 1664, 1716, 1932, 1988, 2220, 2280, 2528, 2592, 2856, 2924, 3204, 3276, 3572, 3648, 3960, 4040, 4368, 4452, 4796, 4884, 5244, 5336, 5712

Offset: 0

Mohamed Bouhamida, Nov 08 2007

X values of solutions to the equation 10*X^3 + X^2 = Y^2.
Polygonal number connection: 2*H_n + 6S_n, where H_n is the n-th hexagonal number and S_n is the n-th square number. This is the base formula that is expanded upon to achieve the full series. See contributing formula below. - William A. Tedeschi, Sep 12 2010
Equivalently, numbers of the form 2*h*(5*h+1), where h = 0, -1, 1, -2, 2, -3, 3, -4, 4, ... . - Bruno Berselli, Feb 02 2017

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

Cf. A054000, A056220, A087475, A028347, A046092, A132209.
Cf. numbers m such that k*m+1 is a square: A005563 (k=1), A046092 (k=2), A001082 (k=3), A002378 (k=4), A036666 (k=5), A062717 (k=6), A132354 (k=7), A000217 (k=8), A132355 (k=9), A219257 (k=11), A152749 (k=12), A219389 (k=13), A219390 (k=14), A204221 (k=15), A074378 (k=16), A219394 (k=17), A219395 (k=18), A219396 (k=19), A219190 (k=20), A219391 (k=21), A219392 (k=22), A219393 (k=23), A001318 (k=24), A219259 (k=25), A217441 (k=26), A219258 (k=27), A219191 (k=28).
Cf. A220082 (numbers k such that 10*k-1 is a square).

CoefficientList[Series[4*x*(2*x^2 + x + 2)/((1 - x)^3*(1 + x)^2), {x, 0, 50}], x] (* G. C. Greubel, Jun 12 2017 *)
LinearRecurrence[{1,2,-2,-1,1},{0,8,12,36,44},50] (* Harvey P. Dale, Dec 15 2023 *)

my(x='x+O('x^50)); concat([0], Vec(4*x*(2*x^2+x+2)/((1-x)^3*(1+x)^2))) \\ G. C. Greubel, Jun 12 2017

a(n) = n^2 + n + 6*((n+1)\2)^2 \\ Charles R Greathouse IV, Sep 11 2022

G.f.: 4*x*(2*x^2+x+2)/((1-x)^3*(1+x)^2). - R. J. Mathar, Apr 07 2008
a(n) = 10*x^2 - 2*x, where x = floor(n/2)*(-1)^n for n >= 1. - William A. Tedeschi, Sep 12 2010
a(n) = ((2*n+1-(-1)^n)*(10*(2*n+1)-2*(-1)^n))/16. - Luce ETIENNE, Sep 13 2014
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n > 4. - Chai Wah Wu, May 24 2016
Sum_{n>=1} 1/a(n) = 5/2 - sqrt(1+2/sqrt(5))*Pi/2. - Amiram Eldar, Mar 15 2022
a(n) = n^2 + n + 6*ceiling(n/2)^2. - Ridouane Oudra, Aug 06 2022

More terms from Max Alekseyev, Nov 13 2009

A132209 a(0) = 0 and a(n) = 2n^2 + 2n - 1, for n>=1.

Original entry on oeis.org

0, 3, 11, 23, 39, 59, 83, 111, 143, 179, 219, 263, 311, 363, 419, 479, 543, 611, 683, 759, 839, 923, 1011, 1103, 1199, 1299, 1403, 1511, 1623, 1739, 1859, 1983, 2111, 2243, 2379, 2519, 2663, 2811, 2963, 3119, 3279, 3443, 3611, 3783, 3959, 4139, 4323, 4511

Offset: 0

Mohamed Bouhamida, Nov 06 2007

nonn

Previous name was: Sequence gives X values that satisfy the integer equation 2*X^3 + 3*X^2 = Y^2.

To find Y values: b(n) = (2*n^2 + 2*n - 1)*(2*n - 1).

G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (3, -3, 1).

Cf. A005563, A046092, A001082, A002378, A036666, A062717, A028347, A087475, A000217.

[0] cat [2*n^2+2*n-1: n in [1..50]]; // Vincenzo Librandi, Sep 22 2015

Join[{0}, LinearRecurrence[{3, -3, 1}, {3, 11, 23}, 40]] (* Vincenzo Librandi, Sep 22 2015 *)

for(n=0,50, print1(if(n==0, 0, 2*n^2 + 2*n -1), ", ")) \\ G. C. Greubel, Jul 13 2017

a(n) = 2*n^2 + 2*n - 1 for n>=1.
G.f.: x*(1+x)*(3-x)/(1-x)^3. - R. J. Mathar, Nov 14 2007
E.g.f.: 1 + (2*x^2 + 4*x -1)*exp(x). - G. C. Greubel, Jul 13 2017
From Amiram Eldar, Mar 07 2021: (Start)
Sum_{n>=1} 1/a(n) = 1 + sqrt(3)*Pi*tan(sqrt(3)*Pi/2)/6.
Product_{n>=1} (1 + 1/a(n)) = -Pi*sec(sqrt(3)*Pi/2)/2.
Product_{n>=1} (1 - 1/a(n)) = cos(sqrt(5)*Pi/2)*sec(sqrt(3)*Pi/2)/2. (End)

Edited by the Associate Editors of the OEIS, Nov 15 2009
More terms from Vincenzo Librandi, Sep 22 2015
Shorter name (using formula given) from Joerg Arndt, Sep 27 2015

A186423 Partial sums of A186421.

Original entry on oeis.org

0, 1, 3, 4, 8, 11, 17, 20, 28, 33, 43, 48, 60, 67, 81, 88, 104, 113, 131, 140, 160, 171, 193, 204, 228, 241, 267, 280, 308, 323, 353, 368, 400, 417, 451, 468, 504, 523, 561, 580, 620, 641, 683, 704, 748, 771, 817, 840, 888, 913, 963, 988, 1040, 1067, 1121, 1148, 1204, 1233, 1291, 1320

Offset: 0

Reinhard Zumkeller, Feb 21 2011

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

A062717 is the subsequence of even terms.

A186424 is the subsequence of odd terms.

List([0..65], n-> (6*n^2 +6*n +3 +(-1)^n*(2*n+1) -4*(-1)^Binomial(n+1, 2))/16); # G. C. Greubel, Oct 09 2019

a186423 n = a186423_list !! n
a186423_list = scanl1 (+) a186421_list

[(6*n^2 +6*n +3 +(-1)^n*(2*n+1) -4*(-1)^Binomial(n+1, 2))/16: n in [0..65]]; // G. C. Greubel, Oct 09 2019

A087960 := proc(n) op((n mod 4)+1,[1,-1,-1,1]) ; end proc:
A186423 := proc(n) 3*n*(n+1)/8 +3/16 +(-1)^n*(2*n+1)/16 -A087960(n)/4 ; end proc: # R. J. Mathar, Feb 28 2011

CoefficientList[Series[x(1+2x+2x^3+x^4)/((1-x)^3(1+x)^2(1+x^2)),{x, 0, 65}],x]  (* Harvey P. Dale, Mar 13 2011 *)
Table[(6*n^2 +6*n +3 +(-1)^n*(2*n+1) -4*(-1)^Binomial[n+1, 2])/16, {n, 0, 65}] (* G. C. Greubel, Oct 09 2019 *)

vector(66, n, my(m=n-1); (6*m^2 +6*m +3 +(-1)^m*(2*m+1) -4*(-1)^binomial(m+1, 2))/16) \\ G. C. Greubel, Oct 09 2019

def A186423(n): return (6*n*(n+1)+3+(-2*n-1 if n&1 else 2*n+1)+(4 if n+1&2 else -4))>>4 # Chai Wah Wu, Jan 31 2023

[(6*n^2 +6*n +3 +(-1)^n*(2*n+1) -4*(-1)^binomial(n+1, 2))/16 for n in (0..65)] # G. C. Greubel, Oct 09 2019

From R. J. Mathar, Feb 28 2011: (Start)
G.f.: x*(1 + 2*x + 2*x^3 + x^4)/( (1+x^2)*(1+x)^2*(1-x)^3 ).
a(n) = (6*n*(n+1) + 3 + (-1)^n*(2*n+1) - 4*A087960(n))/16. (End)
E.g.f.: ((2 + 5*x + 3*x^2)*cosh(x) + (1 + 7*x + 3*x^2)*sinh(x) + 2*sin(x) - 2*cos(x))/8. - G. C. Greubel, Oct 09 2019

More terms added by G. C. Greubel, Oct 09 2019

A164097 Numbers k such that 6*k + 7 is a perfect square.

Original entry on oeis.org

3, 7, 19, 27, 47, 59, 87, 103, 139, 159, 203, 227, 279, 307, 367, 399, 467, 503, 579, 619, 703, 747, 839, 887, 987, 1039, 1147, 1203, 1319, 1379, 1503, 1567, 1699, 1767, 1907, 1979, 2127, 2203, 2359, 2439, 2603, 2687, 2859, 2947, 3127, 3219, 3407, 3503, 3699

Offset: 1

Vincenzo Librandi, Aug 10 2009

The entries are prime, or divisible by 3, or divisible by prime of the form 3*m+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. A062717, A104777 (the squares 6*k+7).

[n: n in [1..4000] | IsSquare(6*n+7)]; // Vincenzo Librandi, Oct 12 2012

Select[Range[4000], IntegerQ[Sqrt[6 # + 7 ]] &] (* or *) LinearRecurrence[{1, 2, -2, -1, 1}, {3, 7, 19, 27, 47}, 50] (* Harvey P. Dale, Apr 29 2011 *)

From R. J. Mathar, Aug 26 2009: (Start)
a(n) = a(n-1) +2*a(n-2) -2*a(n-3) -a(n-4) +a(n-5).
G.f.: x*(-3-4*x-6*x^2+x^4)/((1+x)^2*(x-1)^3).
a(n) = 3*(2*n-1+2*n^2)/4 -(-1)^n*(1+2*n)/4 = A062717(n+1)-1. (End)
Sum_{n>=1} 1/a(n) = 1 + (tan((2+sqrt(7))*Pi/6) - cot((1+sqrt(7))*Pi/6))*Pi/(2*sqrt(7)). - Amiram Eldar, Feb 24 2023

Edited by R. J. Mathar, Aug 26 2009

A132354 Integers m such that 7*m + 1 is a square.

Original entry on oeis.org

0, 5, 9, 24, 32, 57, 69, 104, 120, 165, 185, 240, 264, 329, 357, 432, 464, 549, 585, 680, 720, 825, 869, 984, 1032, 1157, 1209, 1344, 1400, 1545, 1605, 1760, 1824, 1989, 2057, 2232, 2304, 2489, 2565, 2760, 2840, 3045, 3129, 3344, 3432, 3657, 3749, 3984, 4080

Offset: 0

Mohamed Bouhamida, Nov 08 2007

Numbers of the form m*(7*m + 2) for m = 0, -1, 1, -2, 2, -3, 3, ... - Bruno Berselli, Feb 26 2018

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

Cf. A054000, A056220, A000217, A087475, A028347, A062717, A036666, A002378, A001082, A046092, A005563.

[n^2+n+3*Ceiling(n/2)^2: n in [0..50]]; // Wesley Ivan Hurt, Jul 07 2014

seq(n^2+n+3*ceil(n/2)^2, n=0..48); # Gary Detlefs, Feb 23 2010

f[n_]:=IntegerQ[Sqrt[1+7*n]]; Select[Range[0,8! ],f[ # ]&] (* Vladimir Joseph Stephan Orlovsky, Feb 19 2010 *)

a(2*k) = k*(7*k + 2), a(2*k + 1) = 7*k^2 + 12*k + 5.
a(n) = n^2 + n + 3*ceiling(n/2)^2. - Gary Detlefs, Feb 23 2010
G.f.: -x*(5*x^2 + 4*x + 5)/((x - 1)^3*(x + 1)^2). - Colin Barker, Oct 24 2012
Sum_{n>=1} 1/a(n) = 7/4 - cot(2*Pi/7)*Pi/2. - Amiram Eldar, Mar 15 2022

More terms from Max Alekseyev, Nov 13 2009

Better definition from Max Alekseyev, Oct 24 2012

A323674 Square array, read by antidiagonals, of the positive integers 6cd +-c +-d = (6c +- 1)d +- c. Alternate rows (or columns) are numbers that differ by c from multiples of 6c - 1 or 6c + 1.

Original entry on oeis.org

4, 6, 6, 9, 8, 9, 11, 13, 13, 11, 14, 15, 20, 15, 14, 16, 20, 24, 24, 20, 16, 19, 22, 31, 28, 31, 22, 19, 21, 27, 35, 37, 37, 35, 27, 21, 24, 29, 42, 41, 48, 41, 42, 29, 24, 26, 34, 46, 50, 54, 54, 50, 46, 34, 26, 29, 36, 53, 54, 65, 60, 65, 54, 53, 36, 29, 31, 41, 57, 63, 71, 73, 73, 71, 63, 57, 41, 31

Offset: 1

Sally Myers Moite, Jan 23 2019

This sequence without duplicates is A067611, which is the complement of A002822, the positive integers x for which 6x - 1 and 6x + 1 are twin primes.

			Square array begins:
   4,   6,   9,  11,  14,  16,  19,  21,  24,  26, ...
   6,   8,  13,  15,  20,  22,  27,  29,  34,  36, ...
   9,  13,  20,  24,  31,  35,  42,  46,  53,  57, ...
  11,  15,  24,  28,  37,  41,  50,  54,  63,  67, ...
  14,  20,  31,  37,  48,  54,  65,  71,  82,  88, ...
  16,  22,  35,  41,  54,  60,  73,  79,  92,  98, ...
  19,  27,  42,  50,  65,  73,  88,  96, 111, 119, ...
  21,  29,  46,  54,  71,  79,  96, 104, 121, 129, ...
  24,  34,  53,  63,  82,  92, 111, 121, 140, 150, ...
  26,  36,  57,  67,  88,  98, 119, 129, 150, 160, ...
  ...
Note that, for example, the third row (or column) contains numbers that differ by 2 from multiples of 11 = 6*2 - 1, and the eighth row contains numbers that differ by 4 from multiples of 25 = 6*4 + 1.

The first and second rows are A047209 and A047336.

The diagonal is A062717, the numbers x for which 6*x + 1 is a perfect square.

a(m,n) = 6*floor((m+1)/2)*floor((n+1)/2) + ((-1)^n)*floor((m+1)/2) + ((-1)^m)*floor((n+1)/2);
matrix(7, 7, n, k, a(n, k)) \\ Michel Marcus, Jan 25 2019

a(m,n) = 6*floor((m+1)/2)*floor((n+1)/2) + ((-1)^n)*floor((m+1)/2) + ((-1)^m)*floor((n+1)/2), m,n >= 1.

A247215 Integers k such that 3k+1 and 6k+1 are both squares.

Original entry on oeis.org

0, 8, 280, 9520, 323408, 10986360, 373212840, 12678250208, 430687294240, 14630689753960, 497012764340408, 16883803297819920, 573552299361536880, 19483894374994434008, 661878856450449219400, 22484397224940279025600, 763807626791519037651008

Offset: 1

Casey Leung, Nov 26 2014

			When n=1, a(1)=0, 3(0)+1=1, 6(0)+1=1.
When n=2, a(2)=8, 3(8)+1=25, 6(8)+1=49.
When n=3, a(3)=280, 3(280)+1=841=29^2, 6(280)+1=1681=41^2.
When n=4, a(4)=9520, 3(9520)+1=28560=169^2, 6(9520)+1=57121=239^2.

Colin Barker, Table of n, a(n) for n = 1..650
Index entries for linear recurrences with constant coefficients, signature (35,-35,1).

The common terms of A062717 and A001082.

LinearRecurrence[{35,-35,1},{0,8,280},20] (* Harvey P. Dale, Mar 25 2025 *)

concat(0, Vec(-8*x^2/((x-1)*(x^2-34*x+1)) + O(x^100))) \\ Colin Barker, Nov 26 2014

a(n) = (1/72)*(3*(3*(17-12*sqrt(2))^n+2*sqrt(2)*(17-12*sqrt(2))^n+3*(17+12*sqrt(2))^n-2*sqrt(2)*(17+12*sqrt(2))^n)-18).
From Colin Barker, Nov 26 2014: (Start)
a(n) = 8*A029546(n).
a(n) = 35*a(n-1)-35*a(n-2)+a(n-3).
G.f.: -8*x^2 / ((x-1)*(x^2-34*x+1)).
(End)
Lim_{n -> infinity} a(n+1)/a(n) = 33.970562748... = (1+sqrt(2))^4 (the dominant root of x^2-34*x+1). - Joerg Arndt, Dec 01 2014

More terms from Colin Barker, Nov 26 2014

cp's OEIS Frontend

A152749 a(n) = (n+1)*(3*n+1)/4 for n odd, a(n) = n*(3*n+2)/4 for n even.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Magma

Magma

Maple

Mathematica

Formula

Extensions

A132355 Numbers of the form 9*h^2 + 2*h, for h an integer.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

Extensions

A033580 Four times second pentagonal numbers: a(n) = 2*n*(3*n+1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

A132356 a(2*k) = k*(10*k+2), a(2*k+1) = 10*k^2 + 18*k + 8, with k >= 0.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

Extensions

A132209 a(0) = 0 and a(n) = 2*n^2 + 2*n - 1, for n>=1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

A186423 Partial sums of A186421.

Views

Author

Keywords

Links

Crossrefs

Programs

GAP

Haskell

A152749 a(n) = (n+1)(3n+1)/4 for n odd, a(n) = n(3n+2)/4 for n even.

A132355 Numbers of the form 9h^2 + 2h, for h an integer.

A033580 Four times second pentagonal numbers: a(n) = 2n(3*n+1).

A132356 a(2k) = k(10k+2), a(2k+1) = 10k^2 + 18k + 8, with k >= 0.

A132209 a(0) = 0 and a(n) = 2n^2 + 2n - 1, for n>=1.