A132355 -id:A132355 - cp's OEIS frontend

A147601 First differences of A132355.

7, 4, 21, 8, 35, 12, 49, 16, 63, 20, 77, 24, 91, 28, 105, 32, 119, 36, 133, 40, 147, 44, 161, 48, 175, 52, 189, 56, 203, 60, 217, 64, 231, 68, 245, 72, 259, 76, 273, 80, 287, 84, 301, 88, 315, 92, 329, 96, 343, 100, 357

Offset: 0

Paul Curtz, Nov 08 2008

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

Cf. A132355, A147587.

[(9 + 5*(-1)^n)*(1+n)/2: n in [0..60]]; // G. C. Greubel, Oct 25 2022

Differences[Sort[Table[9n^2+2n,{n,-30,30}]]] (* or *) LinearRecurrence[ {0,2,0,-1},{7,4,21,8},60] (* Harvey P. Dale, Jun 04 2017 *)

[(9 + 5*(-1)^n)*(1+n)/2 for n in range(61)] # G. C. Greubel, Oct 25 2022

G.f.: ( 7+4*x+7*x^2 ) / ((1-x)*(1+x))^2. - R. J. Mathar, Jun 21 2011
a(2*n) = A147587(n).
a(2*n+1) = 4*(n+1).
a(n) = (9 + 5*(-1)^n)*(1+n)/2. - Colin Barker, Jun 06 2012
E.g.f.: (1/2)*(5*(1-x)*exp(-x) + 9*(1+x)*exp(x)). - G. C. Greubel, Oct 25 2022

A218864 Numbers of the form 9k^2 + 8k, k an integer.

Original entry on oeis.org

0, 1, 17, 20, 52, 57, 105, 112, 176, 185, 265, 276, 372, 385, 497, 512, 640, 657, 801, 820, 980, 1001, 1177, 1200, 1392, 1417, 1625, 1652, 1876, 1905, 2145, 2176, 2432, 2465, 2737, 2772, 3060, 3097, 3401, 3440, 3760, 3801, 4137, 4180, 4532, 4577, 4945, 4992

Offset: 1

Jason Kimberley, Nov 08 2012

Numbers m such that 9*m + 16 is a square. - Vincenzo Librandi, Apr 07 2013
Equivalently, integers of the form h*(h + 8)/9 (nonnegative values of h are listed in A090570). - Bruno Berselli, Jul 15 2016
Generalized 20-gonal (or icosagonal) numbers: r*(9*r - 8) with r = 0, +1, -1, +2, -2, +3, -3, ... - Omar E. Pol, Jun 06 2018
Partial sums of A317316. - Omar E. Pol, Jul 28 2018
Exponents in expansion of Product_{n >= 1} (1 + x^(18*n-17))*(1 + x^(18*n-1))*(1 - x^(18*n)) = 1 + x + x^17 + x^20 + x^52 + .... - Peter Bala, Dec 10 2020

Jason Kimberley, Table of n, a(n) for n = 1..2000
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 C(q).
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Characteristic function is A205987.
Numbers of the form 9*m^2+k*m, for integer n: A016766 (k=0), A132355 (k=2), A185039 (k=4), A057780 (k=6), this sequence (k=8).
Cf. A074377 (numbers m such that 16*m+9 is a square).
Cf. A317316.
For similar sequences of numbers m such that 9*m+i is a square, see list in A266956.
Cf. sequences of the form m*(m+i)/(i+1) listed in A274978. [Bruno Berselli, Jul 25 2016]
Sequences of generalized k-gonal numbers: A001318 (k=5), A000217 (k=6), A085787 (k=7), A001082 (k=8), A118277 (k=9), A074377 (k=10), A195160 (k=11), A195162 (k=12), A195313 (k=13), A195818 (k=14), A277082 (k=15), A274978 (k=16), A303305 (k=17), A274979 (k=18), A303813 (k=19), this sequence (k=20), A303298 (k=21), A303299 (k=22), A303303 (k=23), A303814 (k=24), A303304 (k=25), A316724 (k=26), A316725 (k=27), A303812 (k=28), A303815 (k=29), A316729 (k=30).

a:=func; [0]cat[a(n*m): m in [-1,1], n in [1..20]];

Array[(18 # (# - 1) - 7 (-1)^#*(2 # - 1) - 7)/8 &, 48] (* or *)
CoefficientList[Series[x (1 + 16 x + x^2)/((1 + x)^2*(1 - x)^3), {x, 0, 47}], x] (* Michael De Vlieger, Jun 06 2018 *)

a(n) = (18*n*(n - 1) - 7*(-1)^n*(2*n - 1) - 7)/8. - Bruno Berselli, Nov 13 2012
G.f.: x*(1 + 16*x + x^2)/((1 + x)^2*(1 - x)^3). - Bruno Berselli, Nov 14 2012
Sum_{n>=2} 1/a(n) = (9 + 8*Pi*cot(Pi/9))/64. - Amiram Eldar, Feb 28 2022

A056020 Numbers that are congruent to +-1 mod 9.

Original entry on oeis.org

1, 8, 10, 17, 19, 26, 28, 35, 37, 44, 46, 53, 55, 62, 64, 71, 73, 80, 82, 89, 91, 98, 100, 107, 109, 116, 118, 125, 127, 134, 136, 143, 145, 152, 154, 161, 163, 170, 172, 179, 181, 188, 190, 197, 199, 206, 208, 215, 217, 224, 226, 233, 235, 242, 244, 251, 253

Offset: 1

Robert G. Wilson v, Jun 08 2000

Or, numbers k such that k^2 == 1 (mod 9).

Or, numbers k such that the iterative cycle j -> sum of digits of j^2 when started at k contains a 1. E.g., 8 -> 6+4 = 10 -> 1+0+0 = 1 and 17 -> 2+8+9 = 19 -> 3+6+1 = 10 -> 1+0+0 = 1. - Asher Auel, May 17 2001

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

Cf. A007953, A047522 (n=1 or 7 mod 8), A090771 (n=1 or 9 mod 10).
Cf. A129805 (primes), A195042 (partial sums).
Cf. A005408, A019676, A019968, A047209, A007310, A047336, A175885, A091998, A175886, A113801, A175887, A332437.
Cf. A381319 (general case mod n^2).

a056020 n = a056020_list !! (n-1)
a05602_list = 1 : 8 : map (+ 9) a056020_list
-- Reinhard Zumkeller, Jan 07 2012

Select[ Range[ 300 ], PowerMod[ #, 2, 3^2 ]==1& ]
(* or *)
LinearRecurrence[{1, 1, -1}, {1, 8, 10}, 67] (* Mike Sheppard, Feb 18 2025 *)

a(n)=9*(n>>1)+if(n%2,1,-1) \\ Charles R Greathouse IV, Jun 29 2011

for(n=1,40,print1(9*n-8,", ",9*n-1,", ")) \\ Charles R Greathouse IV, Jun 29 2011

a(1) = 1; a(n) = 9(n-1) - a(n-1). - Rolf Pleisch, Jan 31 2008 [Offset corrected by Jon E. Schoenfield, Dec 22 2008]
From R. J. Mathar, Feb 10 2008: (Start)
O.g.f.: 1 + 5/(4(x+1)) + 27/(4(-1+x)) + 9/(2(-1+x)^2).
a(n+1) - a(n) = A010697(n). (End)
a(n) = (9*A132355(n) + 1)^(1/2). - Gary Detlefs, Feb 22 2010
From Bruno Berselli, Nov 17 2010: (Start)
a(n) = a(n-2) + 9, for n > 2.
a(n) = 9*A000217(n-1) + 1 - 2*Sum_{i=1..n-1} a(i), n > 1. (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = (Pi/9)*cot(Pi/9) = A019676 * A019968. - Amiram Eldar, Dec 04 2021
E.g.f.: 1 + ((18*x - 9)*exp(x) + 5*exp(-x))/4. - David Lovler, Sep 04 2022
From Amiram Eldar, Nov 22 2024: (Start)
Product_{n>=1} (1 - (-1)^n/a(n)) = 2*cos(Pi/9) (A332437).
Product_{n>=2} (1 + (-1)^n/a(n)) = (Pi/9)*cosec(Pi/9). (End)
From Mike Sheppard, Feb 18 2025: (Start)
a(n) = a(n-1) + a(n-2) - a(n-3).
a(n) ~ (3^2/2)*n. (End)

A016766 a(n) = (3*n)^2.

Original entry on oeis.org

0, 9, 36, 81, 144, 225, 324, 441, 576, 729, 900, 1089, 1296, 1521, 1764, 2025, 2304, 2601, 2916, 3249, 3600, 3969, 4356, 4761, 5184, 5625, 6084, 6561, 7056, 7569, 8100, 8649, 9216, 9801, 10404, 11025, 11664, 12321, 12996, 13689, 14400, 15129, 15876, 16641, 17424

Offset: 0

N. J. A. Sloane

Number of edges of the complete tripartite graph of order 6n, K_n, n, 4n. - Roberto E. Martinez II, Jan 07 2002
Area of a square with side 3n. - Wesley Ivan Hurt, Sep 24 2014
Right-hand side of the binomial coefficient identity Sum_{k = 0..3*n} (-1)^(n+k+1)* binomial(3*n,k)*binomial(3*n + k,k)*(3*n - k) = a(n). - Peter Bala, Jan 12 2022

Ivan Panchenko, Table of n, a(n) for n = 0..200
John Elias, Illustration of initial terms.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Numbers of the form 9*n^2 + k*n, for integer n: this sequence (k = 0), A132355 (k = 2), A185039 (k = 4), A057780 (k = 6), A218864 (k = 8). - Jason Kimberley, Nov 09 2012

Cf. A000217, A000290, A008585, A033428, A051624, A086089, A195042.

[(3*n)^2 : n in [0..50]]; // Wesley Ivan Hurt, Sep 24 2014

A016766:=n->(3*n)^2: seq(A016766(n), n=0..50); # Wesley Ivan Hurt, Sep 24 2014

(3Range[0, 49])^2 (* Alonso del Arte, Sep 24 2014 *)

A016766(n):=(3*n)^2$
makelist(A016766(n),n,0,20); /* Martin Ettl, Nov 12 2012 */

a(n)=9*n^2 \\ Charles R Greathouse IV, Sep 28 2015

a(n) = 9*n^2 = 9*A000290(n). - Omar E. Pol, Dec 11 2008
a(n) = 3*A033428(n). - Omar E. Pol, Dec 13 2008
a(n) = a(n-1) + 9*(2*n-1) for n > 0, a(0)=0. - Vincenzo Librandi, Nov 19 2010
From Wesley Ivan Hurt, Sep 24 2014: (Start)
G.f.: 9*x*(1 + x)/(1 - x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3), n >= 3.
a(n) = A000290(A008585(n)). (End)
From Amiram Eldar, Jan 25 2021: (Start)
Sum_{n>=1} 1/a(n) = Pi^2/54.
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/108.
Product_{n>=1} (1 + 1/a(n)) = sinh(Pi/3)/(Pi/3).
Product_{n>=1} (1 - 1/a(n)) = sinh(Pi/2)/(Pi/2) = 3*sqrt(3)/(2*Pi) (A086089). (End)
a(n) = A051624(n) + 8*A000217(n).  In general, if P(k,n) = the k-th n-gonal number, then (k*n)^2 = P(k^2 + 3,n) + (k^2 - 1)*A000217(n). - Charlie Marion, Mar 09 2022
From Elmo R. Oliveira, Nov 30 2024: (Start)
E.g.f.: 9*x*(1 + x)*exp(x).
a(n) = n*A008591(n) = A195042(2*n). (End)

More terms from Zerinvary Lajos, May 30 2006

A219257 Numbers k such that 11*k+1 is a square.

Original entry on oeis.org

0, 9, 13, 40, 48, 93, 105, 168, 184, 265, 285, 384, 408, 525, 553, 688, 720, 873, 909, 1080, 1120, 1309, 1353, 1560, 1608, 1833, 1885, 2128, 2184, 2445, 2505, 2784, 2848, 3145, 3213, 3528, 3600, 3933, 4009, 4360, 4440, 4809, 4893, 5280, 5368, 5773, 5865

Offset: 1

Bruno Berselli, Nov 16 2012

Equivalently, numbers of the form m*(11*m+2), where m = 0,-1,1,-2,2,-3,3,...

Also, integer values of h*(h+2)/11.

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

Cf. numbers k such that h*k+1 is a square: A005563 (h=1), A046092 (h=2), A001082 (h=3), A002378 (h=4), A036666 (h=5), A062717 (h=6), A132354 (h=7), A000217 (h=8), A132355 (h=9), A132356 (h=10), A152749 (h=12), A219389 (h=13), A219390 (h=14), A204221 (h=15), A074378 (h=16), A219394 (h=17), A219395 (h=18), A219396 (h=19), A219190 (h=20), A219391 (h=21), A219392 (h=22), A219393 (h=23), A001318 (h=24), A219259 (h=25), A217441 (h=26), A219258 (h=27), A219191 (h=28).

Cf. A175885 (square roots of 11*a(n)+1).

[n: n in [0..7000] | IsSquare(11*n+1)];

I:=[0,9,13,40,48]; [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, Aug 18 2013

Select[Range[0, 7000], IntegerQ[Sqrt[11 # + 1]] &]
CoefficientList[Series[x (9 + 4 x + 9 x^2)/((1 + x)^2 (1 - x)^3), {x, 0, 50}], x] (* Vincenzo Librandi, Aug 18 2013 *)

G.f.: x^2*(9+4*x+9*x^2)/((1+x)^2*(1-x)^3).
a(n) = a(-n+1) = (22*n*(n-1)+7*(-1)^n*(2*n-1)-1)/8 + 1 = (1/176)*(22*n+7*(-1)^n-15)*(22*n+7*(-1)^n-7).
Sum_{n>=2} 1/a(n) = 11/4 - cot(2*Pi/11)*Pi/2. - Amiram Eldar, Mar 15 2022

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

A185039 Numbers of the form 9m^2 + 4m, m an integer.

Original entry on oeis.org

0, 5, 13, 28, 44, 69, 93, 128, 160, 205, 245, 300, 348, 413, 469, 544, 608, 693, 765, 860, 940, 1045, 1133, 1248, 1344, 1469, 1573, 1708, 1820, 1965, 2085, 2240, 2368, 2533, 2669, 2844, 2988, 3173, 3325, 3520, 3680, 3885, 4053, 4268, 4444, 4669, 4853, 5088

Offset: 1

N. J. A. Sloane, Feb 04 2012

Also, numbers m such that 9*m+4 is a square. After 0, therefore, there are no squares in this sequence. - Bruno Berselli, Jan 07 2016

Bruno Berselli, Table of n, a(n) for n = 1..1000
S. Cooper and M. D. Hirschhorn, Results of Hurwitz type for three squares. Discrete Math. 274 (2004), no. 1-3, 9-24. See B(q).
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Characteristic function is A205809.
Numbers of the form 9*n^2+k*n, for integer n: A016766 (k=0), A132355 (k=2), this sequence (k=4), A057780 (k=6), A218864 (k=8). [Jason Kimberley, Nov 08 2012]
For similar sequences of numbers m such that 9*m+k is a square, see list in A266956.

[0] cat &cat[[9*n^2-4*n,9*n^2+4*n]: n in [1..32]]; // Bruno Berselli, Feb 04 2011

CoefficientList[Series[x*(5+8*x+5*x^2)/((x+1)^2*(1-x)^3), {x,0,50}], x] (* G. C. Greubel, Jun 20 2017 *)
LinearRecurrence[{1,2,-2,-1,1},{0,5,13,28,44},50] (* Harvey P. Dale, Jan 23 2018 *)

x='x+O('x^50); Vec(x*(5+8*x+5*x^2)/((x+1)^2*(1-x)^3)) \\ G. C. Greubel, Jun 20 2017

From Bruno Berselli, Feb 04 2012: (Start)
G.f.: x*(5+8*x+5*x^2)/((x+1)^2*(1-x)^3).
a(n) = a(-n+1) = (18*n*(n-1)+(2*n-1)*(-1)^n+1)/8 = A004526(n)*A156638(n). (End).

A266956 Numbers m such that 9*m+7 is a square.

Original entry on oeis.org

1, 2, 18, 21, 53, 58, 106, 113, 177, 186, 266, 277, 373, 386, 498, 513, 641, 658, 802, 821, 981, 1002, 1178, 1201, 1393, 1418, 1626, 1653, 1877, 1906, 2146, 2177, 2433, 2466, 2738, 2773, 3061, 3098, 3402, 3441, 3761, 3802, 4138, 4181, 4533, 4578, 4946, 4993, 5377, 5426

Offset: 1

Bruno Berselli, Jan 07 2016

Equivalently, numbers of the form h*(9*h+8)+1, where h = 0, -1, 1, -2, 2, -3, 3, -4, 4, ...
Also, integer values of k*(k+8)/9 plus 1.
It is easy to see that the Diophantine equation 9*x+3*j+1 = y^2 has infinitely many solutions in integers (x,y) for any j in Z. It follows a table with j = -5..5:
...
j = -5, x:  2, 7, 15, 30, 46, 71,  95, 130, 162, 207, 247, ...
j = -4, x:  3, 4, 20, 23, 55, 60, 108, 115, 179, 188, 268, ...
j = -3, x:  1, 8, 12, 33, 41, 76,  88, 137, 153, 216, 236, ...
j = -2, x:  1, 6, 14, 29, 45, 70,  94, 129, 161, 206, 246, ...
j = -1, x:  2, 3, 19, 22, 54, 59, 107, 114, 178, 187, 267, ...
j =  0, x:  0, 7, 11, 32, 40, 75,  87, 136, 152, 215, 235, ... (A132355)
j =  1, x:  0, 5, 13, 28, 44, 69,  93, 128, 160, 205, 245, ... (A185039)
j =  2, x:  1, 2, 18, 21, 53, 58, 106, 113, 177, 186, 266, ... (A266956)
j =  3, x: -1, 6, 10, 31, 39, 74,  86, 135, 151, 214, 234, ... (A266957)
j =  4, x: -1, 4, 12, 27, 43, 68,  92, 127, 159, 204, 244, ... (A266958)
j =  5, x:  0, 1, 17, 20, 52, 57, 105, 112, 176, 185, 265, ... (A218864)
...
The general closed form of these sequences is:
b(n,j) = (18*(n-1)*n + s(j)*(2*n-1)*(-1)^n + t(j))/8, where s(j) = 6*(-j) + 18*floor(j/3) - (-1)^floor(2*(j+1)/3) + 6 and t(j) = 4*(-j) + 4*floor((j+1)/3) + 5.
a(2m) - a(2m-1) gives the odd numbers (A005408); a(2m+1) - a(2m) gives the multiples of 16 (A008598).

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

Cf. numbers m such that 9*m+i: A132355 (i=1), A185039 (i=4), this sequence (i=7), A005563 (i=9), A266957 (i=10), A266958 (i=13), A218864 (i=16), A008865 (i=18, without -2).
Cf. A156638: square roots of 9*a(n)+7.
Cf. A005408, A008598.

Magma
```
[n: n in [0..6000] | IsSquare(9*n+7)];
```

[(18*(n-1)*n-7*(2*n-1)*(-1)^n+1)/8: n in [1..50]];

Select[Range[0, 6000], IntegerQ[Sqrt[9 # + 7]] &]
Table[(18 (n - 1) n - 7 (2 n - 1) (-1)^n + 1)/8, {n, 1, 50}]

for(n=0, 6000, if(issquare(9*n+7), print1(n, ", ")))

vector(50, n, n; (18*(n-1)*n-7*(2*n-1)*(-1)^n+1)/8)

from gmpy2 import is_square
[n for n in range(6000) if is_square(9*n+7)]

[(18*(n-1)*n-7*(2*n-1)*(-1)**n+1)/8 for n in range(1, 60)]

[n for n in (0..6000) if is_square(9*n+7)]

[(18*(n-1)*n-7*(2*n-1)*(-1)^n+1)/8 for n in (1..50)]

G.f.: x*(1 + x + 14*x^2 + x^3 + x^4)/((1 + x)^2*(1 - x)^3).
a(n) = a(-n+1) = (18*(n-1)*n - 7*(2*n-1)*(-1)^n + 1)/8.
a(n) = A218864(n) + 1.

A057780 Multiples of 3 that are one less than a perfect square.

Original entry on oeis.org

0, 3, 15, 24, 48, 63, 99, 120, 168, 195, 255, 288, 360, 399, 483, 528, 624, 675, 783, 840, 960, 1023, 1155, 1224, 1368, 1443, 1599, 1680, 1848, 1935, 2115, 2208, 2400, 2499, 2703, 2808, 3024, 3135, 3363, 3480, 3720, 3843, 4095, 4224, 4488, 4623, 4899, 5040

Offset: 1

Benjamin Geiger (benjamin_geiger(AT)yahoo.com), Nov 02 2000

Also, numbers of the form 9*m^2+6*m, m an integer. - Jason Kimberley, Nov 08 2012
k is in this list iff k+1 is in the support of A033684. - Jason Kimberley, Nov 13 2012
Exponents in the expansion of Product_{n >= 1} (1 - q^(6*n))^2 * (1 - q ^(9*n)) * (1 - q^(36*n))/((1 - q^(3*n))*(1 - q^(12*n))*(1 - q^(18*n))) = 1 + q^3 + q^15 + q^24 + q^48 + q^63 + q^99 + ... (see Oliver, Theorem 1.1). - Peter Bala, Jan 06 2025

Jason Kimberley, Table of n, a(n) for n = 1..2001
Robert J. Lemke Oliver, Eta quotients and theta functions, Advances in Mathematics, Vol. 241, Jul. 2013, pp. 1-17.
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Numbers of the form 9n^2+kn, for integer n: A016766 (k=0), A132355 (k=2), A185039 (k=4), this sequence (k=6), A218864 (k=8). - Jason Kimberley, Nov 08 2012

a:=func;[0]cat[a(n*m):m in[-1, 1],n in[1..24]]; // Jason Kimberley, Nov 09 2012

Select[3*Range[0,2000],IntegerQ[Sqrt[#+1]]&] (* or *) LinearRecurrence[ {1,2,-2,-1,1},{0,3,15,24,48},50] (* Harvey P. Dale, Sep 10 2019 *)

concat(0, Vec(3*x^2*(1+4*x+x^2)/((1-x)^3*(1+x)^2) + O(x^100))) \\ Colin Barker, Dec 26 2015

a(n) = A001651(n)^2 - 1 = 3 * A001082(n).
G.f.: 3*x^2*(1+4*x+x^2) / ((1-x)^3*(1+x)^2). - Colin Barker, Nov 24 2012
From Colin Barker, Dec 26 2015: (Start)
a(n) = 3/8*(6*n^2-2*((-1)^n+3)*n+(-1)^n-1).
a(n) = a(n-1)+2*a(n-2)-2*a(n-3)-a(n-4)+a(n-5) for n>5. (End)

Since this is a list, offset corrected to 1 by Jason Kimberley, Nov 09 2012

A147587 a(n) = 14*n + 7.

Original entry on oeis.org

7, 21, 35, 49, 63, 77, 91, 105, 119, 133, 147, 161, 175, 189, 203, 217, 231, 245, 259, 273, 287, 301, 315, 329, 343, 357, 371, 385, 399, 413, 427, 441, 455, 469, 483, 497, 511, 525, 539, 553, 567, 581, 595, 609, 623, 637, 651, 665, 679, 693, 707, 721, 735

Offset: 0

Paul Curtz, Nov 08 2008

a(n+3) = 14*n + 49 is the sum of seven consecutive odd numbers starting with 2*n+1. - Wesley Ivan Hurt, Apr 11 2015
Numbers k such that 3^k + 1 is divisible by 547. - Bruno Berselli, Aug 22 2018
Sum of the numbers from 2*(n-1) to 2*(n+2). - Bruno Berselli, Oct 25 2018

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

Cf. A005408, A008596, A131877, A132355, A132744.

[14*n+7 : n in [0..100]]; // Wesley Ivan Hurt, Apr 11 2015

A147587:=n->14*n+7: seq(A147587(n), n=0..100); # Wesley Ivan Hurt, Apr 11 2015

Range[7, 1000, 14] (* Vladimir Joseph Stephan Orlovsky, May 31 2011 *)

a(n)=14*n+7 \\ Charles R Greathouse IV, Oct 16 2015

a(n) = a(n-1) + 14.
a(n) = A132355(2*n+2) - A132355(2*n+1) = 7*A005408(n).
a(n) = 28*n - a(n-1) for n>0, a(0)=7. - Vincenzo Librandi, Nov 24 2010
From Wesley Ivan Hurt, Apr 11 2015: (Start)
G.f.: 7*(1 + x)/(1 - x)^2.
a(n) = 2*a(n-1) - a(n-2). (End)
Sum_{n>=0} (-1)^n/a(n) = Pi/28 (A132744). - Amiram Eldar, Dec 13 2021
From Amiram Eldar, Nov 25 2024: (Start)
Product_{n>=0} (1 - (-1)^n/a(n)) = sqrt(2)*sin(3*Pi/14).
Product_{n>=0} (1 + (-1)^n/a(n)) = sqrt(2)*cos(3*Pi/14). (End)
a(n) = (n+4)^2 - (n-3)^2. - Alexander Yutkin, Mar 16 2025
E.g.f.: 7*exp(x)*(1 + 2*x). - Stefano Spezia, Mar 18 2025

More terms from Vincenzo Librandi, Oct 23 2009

cp's OEIS Frontend

A147601 First differences of A132355.

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A218864 Numbers of the form 9*k^2 + 8*k, k an integer.

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A056020 Numbers that are congruent to +-1 mod 9.

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A016766 a(n) = (3*n)^2.

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A219257 Numbers k such that 11*k+1 is a square.

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

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A185039 Numbers of the form 9*m^2 + 4*m, m an integer.

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A218864 Numbers of the form 9k^2 + 8k, k an integer.

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

A185039 Numbers of the form 9m^2 + 4m, m an integer.