A017197 -id:A017197 - cp's OEIS frontend

A017198 a(n) = (9*n + 3)^2.

9, 144, 441, 900, 1521, 2304, 3249, 4356, 5625, 7056, 8649, 10404, 12321, 14400, 16641, 19044, 21609, 24336, 27225, 30276, 33489, 36864, 40401, 44100, 47961, 51984, 56169, 60516, 65025, 69696

Offset: 0

N. J. A. Sloane

a(n) = A000290(A017197(n)) = A156677(n+2) + A017305(n). - Reinhard Zumkeller, Jul 13 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

[(9*n+3)^2: n in [0..35]]; // Vincenzo Librandi, Jul 23 2011

(9*Range[0, 30] + 3)^2 (* or *) LinearRecurrence[{3,-3,1},{9,144,441},30] (* Harvey P. Dale, Dec 25 2015 *)

a(n)=(9*n+3)^2 \\ Charles R Greathouse IV, Jun 17 2017

a(0)=9, a(1)=144, a(2)=441, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Dec 25 2015

G.f.: (-9 - 117*x - 36*x^2) / (x-1)^3. - R. J. Mathar, Jul 14 2016

A031913 a(n) = prime(9*n - 6).

Original entry on oeis.org

5, 37, 73, 113, 167, 223, 269, 317, 379, 433, 487, 557, 607, 659, 727, 787, 853, 911, 977, 1033, 1093, 1163, 1229, 1291, 1367, 1439, 1489, 1559, 1613, 1693, 1753, 1831, 1901, 1987, 2039, 2111, 2179, 2267, 2333, 2383, 2447, 2543, 2621

Offset: 1

Jeff Burch

nonn

Ivan Panchenko, Table of n, a(n) for n = 1..1000

Cf. A000040, A017197, A031911, A031912, A031914.

[ NthPrime(9*n-6): n in [1..1000] ]; // Vincenzo Librandi, Apr 09 2011

Prime[9Range[50]-6] (* Harvey P. Dale, Jul 27 2011 *)

[nth_prime(9*n-6) for n in range(1,61)] # G. C. Greubel, Feb 18 2024

a(n) = A000040(A017197(n-1)). - G. C. Greubel, Feb 18 2024

A145910 a(n) = (1 + 3n)(4 + 3*n)/2.

Original entry on oeis.org

2, 14, 35, 65, 104, 152, 209, 275, 350, 434, 527, 629, 740, 860, 989, 1127, 1274, 1430, 1595, 1769, 1952, 2144, 2345, 2555, 2774, 3002, 3239, 3485, 3740, 4004, 4277, 4559, 4850, 5150, 5459, 5777, 6104, 6440, 6785, 7139, 7502, 7874

Offset: 0

Paul Curtz, Oct 24 2008

R. C. Alperin, A nonlinear recurrence and its relations to Chebyshev polynomials, Fib. Q., Vol. 58, No. 2 (2020), 140-142.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A017197, A085001, A144449.

A145910:=n->(1+3*n)*(4+3*n)/2: seq(A145910(n), n=0..100); # Wesley Ivan Hurt, Jul 25 2017

Table[(1+3n)(4+3n)/2, {n,0,50}] (* Harvey P. Dale, Feb 23 2011 *)

a(n)=(1+3*n)*(4+3*n)/2 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = a(n-1) + 3*(3*n+1) = a(n-1) + A017197(n+1).
G.f.: (-2 - 8*x + x^2)/(x-1)^3. - R. J. Mathar, Jan 06 2011
a(n) = A144449(n)/8.
a(n) = 2*a(n-1) - a(n-2) + 9.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
From Amiram Eldar, Mar 11 2022: (Start)
Sum_{n>=0} 1/a(n) = 2/3.
Sum_{n>=0} (-1)^n/a(n) = 4*Pi/(9*sqrt(3)) + 4*log(2)/9 - 2/3. (End)
From Elmo R. Oliveira, Nov 15 2024: (Start)
E.g.f.: exp(x)*(4 + 24*x + 9*x^2)/2.
a(n) = A085001(n)/2. (End)

Terms a(11)-a(42) from Vincenzo Librandi, Nov 17 2009

A154266 a(n) = 27*n + 12.

Original entry on oeis.org

12, 39, 66, 93, 120, 147, 174, 201, 228, 255, 282, 309, 336, 363, 390, 417, 444, 471, 498, 525, 552, 579, 606, 633, 660, 687, 714, 741, 768, 795, 822, 849, 876, 903, 930, 957, 984, 1011, 1038, 1065, 1092, 1119, 1146, 1173, 1200, 1227, 1254, 1281, 1308, 1335

Offset: 0

Vincenzo Librandi, Jan 06 2009

The identity (81*n^2 + 72*n + 17)^2 - (9*n^2 + 8*n + 2)*(27*n + 12)^2 = 1 can be written as A154295(n+1)^2 - A154262(n+1)*a(n)^2 = 1. - Vincenzo Librandi, Feb 03 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A008585, A017197, A017209, A154262, A154295.

I:=[12, 39]; [n le 2 select I[n] else 2*Self(n-1)-Self(n-2): n in [1..40]]; // Vincenzo Librandi, Feb 02 2012

Range[12, 7000, 27] (* Vladimir Joseph Stephan Orlovsky, Jul 13 2011 *)
LinearRecurrence[{2, -1}, {12, 39}, 50] (* Vincenzo Librandi, Feb 02 2012 *)

a(n)=27*n+12 \\ Charles R Greathouse IV, Dec 28 2011

From R. J. Mathar, Jan 05 2011: (Start)
G.f.: 3*(4 + 5*x)/(1-x)^2.
a(n) = 3*A017209(n). (End)
a(n) = 2*a(n-1) - a(n-2). - Vincenzo Librandi, Feb 02 2012
E.g.f.: (27*x + 12)*exp(x). - G. C. Greubel, Sep 08 2016
a(n) = A017197(3*n+1) = A008585(9*n+4). - Elmo R. Oliveira, Apr 12 2025

119 replaced by 1119 - R. J. Mathar, Jan 07 2009

A354984 Numbers that are 3 * prime powers congruent to 1 (mod 3).

Original entry on oeis.org

12, 21, 39, 48, 57, 75, 93, 111, 129, 147, 183, 192, 201, 219, 237, 291, 309, 327, 363, 381, 417, 453, 471, 489, 507, 543, 579, 597, 633, 669, 687, 723, 768, 813, 831, 849, 867, 921, 939, 993, 1011, 1029, 1047, 1083, 1101, 1119, 1137, 1191, 1227, 1263, 1299, 1317, 1371, 1389, 1461, 1497, 1569, 1587, 1623, 1641, 1713

Offset: 1

Antti Karttunen, Jun 15 2022

nonn

Numbers k of the form 9m+3 such that k/3 = p^k, with p a prime and k >= 1.

Intersection of A017197 and 3*A246655.
Cf. A137827, A354983 (characteristic function).
Row 3 of A354930 (conjectured).

Select[Range[2000], Mod[#, 9] == 3 && PrimePowerQ[#/3] &] (* Amiram Eldar, Jun 15 2022 *)

A354983(n) = ((3==(n%9)) && isprimepower(n/3));
isA354984(n) = A354983(n);

a(n) = 3 * A137827(n).

A264938 a(n) = n(2n-1) + floor(n/3).

Original entry on oeis.org

0, 1, 6, 16, 29, 46, 68, 93, 122, 156, 193, 234, 280, 329, 382, 440, 501, 566, 636, 709, 786, 868, 953, 1042, 1136, 1233, 1334, 1440, 1549, 1662, 1780, 1901, 2026, 2156, 2289, 2426, 2568, 2713, 2862, 3016, 3173, 3334, 3500, 3669, 3842, 4020, 4201, 4386, 4576, 4769

Offset: 0

Paul Curtz, Nov 29 2015

Sequence extended to the left:
..., 133, 102, 76, 53, 34, 20, 9, 2, 0, 1, 6, 16, 29, 46, 68, 93, ...
Conjecture: after 0, a(n) provides the first bisection of A264041.
Conjecture: 2, 9, 20, 34, 53, 76, 102, 133, ... is A248121.

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

Second bisection of A260708.
Cf. A000217, A002264, A016777, A016969, A017197, A017641, A099837, A248121, A256320, A260699, A264041.
Cf. A171452: A000217(n-1) + A002264(n).

[n*(2*n-1)+Floor(n/3): n in [0..60]]; // Vincenzo Librandi, Dec 02 2015

seq(n*(2*n-1) + floor(n/3), n=0..100); # Robert Israel, Dec 02 2015

Table[n (2 n - 1) + Floor[n/3], {n, 0, 50}] (* Vincenzo Librandi, Dec 02 2015 *)
LinearRecurrence[{2,-1,1,-2,1},{0,1,6,16,29},60] (* Harvey P. Dale, Oct 13 2020 *)

concat(0, Vec(x*(1+x)^2*(1+2*x)/((1-x)^3*(1+x+x^2)) + O(x^100))) \\ Colin Barker, Dec 02 2015

a(n) = n*(2*n-1) + n\3; \\ Altug Alkan, Dec 01 2015

a(n) = a(n-3) + 12*n - 20 for n>2.
From Colin Barker, Dec 02 2015: (Start)
a(n) = 2*a(n-1) - a(n-2) + a(n-3) - 2*a(n-4) + a(n-5) for n>4.
G.f.: x*(1+x)^2*(1+2*x) / ((1-x)^3*(1+x+x^2)).
(End)
a(n) = A000217(2n-1) + A002264(n).
a(n) + a(-n) = 3*A256320(n).
a(n +8) - a(n -7) = 20*A016777(n).
a(n+16) - a(n-14) = 20*A016969(n).
a(n+23) - a(n-22) = 20*A017197(n).
a(n+31) - a(n-29) = 20*A017641(n).
Generalization of the previous four formulas:
a(n+30*k +8) - a(n-30*k -7) = 20*(4*k+1)*(3*n+1).
a(n+30*k+16) - a(n-30*k-14) = 20*(2*k+1)*(6*n+5).
a(n+30*k+24) - a(n-30*k-21) = 20*(4*k+3)*(3*n+4).
a(n+30*k+32) - a(n-30*k-28) = 20*(2*k+2)*(6*n+11).
E.g.f.: (6*x^2+4*x-1)*exp(x)/3 + (cos(sqrt(3)*x/2)/3 +sqrt(3)*sin(sqrt(3)*x/2)/9)*exp(-x/2). - Robert Israel, Dec 02 2015

Edited by Bruno Berselli, Dec 01 2015

A304504 a(n) = 3(3n+1)(9n+8)/2.

Original entry on oeis.org

12, 102, 273, 525, 858, 1272, 1767, 2343, 3000, 3738, 4557, 5457, 6438, 7500, 8643, 9867, 11172, 12558, 14025, 15573, 17202, 18912, 20703, 22575, 24528, 26562, 28677, 30873, 33150, 35508, 37947, 40467, 43068, 45750, 48513, 51357, 54282, 57288, 60375, 63543, 66792

Offset: 0

Emeric Deutsch, May 13 2018

The second Zagreb index of the single-defect 3-gonal nanocone CNC(3,n) (see definition in the Doslic et al. reference, p. 27).
The second Zagreb index of a simple connected graph is the sum of the degree products d(i)d(j) over all edges ij of the graph.
The M-polynomial of CNC(3,n) is M(CNC(3,n); x,y) = 3*x^2*y^2 + 6*n*x^2*y^3 + 3*n*(3*n+1)*x^3*y^3/2.
More generally, the M-polynomial of CNC(k,n) is M(CNC(k,n); x,y) = k*x^2*y^2 + 2*k*n*x^2*y^3 + k*n*(3*n + 1)*x^3*y^3/2.
8*a(n) + 25 is a square. - Bruno Berselli, May 14 2018

Colin Barker, Table of n, a(n) for n = 0..1000
E. Deutsch and Sandi Klavzar, M-polynomial and degree-based topological indices, Iranian Journal of Mathematical Chemistry, Vol. 6, No. 2, 2015, pp. 93-102.
T. Doslic and M. Saheli, Augmented eccentric connectivity index of single-defect nanocones, Journal of Mathematical Nanoscience, Vol. 1, No. 1, 2011, pp. 25-31.
A. Khaksar, M. Ghorbani, and H. R. Maimani, On atom bond connectivity and GA indices of nanocones, Optoelectronics and Advanced Materials - Rapid Communications, Vol. 4, No. 11, 2010, pp. 1868-1870.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A017197, A017257, A304503.

seq((1/2)*(3*(9*n+8))*(3*n+1), n = 0 .. 40);

Vec(3*(4 + 22*x + x^2) / (1 - x)^3 + O(x^40)) \\ Colin Barker, May 14 2018

From Colin Barker, May 14 2018: (Start)
G.f.: 3*(4 + 22*x + x^2)/(1 - x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)
From Elmo R. Oliveira, Nov 15 2024: (Start)
E.g.f.: 3*exp(x)*(8 + 60*x + 27*x^2)/2.
a(n) = A017197(n)*A017257(n)/2. (End)

A017199 a(n) = (9*n + 3)^3.

Original entry on oeis.org

27, 1728, 9261, 27000, 59319, 110592, 185193, 287496, 421875, 592704, 804357, 1061208, 1367631, 1728000, 2146689, 2628072, 3176523, 3796416, 4492125, 5268024, 6128487, 7077888, 8120601, 9261000, 10503459, 11852352, 13312053, 14886936, 16581375, 18399744, 20346417

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A002117, A016779, A017197.

[(9*n+3)^3: n in [0..30]]; // Vincenzo Librandi, Jul 23 2011

(9Range[0,30]+3)^3 (* or *) LinearRecurrence[{4,-6,4,-1},{27,1728,9261,27000},30] (* Harvey P. Dale, Jun 20 2024 *)

G.f.: 27*(1 + 60*x + 93*x^2 + 8*x^3)/ (x-1)^4. - R. J. Mathar, Aug 01 2014
a(n) = 27*A016779(n). - R. J. Mathar, Aug 01 2014
From Amiram Eldar, Oct 03 2024: (Start)
a(n) = A017197(n)^3.
Sum_{n>=0} 1/a(n) = 2*Pi^3/(2187*sqrt(3)) + 13*zeta(3)/729. (End)

A017200 a(n) = (9*n+3)^4.

Original entry on oeis.org

81, 20736, 194481, 810000, 2313441, 5308416, 10556001, 18974736, 31640625, 49787136, 74805201, 108243216, 151807041, 207360000, 276922881, 362673936, 466948881, 592240896, 741200625, 916636176

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1)

Cf. A000583 (n^4), A016780 ((3n+1)^4), A017197 (9n+3).

[(9*n+3)^4: n in [0..30]]; // Vincenzo Librandi, Jul 23 2011

Table[(9n+3)^4,{n,0,100}] (* Mohammad K. Azarian, Jun 15 2016 *)
LinearRecurrence[{5,-10,10,-5,1},{81,20736,194481,810000,2313441},20] (* Harvey P. Dale, May 26 2023 *)

a(n) = A000583(A017197(n)). - Michel Marcus, Nov 06 2015
a(n) = 81*A016780(n). - Michel Marcus, Nov 06 2015
From Ilya Gutkovskiy, Jun 16 2016: (Start)
G.f.: 81*(1 + 251*x + 1131*x^2 + 545*x^3 + 16*x^4)/(1 - x)^5.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). (End)

A017201 a(n) = (9*n + 3)^5.

Original entry on oeis.org

243, 248832, 4084101, 24300000, 90224199, 254803968, 601692057, 1252332576, 2373046875, 4182119424, 6956883693, 11040808032, 16850581551, 24883200000, 35723051649, 50049003168, 68641485507, 92389579776, 122298103125, 159494694624, 205236901143, 260919263232

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A013663, A016781, A017197.

[(9*n+3)^5: n in [0..30]]; // Vincenzo Librandi, Jul 23 2011

(9*Range[0,20]+3)^5 (* or *) LinearRecurrence[{6,-15,20,-15,6,-1},{243,248832,4084101,24300000,90224199,254803968},20] (* Harvey P. Dale, Jul 27 2019 *)

G.f.: 243*(1 + 1018*x + 10678*x^2 + 14498*x^3 + 2933*x^4 + 32*x^5)/(x-1)^6. - R. J. Mathar, Jul 14 2016
From Amiram Eldar, Oct 03 2024: (Start)
a(n) = A017197(n)^5 = 3^5 * A016781(n).
Sum_{n>=0} 1/a(n) = 2*Pi^5/(177147*sqrt(3)) + 121*zeta(5)/59049. (End)

cp's OEIS Frontend

A017198 a(n) = (9*n + 3)^2.

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A031913 a(n) = prime(9*n - 6).

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A145910 a(n) = (1 + 3*n)*(4 + 3*n)/2.

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A154266 a(n) = 27*n + 12.

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A354984 Numbers that are 3 * prime powers congruent to 1 (mod 3).

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A264938 a(n) = n*(2*n-1) + floor(n/3).

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A304504 a(n) = 3*(3*n+1)*(9*n+8)/2.

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A145910 a(n) = (1 + 3n)(4 + 3*n)/2.

A264938 a(n) = n(2n-1) + floor(n/3).

A304504 a(n) = 3(3n+1)(9n+8)/2.