A017257 -id:A017257 - cp's OEIS frontend

A224829 Numbers m, such that there is no solution m = x + y + 3*z, with triangular numbers x, y, z.

8, 17, 26, 35, 44, 53, 62, 71, 77, 80, 89, 98, 107, 116, 125, 134, 143, 152, 158, 161, 170, 179, 188, 197, 206, 215, 224, 233, 239, 242, 251, 260, 269, 278, 287, 296, 305, 314, 320, 323, 332, 341, 350, 359, 368, 377, 386, 395, 401, 404, 413, 422, 431, 440

Offset: 1

Reinhard Zumkeller, Jul 21 2013

nonn

A224823(a(n)) = 0;

terms not of the form 9*k+8: 77,158,239,320,401,482,563,644,698,... .

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A017257 (subsequence), A000217.

a224829 n = a224829_list !! n
a224829_list = filter ((== 0) . a224823) [0..]

A350515 a(n) = (n-1)/3 if n mod 3 = 1; a(n) = n/2 if n mod 6 = 0 or n mod 6 = 2; a(n) = (3n+1)/2 if n mod 6 = 3 or n mod 6 = 5.

Original entry on oeis.org

0, 0, 1, 5, 1, 8, 3, 2, 4, 14, 3, 17, 6, 4, 7, 23, 5, 26, 9, 6, 10, 32, 7, 35, 12, 8, 13, 41, 9, 44, 15, 10, 16, 50, 11, 53, 18, 12, 19, 59, 13, 62, 21, 14, 22, 68, 15, 71, 24, 16, 25, 77, 17, 80, 27, 18, 28, 86, 19, 89, 30, 20, 31, 95, 21, 98, 33, 22, 34, 104

Offset: 0

Paolo Xausa, Jan 02 2022

This is a variant of the Farkas map (A349407).
Yolcu, Aaronson and Heule prove that the trajectory of the iterates of the map starting from any nonnegative integer always reaches 0.
If displayed as a rectangular array with six columns, the columns are A008585, A005843, A016777, A017221, A005408, A017257 (see example). - Omar E. Pol, Jan 02 2022

			From _Omar E. Pol_, Jan 02 2022: (Start)
Written as a rectangular array with six columns read by rows the sequence begins:
   0,  0,  1,  5,  1,  8;
   3,  2,  4, 14,  3, 17;
   6,  4,  7, 23,  5, 26;
   9,  6, 10, 32,  7, 35;
  12,  8, 13, 41,  9, 44;
  15, 10, 16, 50, 11, 53;
  18, 12, 19, 59, 13, 62;
  21, 14, 22, 68, 15, 71;
  24, 16, 25, 77, 17, 80;
  27, 18, 28, 86, 19, 89;
  30, 20, 31, 95, 21, 98;
...
(End)

H. M. Farkas, "Variants of the 3N+1 Conjecture and Multiplicative Semigroups", in Entov, Pinchover and Sageev, Geometry, Spectral Theory, Groups, and Dynamics, Contemporary Mathematics, vol. 387, American Mathematical Society, 2005, p. 121.
Emre Yolcu, Scott Aaronson and Marijn J. H. Heule, An Automated Approach to the Collatz Conjecture, arXiv:2105.14697 [cs.LO], 2021, pp. 21-25.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,2,0,0,0,0,0,-1).

Cf. A005408, A005843, A006370, A014682, A016777, A017221, A017257, A349407, A350279.

nterms=100;Table[If[Mod[n,3]==1,(n-1)/3,If[Mod[n,6]==0||Mod[n,6]==2,n/2,(3n+1)/2]],{n,0,nterms-1}]
(* Second program *)
nterms=100;LinearRecurrence[{0,0,0,0,0,2,0,0,0,0,0,-1},{0,0,1,5,1,8,3,2,4,14,3,17},nterms]

def a(n):
    r = n%6
    if r == 1 or r == 4: return (n-1)//3
    if r == 0 or r == 2: return n//2
    if r == 3 or r == 5: return (3*n+1)//2
print([a(n) for n in range(70)]) # Michael S. Branicky, Jan 02 2022

a(n) = (A349407(n+1)-1)/2.

a(n) = 2*a(n-6)-a(n-12). - Wesley Ivan Hurt, Jan 03 2022

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)

A304506 a(n) = 2(3n+1)(9n+8).

Original entry on oeis.org

16, 136, 364, 700, 1144, 1696, 2356, 3124, 4000, 4984, 6076, 7276, 8584, 10000, 11524, 13156, 14896, 16744, 18700, 20764, 22936, 25216, 27604, 30100, 32704, 35416, 38236, 41164, 44200, 47344, 50596, 53956, 57424, 61000, 64684, 68476, 72376, 76384, 80500, 84724, 89056

Offset: 0

Emeric Deutsch, May 14 2018

a(n) is the second Zagreb index of the single-defect 4-gonal nanocone CNC(4,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(4,n) is M(CNC(4,n);x,y) = 4*x^2*y^2 + 8*n*x^2*y^3 + 2*n*(3*n+1)*x^3*y^3.
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.
6*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. A016933, A017257, A304505.

List([0..50],n->2*(3*n+1)*(9*n+8)); # Muniru A Asiru, May 14 2018

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

Table[2(3n+1)(9n+8),{n,0,40}] (* or *) LinearRecurrence[{3,-3,1},{16,136,364},50] (* Harvey P. Dale, Aug 15 2022 *)

a(n) = 2*(3*n+1)*(9*n+8); \\ Altug Alkan, May 14 2018

Vec(4*(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.: 4*(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.: 2*exp(x)*(8 + 60*x + 27*x^2).
a(n) = A016933(n)*A017257(n). (End)

A304508 a(n) = 5(3n+1)(9n+8)/2 (n>=0).

Original entry on oeis.org

20, 170, 455, 875, 1430, 2120, 2945, 3905, 5000, 6230, 7595, 9095, 10730, 12500, 14405, 16445, 18620, 20930, 23375, 25955, 28670, 31520, 34505, 37625, 40880, 44270, 47795, 51455, 55250, 59180, 63245, 67445, 71780, 76250, 80855, 85595, 90470, 95480, 100625, 105905, 111320

Offset: 0

Emeric Deutsch, May 14 2018

The second Zagreb index of the single-defect  5-gonal nanocone CNC(5,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(5,n) is M(CNC(5,n); x,y) = 5*x^2*y^2 + 10*n*x^2*y^3 + 5*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.

Colin Barker, Table of n, a(n) for n = 0..1000
Emeric Deutsch and Sandi Klavzar, M-polynomial and degree-based topological indices, Iranian J. Math. Chemistry, Vol. 6, No. 2, 2015, pp. 93-102.
T. Doslic and M. Saheli, Augmented eccentric connectivity index of single-defect nanocones, J. 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. A016777, A017257, A304507.

seq((1/2)*(5*(3*n+1))*(9*n+8), n = 0 .. 40);

Array[5 (3 # + 1) (9 # + 8)/2 &, 41, 0] (* or *)
LinearRecurrence[{3, -3, 1}, {20, 170, 455}, 41] (* or *)
CoefficientList[Series[5 (4 + 22 x + x^2)/(1 - x)^3, {x, 0, 40}], x] (* Michael De Vlieger, May 14 2018 *)

a(n) = 5*(3*n+1)*(9*n+8)/2; \\ Altug Alkan, May 14 2018

Vec(5*(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.: 5*(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.: 5*exp(x)*(8 + 60*x + 27*x^2)/2.
a(n) = 5*A016777(n)*A017257(n)/2. (End)

A350522 a(n) = 18*n + 16.

Original entry on oeis.org

16, 34, 52, 70, 88, 106, 124, 142, 160, 178, 196, 214, 232, 250, 268, 286, 304, 322, 340, 358, 376, 394, 412, 430, 448, 466, 484, 502, 520, 538, 556, 574, 592, 610, 628, 646, 664, 682, 700, 718, 736, 754, 772, 790, 808, 826, 844, 862, 880, 898, 916, 934, 952, 970

Offset: 0

Omar E. Pol, Jan 03 2022

Sixth column of A006370 (the Collatz or 3x+1 map) when it is interpreted as a rectangular array with six columns read by rows.

Leo Tavares, Illustration: Triple Hexagonal Rings
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Bisection of A017245.

Cf. A006370, A008588, A008600, A016969, A017257, A062728, A239129.

GAP
```
List([0..53], n-> 18*n+16)
```
Magma
```
[18*n+16: n in [0..53]];
```
Maple
```
seq(18*n+16, n=0..53);
```
Mathematica
```
Table[18n+16, {n, 0, 53}]
```
Maxima
```
makelist(18*n+16, n, 0, 53);
```
PARI
```
a(n)=18*n+16
```
Python
```
[18*n+16 for n in range(53)]
```

a(n) = A239129(n+1) - 1.
From Stefano Spezia, Jan 04 2022: (Start)
O.g.f.: 2*(8 + x)/(1 - x)^2.
E.g.f.: 2*exp(x)*(8 + 9*x).
a(n) = 2*a(n-1) - a(n-2) for n > 1. (End)
a(n) = 3*A008588(n+1) - 2. - Leo Tavares, Sep 14 2022
From Elmo R. Oliveira, Apr 12 2024: (Start)
a(n) = 2*A017257(n) = A006370(A016969(n)).
a(n) = 2*(A062728(n+1) - A062728(n)). (End)

A361692 a(n) = 17*n - 1.

Original entry on oeis.org

16, 33, 50, 67, 84, 101, 118, 135, 152, 169, 186, 203, 220, 237, 254, 271, 288, 305, 322, 339, 356, 373, 390, 407, 424, 441, 458, 475, 492, 509, 526, 543, 560, 577, 594, 611, 628, 645, 662, 679, 696, 713, 730, 747, 764, 781, 798, 815, 832, 849, 866, 883, 900, 917, 934, 951, 968, 985, 1002, 1019

Offset: 1

Leo Tavares, Mar 20 2023

Leo Tavares, Illustration: Double Diamonds.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A008590, A008599, A017257.

17*Range[100] - 1 (* Paolo Xausa, Aug 30 2024 *)
LinearRecurrence[{2,-1},{16,33},90] (* Harvey P. Dale, Jun 03 2025 *)

a(n) = 17*n - 1 = A008599(n) - 1.
a(n) = 2*A008590(n) + n - 1.
a(n) = A008590(n) + A017257(n-1).
From Elmo R. Oliveira, Apr 03 2025: (Start)
G.f.: x*(16 + x)/(x - 1)^2.
E.g.f.: exp(x)*(17*x - 1) + 1.
a(n) = 2*a(n-1) - a(n-2) for n > 2. (End)

A001534 a(n) = (9n+1)(9*n+8).

Original entry on oeis.org

8, 170, 494, 980, 1628, 2438, 3410, 4544, 5840, 7298, 8918, 10700, 12644, 14750, 17018, 19448, 22040, 24794, 27710, 30788, 34028, 37430, 40994, 44720, 48608, 52658, 56870, 61244, 65780, 70478, 75338, 80360, 85544, 90890, 96398, 102068, 107900, 113894, 120050

Offset: 0

N. J. A. Sloane

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

Cf. A017173, A017257.

f[n_]:=Module[{n9=9n},(n9+1)(n9+8)];Array[f,40,0] (* or *) LinearRecurrence[ {3,-3,1},{8,170,494},50] (* Harvey P. Dale, Aug 20 2011 *)

a(n)=(9*n+1)*(9*n+8) \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 162*n + a(n-1) with a(0)=8. - Vincenzo Librandi, Nov 12 2010
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) with a(0)=8, a(1)=170, a(2)=494. - Harvey P. Dale, Aug 20 2011
G.f.: -((2*(x*(4*x+73)+4))/(x-1)^3). - Harvey P. Dale, Aug 20 2011
Sum_{n>=0} 1/a(n) = (Psi(8/9)-Psi(1/9))/63 = 0.13700722.. - R. J. Mathar, May 30 2022
Sum_{n>=0} 1/a(n) = cot(Pi/9)*Pi/63. - Amiram Eldar, Sep 10 2022
From Amiram Eldar, Feb 19 2023: (Start)
a(n) = A017173(n)*A017257(n).
Product_{n>=0} (1 - 1/a(n)) = cosec(Pi/9)*cos(sqrt(53)*Pi/18).
Product_{n>=0} (1 + 1/a(n)) = cosec(Pi/9)*cos(sqrt(5)*Pi/6). (End)
E.g.f.: exp(x)*(8 + 81*x*(2 + x)). - Elmo R. Oliveira, Oct 18 2024

A017263 a(n) = (9*n + 8)^7.

Original entry on oeis.org

2097152, 410338673, 8031810176, 64339296875, 319277809664, 1174711139837, 3521614606208, 9095120158391, 20971520000000, 44231334895529, 86812553324672, 160578147647843, 282621973446656, 476837158203125, 775771085481344, 1222791080775407, 1874584905187328

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 (8, -28, 56, -70, 56, -28, 8, -1).

Cf. A001015 (n^7), A017257 (9*n+8).

[(9*n+8)^7: n in [0..20]]; // Vincenzo Librandi, Jul 28 2011

(9*Range[0,20]+8)^7 (* or *) LinearRecurrence[{8,-28,56,-70,56,-28,8,-1},{2097152,410338673,8031810176,64339296875,319277809664,1174711139837,3521614606208,9095120158391},30] (* Harvey P. Dale, Apr 06 2013 *)

a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8); a(0)=2097152, a(1)=410338673, a(2)=8031810176, a(3)=64339296875, a(4)=319277809664, a(5)=1174711139837, a(6)=3521614606208, a(7)=9095120158391. - Harvey P. Dale, Apr 06 2013

More terms from Harvey P. Dale, Apr 06 2013

A017267 a(n) = (9*n + 8)^11.

Original entry on oeis.org

8589934592, 34271896307633, 3670344486987776, 96549157373046875, 1196683881290399744, 9269035929372191597, 52036560683837093888, 231122292121701565271, 858993459200000000000, 2775173073766990340489

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 (12, -66, 220, -495, 792, -924, 792, -495, 220, -66, 12, -1).

Cf. A008455 (n^11), A017257 (9*n+8).

[(9*n+8)^11: n in [0..10]]; // Vincenzo Librandi, Jul 28 2011

(9*Range[0,30]+8)^11 (* or *) LinearRecurrence[ {12,-66,220,-495,792,-924,792,-495,220,-66,12,-1},{8589934592,34271896307633,3670344486987776,96549157373046875,1196683881290399744,9269035929372191597,52036560683837093888,231122292121701565271,858993459200000000000,2775173073766990340489,8007313507497959524352,21048519522998348950643},30] (* Harvey P. Dale, Oct 21 2013 *)

a(n) = 12*a(n-1) - 66*a(n-2) + 220*a(n-3) - 495*a(n-4) + 792*a(n-5) - 924*a(n-6) + 792*a(n-7) - 495*a(n-8) + 220*a(n-9) - 66*a(n-10) + 12*a(n-11) - a(n-12); a(0)=8589934592, a(1)=34271896307633, a(2)=3670344486987776, a(3)=96549157373046875, a(4)=1196683881290399744, a(5)=9269035929372191597, a(6)=52036560683837093888, a(7)=231122292121701565271, a(8)=858993459200000000000, a(9)=2775173073766990340489, a(10)=8007313507497959524352, a(11)=21048519522998348950643. - Harvey P. Dale, Oct 21 2013

cp's OEIS Frontend

A224829 Numbers m, such that there is no solution m = x + y + 3*z, with triangular numbers x, y, z.

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A350515 a(n) = (n-1)/3 if n mod 3 = 1; a(n) = n/2 if n mod 6 = 0 or n mod 6 = 2; a(n) = (3n+1)/2 if n mod 6 = 3 or n mod 6 = 5.

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

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A304506 a(n) = 2*(3*n+1)*(9*n+8).

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PARI

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A304508 a(n) = 5*(3*n+1)*(9*n+8)/2 (n>=0).

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PARI

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A350522 a(n) = 18*n + 16.

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A361692 a(n) = 17*n - 1.

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

A304506 a(n) = 2(3n+1)(9n+8).

A304508 a(n) = 5(3n+1)(9n+8)/2 (n>=0).

A001534 a(n) = (9n+1)(9*n+8).