A017221 -id:A017221 - cp's OEIS frontend

A017224 a(n) = (9*n + 5)^4.

625, 38416, 279841, 1048576, 2825761, 6250000, 12117361, 21381376, 35153041, 54700816, 81450625, 116985856, 163047361, 221533456, 294499921, 384160000, 492884401, 623201296, 777796321, 959512576

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).

Sequences of the form (9*n+5)^k: A017221 (k=1), A017222 (k=2), A017223 (k=3), this sequence (k=4), A017225 (k=5), A017226 (k=6), A017227 (k=7), A017228 (k=8), A017229 (k=9), A017230 (k=10), A017231 (k=11).

Cf. A000583 (n^4).

[(9*n+5)^4: n in [0..35]] ; // Vincenzo Librandi, Jul 24 2011

(9*Range[0,20]+5)^4 (* Harvey P. Dale, Apr 27 2016 *)

[(9*n+5)^4 for n in range(41)] # G. C. Greubel, Jan 06 2023

a(n) = A000583(A017221(n)).
From Harvey P. Dale, Apr 27 2016: (Start)
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
G.f.: (625 + 35291*x + 94011*x^2 + 27281*x^3 + 256*x^4)/(1-x)^5. (End)
E.g.f.: (625 + 37791*x + 101817*x^2 + 53946*x^3 + 6561*x^4)*exp(x). - G. C. Greubel, Jan 06 2023

A130877 Numbers that are congruent to {0, 5} mod 9.

Original entry on oeis.org

0, 5, 9, 14, 18, 23, 27, 32, 36, 41, 45, 50, 54, 59, 63, 68, 72, 77, 81, 86, 90, 95, 99, 104, 108, 113, 117, 122, 126, 131, 135, 140, 144, 149, 153, 158, 162, 167, 171, 176, 180, 185, 189, 194, 198, 203, 207, 212, 216, 221, 225, 230, 234, 239, 243, 248, 252, 257

Offset: 1

Paolo P. Lava and Giorgio Balzarotti, Jul 25 2007

Numbers m such that m = digitsum(k*(m+k)) for some k>=0.

The first differences are 2-periodic: 5, 4, 5, 4, etc. The minimum numbers k associated to the first elements of the sequence are (m,k): (0,0), (5,2), (9,3), (14,5), (18,15), (23,44), (27,42), (32,119), etc.

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

Cf. A008591, A017221.

op(select(n->n mod 9=0 or n mod 9=5,[$0..257])); # Paolo P. Lava, Jul 12 2018
# second Maple program:
a:= n-> ceil(9*(n-1)/2):
seq(a(n), n=1..58);  # Alois P. Heinz, Apr 12 2025

Table[5n-5-Floor[(n-1)/2], {n,100}] (* Wesley Ivan Hurt, Oct 25 2013 *)
Select[Range[0,300],MemberQ[{0,5},Mod[#,9]]&] (* or *) LinearRecurrence[ {1,1,-1},{0,5,9},60] (* Harvey P. Dale, Aug 04 2019 *)

forstep(n=0,200,[5,4],print1(n", ")) \\ Charles R Greathouse IV, Oct 17 2011

a(n) = a(n-2) + 9 for n >= 3.
a(n) = 9/2*(n+1) - 4 + Sum{j=0..n} (-1)^j/2.
O.g.f.: x^2(5+4x)/((1+x)(1-x)^2). a(n) = 9(n-1)/2+(1+(-1)^n)/4. - R. J. Mathar, Jun 13 2008
a(n+1) = Sum_{k>=0} A030308(n,k)*A116453(k+1). - Philippe Deléham, Oct 17 2011
a(n) = 5n - 5 - floor((n-1)/2). - Wesley Ivan Hurt, Oct 25 2013
a(n) = ceiling(9*(n-1)/2). - Alois P. Heinz, Apr 12 2025

A154267 a(n) = 27*n + 15.

Original entry on oeis.org

15, 42, 69, 96, 123, 150, 177, 204, 231, 258, 285, 312, 339, 366, 393, 420, 447, 474, 501, 528, 555, 582, 609, 636, 663, 690, 717, 744, 771, 798, 825, 852, 879, 906, 933, 960, 987, 1014, 1041, 1068, 1095, 1122, 1149, 1176, 1203, 1230, 1257, 1284, 1311, 1338

Offset: 0

Vincenzo Librandi, Jan 06 2009

The identity (81*n^2 + 90*n + 26)^2 - (9*n^2 + 10*n + 3)*(27*n + 15)^2 = 1 can be written as A154277(n+1)^2 - A154254(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. A154254, A154277.

I:=[15, 42]; [n le 2 select I[n] else 2*Self(n-1)-Self(n-2): n in [1..50]]; // Vincenzo Librandi, Feb 02 2012

Range[15, 7000, 27] (* Vladimir Joseph Stephan Orlovsky, Jul 13 2011 *)
LinearRecurrence[{2, -1}, {15, 42}, 40] (* Vincenzo Librandi, Feb 02 2012 *)
27*Range[0,50]+15 (* Harvey P. Dale, Feb 26 2017 *)

a(n)=27*n+15 \\ Charles R Greathouse IV, Dec 28 2011

G.f.: 3*(5 + 4*x)/(1-x)^2. - R. J. Mathar, Jan 05 2011
a(n) = 3*A017221(n). - R. J. Mathar, Jan 05 2011
a(n) = 2*a(n-1) - a(n-2). - Vincenzo Librandi, Feb 02 2012
E.g.f.: (27*x + 15)*exp(x). - G. C. Greubel, Sep 08 2016

A155704 Triangle read by rows where T(m,n)=2mn + m + n + 10.

Original entry on oeis.org

14, 17, 22, 20, 27, 34, 23, 32, 41, 50, 26, 37, 48, 59, 70, 29, 42, 55, 68, 81, 94, 32, 47, 62, 77, 92, 107, 122, 35, 52, 69, 86, 103, 120, 137, 154, 38, 57, 76, 95, 114, 133, 152, 171, 190, 41, 62, 83, 104, 125, 146, 167, 188, 209, 230, 44, 67, 90, 113, 136, 159, 182

Offset: 1

Vincenzo Librandi, Jan 25 2009

The numbers 2*T(m,n)-19 = (2*m+1)*(2*n+1) are not prime.

First column: A016789, second column: A016873, third column: A017053, fourth column: A017221. - Vincenzo Librandi, Nov 20 2012

			Triangle begins:
14;
17, 22;
20, 27, 34;
23, 32, 41, 50;
26, 37, 48, 59, 70;
29, 42, 55, 68, 81, 94;
32, 47, 62, 77, 92, 107, 122;
35, 52, 69, 86, 103, 120, 137, 154;
38, 57, 76, 95, 114, 133, 152, 171, 190;
41, 62, 83, 104, 125, 146, 167, 188, 209, 230;

Vincenzo Librandi, Rows n = 1..100, flattened

Cf. A153041, A016789, A016873, A017053, A017221.

[2*n*k + n + k + 10: k in [1..n], n in [1..11]]; // Vincenzo Librandi, Nov 20 2012

t[n_,k_]:=2 n*k + n + k +  10; Table[t[n, k], {n, 15}, {k, n}]//Flatten (* Vincenzo Librandi, Nov 20 2012 *)

A330613 Triangle read by rows: T(n, k) = 1 + k - 2n - 2kn + 2n^2, with 0 <= k < n.

Original entry on oeis.org

1, 5, 2, 13, 8, 3, 25, 18, 11, 4, 41, 32, 23, 14, 5, 61, 50, 39, 28, 17, 6, 85, 72, 59, 46, 33, 20, 7, 113, 98, 83, 68, 53, 38, 23, 8, 145, 128, 111, 94, 77, 60, 43, 26, 9, 181, 162, 143, 124, 105, 86, 67, 48, 29, 10, 221, 200, 179, 158, 137, 116, 95, 74, 53, 32, 11

Offset: 1

Stefano Spezia, Dec 20 2019

T(n, k) is the k-th super- and subdiagonal sum of the matrix M(n) whose permanent is A330287(n).

			n\k|   0   1   2   3   4   5
---+------------------------
1  |   1
2  |   5   2
3  |  13   8   3
4  |  25  18  11   4
5  |  41  32  23  14   5
6  |  61  50  39  28  17   6
...
For n = 3 the matrix M is
      1, 2, 3
      2, 4, 6
      3, 6, 8
and therefore T(3, 0) = 1 + 4 + 8 = 13, T(3, 1) = 2 + 6 = 8 and T(3, 2) = 3.

Stefano Spezia, First 150 rows of the triangle, flattened

Cf. A005408, A330287.

Cf. A000027: diagonal; A001105: 2nd column; A001844: 1st column; A016789: 1st subdiagonal; A016885: 2nd subdiagonal; A017029: 3rd subdiagonal; A017221: 4th subdiagonal; A017461: 5th subdiagonal; A081436: row sums; A132209: 3rd column; A164284: 7th subdiagonal; A269044: 6th subdiagonal.

Flatten[Table[1+k-2n-2k*n+2n^2,{n,1,11},{k,0,n-1}]] (* or *)
r[n_] := Table[SeriesCoefficient[(1-x*(2-5x+2(1+x)y))/((1-x)^3*(1-y)^2), {x, 0, i}, {y, 0, j}], {i, n, n}, {j, 0, n-1}]; Flatten[Array[r, 11]] (* or *)
r[n_] := Table[SeriesCoefficient[Exp[x+y]*(1+2x(x-y)+y), {x, 0, i}, {y, 0, j}]*i!*j!, {i, n, n}, {j, 0, n-1}]; Flatten[Array[r, 11]]

O.g.f.: (1 - x*(2 - 5*x + 2*(1 + x)*y))/((1 - x)^3*(1 - y)^2).
E.g.f.: exp(x+y)*(1 + 2*x*(x - y) + y).
T(n, k) = A001844(n-1) - k*A005408(n-1), with 0 <= k < n. [Typo corrected by Stefano Spezia, Feb 14 2020]

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

A122485 Values of A083097(k) such that A083097(k) = A083097(k+1) - 1.

Original entry on oeis.org

5, 14, 41, 59, 122, 140, 167, 176, 365, 383, 410, 419, 491, 500, 527, 545, 1094, 1112, 1139, 1148, 1220, 1229, 1256, 1274, 1463, 1472, 1499, 1517, 1580, 1598, 1625, 1634, 3281, 3299, 3326, 3335, 3407, 3416, 3443, 3461, 3650, 3659, 3686, 3704, 3767, 3785, 3812

Offset: 1

Alexander Adamchuk, Sep 15 2006

nonn

A083097(n) = A083095(n) = A083096(n)/6 = A083094(n)/4, where A083096 are the Numbers k such that 3 divides Sum_{j=1..k} C(2*j,j) = A066796(k).
All terms are of the form 9*m + 5 and belong to A017221 with m = {0, 1, 4, 6, 13, 15, 18, 19, 40, 42, ...}.
Corresponding numbers m such that a(m) = A083097(m) are A129771 (evil odd numbers).

			A083097 begins {0, 2, 5, 6, 14, 15, 18, 20, 41, 42, 45, 47, 54, 56, 59, 60, ...}.
So a(1) = 5 because 5 = A083097(3) = A083097(3+1) - 1.
a(2) = 14 because 14 = A083097(5) = A083097(5+1) - 1.

Cf. A000069, A000120, A066796, A083094, A083095, A083096, A083097, A010060, A129771.

a(n) = A083097(A129771(n)).

More terms from R. J. Mathar, Jan 17 2008
More terms from Jinyuan Wang, Jan 22 2022
Edited by Michel Marcus, Jan 22 2022

A177073 a(n) = (9n+4)(9*n+5).

Original entry on oeis.org

20, 182, 506, 992, 1640, 2450, 3422, 4556, 5852, 7310, 8930, 10712, 12656, 14762, 17030, 19460, 22052, 24806, 27722, 30800, 34040, 37442, 41006, 44732, 48620, 52670, 56882, 61256, 65792, 70490, 75350, 80372, 85556, 90902, 96410, 102080, 107912, 113906, 120062

Offset: 0

Vincenzo Librandi, May 31 2010

Cf. comment of Reinhard Zumkeller in A177059: in general, (h*n+h-k)*(h*n+k) = h^2*A002061(n+1) + (h-k)*k - h^2; therefore a(n) = 81*A002061(n+1) - 61. - Bruno Berselli, Aug 24 2010

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

Cf. A002061, A017209, A017221, A177059.

[(9*n+4)*(9*n+5): n in [0..50]]; // Vincenzo Librandi, Apr 08 2013

f[n_] := Module[{c = 9n}, (c+4)(c+5)]; Array[f, 40, 0] (* or *) LinearRecurrence[{3, -3, 1}, {20, 182, 506}, 40] (* Harvey P. Dale, Jun 24 2011 *)

a(n)=(9*n+4)*(9*n+5) \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 162*n + a(n-1) with n > 0, a(0)=20.
From Harvey P. Dale, Jun 24 2011: (Start)
a(0)=20, a(1)=182, a(2)=506, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f.: -2*(x*(10*x+61)+10)/(x-1)^3. (End)
From Amiram Eldar, Feb 19 2023: (Start)
a(n) = A017209(n)*A017221(n).
Sum_{n>=0} 1/a(n) = tan(Pi/18)*Pi/9.
Product_{n>=0} (1 - 1/a(n)) = sec(Pi/18)*cos(sqrt(5)*Pi/18).
Product_{n>=0} (1 + 1/a(n)) = sec(Pi/18)*cosh(sqrt(3)*Pi/18). (End)
E.g.f.: exp(x)*(20 + 81*x*(2 + x)). - Elmo R. Oliveira, Oct 18 2024

Edited by N. J. A. Sloane, Jun 22 2010

A354939 Row 9 of A354940: Numbers k for which A345992(k) = 9, divided by 9.

Original entry on oeis.org

5, 7, 10, 14, 16, 19, 23, 25, 28, 32, 37, 41, 43, 46, 50, 59, 61, 64, 68, 73, 79, 82, 86, 91, 97, 100, 109, 113, 118, 122, 127, 131, 136, 145, 149, 151, 158, 163, 167, 169, 172, 181, 185, 194, 199, 212, 221, 223, 226, 235, 239, 241, 244, 253, 257, 262, 271, 277, 289, 293, 298, 302, 307, 311, 313, 316, 325, 331, 334

Offset: 1

Antti Karttunen, Jun 15 2022

nonn

Apparently, all terms are either of the form 9k+1 (in A017173), or 9k+5 (in A017221), or 9k+7 (in A017245).

Row 9 of A354940.

Cf. A017173, A017221, A017245, A344005, A345992.

q[n_] := Module[{m = 1}, While[!Divisible[m*(m + 1), 9*n], m++]; GCD[9*n, m] == 9]; Select[Range[335], q] (* Amiram Eldar, Jun 17 2022 *)

A354939(n) = A354940sq(9,n); \\ See the program in A354940.

A380820 a(0) = 0, a(1) = 1, and for n >= 2, a(n) = a(n-1) + a(n-2) if a(n-1) < n, otherwise a(n-1) - n.

Original entry on oeis.org

0, 1, 1, 2, 3, 5, 8, 1, 9, 0, 9, 9, 18, 5, 23, 8, 31, 14, 45, 26, 6, 32, 10, 42, 18, 60, 34, 7, 41, 12, 53, 22, 75, 42, 8, 50, 14, 64, 26, 90, 50, 9, 59, 16, 75, 30, 105, 58, 10, 68, 18, 86, 34, 120, 66, 11, 77, 20, 97, 38, 135, 74, 12, 86, 22, 108, 42, 150

Offset: 0

Ya-Ping Lu, Feb 04 2025

Sequence starts with the first 7 Fibonacci numbers. For n >= 12, a(n) takes the values of (8*n+30)/7, (n+22)/7, (9*n+35)/7, (2*n+26)/7, (11*n+41)/7, (4*n+30)/7, and (15*n+45)/7 sequentially for n = 5, 6, 0, 1, 2, 3, 4 mod 7 (see plot in Links), which correspond to A017089 (n>=2), A000027 (n>=5), A017221 (n>=2), A005843 (n>=4), A017497 (n>=2), A016825 (n>=3), and A008597 (n>=3), respectively.

Terms for n >= 16 are the same as A322558(n) for n >= 17.

Ya-Ping Lu, Plot of A380820
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,2,0,0,0,0,0,0,-1).

Cf. A000027, A000045, A005843, A008597, A016825, A017089, A017221, A017497, A322558.

s={0,1};Do[AppendTo[s,If[s[[-1]]James C. McMahon, Feb 14 2025 *)

def A380820(n): R = [0, 1, 1, 2, 3, 5, 8, 1, 9, 0, 9, 9]; X = [9, 2, 11, 4, 15, 8, 1]; Y = [35, 26, 41, 30, 45, 30, 22]; return R[n] if n < 12 else (X[n%7]*n + Y[n%7])//7

a(n) = A322558(n+1) for n >= 16.

cp's OEIS Frontend

A017224 a(n) = (9*n + 5)^4.

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A130877 Numbers that are congruent to {0, 5} mod 9.

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A154267 a(n) = 27*n + 15.

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A155704 Triangle read by rows where T(m,n)=2*m*n + m + n + 10.

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A330613 Triangle read by rows: T(n, k) = 1 + k - 2*n - 2*k*n + 2*n^2, with 0 <= k < n.

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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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A122485 Values of A083097(k) such that A083097(k) = A083097(k+1) - 1.

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A177073 a(n) = (9*n+4)*(9*n+5).

A155704 Triangle read by rows where T(m,n)=2mn + m + n + 10.

A330613 Triangle read by rows: T(n, k) = 1 + k - 2n - 2kn + 2n^2, with 0 <= k < n.

A177073 a(n) = (9n+4)(9*n+5).