A017209 -id:A017209 - cp's OEIS frontend

A144829 Partial products of successive terms of A017209; a(0)=1 .

1, 4, 52, 1144, 35464, 1418560, 69509440, 4031547520, 270113683840, 20528639971840, 1744934397606400, 164023833375001600, 16894454837625164800, 1892178941814018457600, 228953651959496233369600, 29763974754734510338048000, 4137192490908096936988672000

Offset: 0

Philippe Deléham, Sep 21 2008

			a(0)=1, a(1)=4, a(2)=4*13=52, a(3)=4*13*22=1144, a(4)=4*13*22*31=35464, ...

G. C. Greubel, Table of n, a(n) for n = 0..300

Cf. A001715, A002866, A007559, A008546, A047053.

Cf. A048994, A049308, A132393, A144827, A144828.

[n le 2 select 4^(n-1) else (9*n-14)*Self(n-1): n in [1..30]]; // G. C. Greubel, May 26 2022

Table[4*9^(n-1)*Pochhammer[13/9, n-1], {n, 0, 20}] (* Vaclav Kotesovec, Nov 29 2021 *)

a(n) = (-5)^n*sum(k=0, n, (9/5)^k*stirling(n+1,n+1-k, 1)); \\ Michel Marcus, Feb 20 2015

[9^n*rising_factorial(4/9, n) for n in (0..30)] # G. C. Greubel, May 26 2022

a(n) = Sum_{k=0..n} A132393(n,k)*4^k*9^(n-k).
a(n) = (-5)^n*Sum_{k=0..n} (9/5)^k*s(n+1,n+1-k), where s(n,k) are the Stirling numbers of the first kind, A048994. - Mircea Merca, May 03 2012
a(n) + (5-9*n)*a(n-1) = 0. - R. J. Mathar, Sep 04 2016
From Vaclav Kotesovec, Nov 29 2021: (Start)
a(n) = 9^n * Gamma(n + 4/9) / Gamma(4/9).
a(n) ~ sqrt(2*Pi) * 9^n * n^(n - 1/18) / (Gamma(4/9) * exp(n)). (End)
From G. C. Greubel, May 26 2022: (Start)
G.f.: hypergeometric2F0([1, 4/9], [], 9*x).
E.g.f.: (1-9*x)^(-4/9). (End)
Sum_{n>=0} 1/a(n) = 1 + (e/9^5)^(1/9)*(Gamma(4/9) - Gamma(4/9, 1/9)). - Amiram Eldar, Dec 21 2022

a(9) originally given incorrectly as 20520639971840 corrected by Peter Bala, Feb 20 2015

A017173 a(n) = 9*n + 1.

Original entry on oeis.org

1, 10, 19, 28, 37, 46, 55, 64, 73, 82, 91, 100, 109, 118, 127, 136, 145, 154, 163, 172, 181, 190, 199, 208, 217, 226, 235, 244, 253, 262, 271, 280, 289, 298, 307, 316, 325, 334, 343, 352, 361, 370, 379, 388, 397, 406, 415, 424, 433, 442, 451, 460, 469, 478

Offset: 0

N. J. A. Sloane

Also all the numbers with digital root 1; A010888(a(n)) = 1. - Rick L. Shepherd, Jan 12 2009
A116371(a(n)) = A156144(a(n)); positions where records occur in A156144: A156145(n+1) = A156144(a(n)). - Reinhard Zumkeller, Feb 05 2009
If A=[A147296] 9*n^2+2*n (n>0, 11, 40, 87, ...); Y=[A010701] 3 (3, 3, 3, ...); X=[A017173] 9*n+1 (n>0, 10, 19, 28, ...), we have, for all terms, Pell's equation X^2 - A*Y^2 = 1. Example: 10^2 - 11*3^2 = 1; 19^2 - 40*3^2 = 1; 28^2 - 87*3^2 = 1. - Vincenzo Librandi, Aug 01 2010

Mohammed Yaseen, Table of n, a(n) for n = 0..10000
Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A093644 ((9,1) Pascal, column m=1).
Cf. A010701, A010888, A016777, A116371, A147296, A156144, A156145.
Numbers with digital root m: this sequence (m=1), A017185 (m=2), A017197 (m=3), A017209 (m=4), A017221 (m=5), A017233 (m=6), A017245 (m=7), A017257 (m=8), A008591 (m=9).

a017173 = (+ 1) . (* 9)
a017173_list = [1, 10 ..]  -- Reinhard Zumkeller, Feb 04 2014

Range[1, 1000, 9] (* Vladimir Joseph Stephan Orlovsky, May 28 2011 *)
LinearRecurrence[{2,-1},{1,10},60] (* Harvey P. Dale, Dec 27 2014 *)

forstep(n=1,500,9,print1(n", ")) \\ Charles R Greathouse IV, May 28 2011

[i+1 for i in range(480) if gcd(i,9) == 9] # Zerinvary Lajos, May 20 2009

G.f.: (1 + 8*x)/(1 - x)^2.
a(n) = 2*a(n-1) - a(n-2) with a(0)=1, a(1)=10. - Vincenzo Librandi, Aug 01 2010
E.g.f.: exp(x)*(1 + 9*x). - Stefano Spezia, Apr 20 2023
a(n) = A016777(3*n). - Elmo R. Oliveira, Apr 12 2025

A002278 a(n) = 4*(10^n - 1)/9.

Original entry on oeis.org

0, 4, 44, 444, 4444, 44444, 444444, 4444444, 44444444, 444444444, 4444444444, 44444444444, 444444444444, 4444444444444, 44444444444444, 444444444444444, 4444444444444444, 44444444444444444, 444444444444444444, 4444444444444444444, 44444444444444444444, 444444444444444444444

Offset: 0

N. J. A. Sloane

Ivan Panchenko, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (11,-10).

Cf. A002275, A002276, A002277, A002279, A002280, A002281, A002282, A002283, A075415, A178632.

Cf. A007091, A010785, A017209, A024049.

LinearRecurrence[{11, -10}, {0, 4}, 20] (* Robert G. Wilson v, Jul 06 2013 *)

a(n)=4*(10^n-1)/9 \\ Charles R Greathouse IV, Sep 24 2015

a(n) = A075415(n)/A002283(n). - Reinhard Zumkeller, May 31 2010
From Vincenzo Librandi, Jul 22 2010: (Start)
a(n) = a(n-1) + 4*10^(n-1) with a(0)=0;
a(n) = 11*a(n-1) - 10*a(n-2) with a(0)=0, a(1)=4. (End)
G.f.: 4*x/((1 - x)*(1 - 10*x)). - Ilya Gutkovskiy, Feb 24 2017
E.g.f.: 4*exp(x)*(exp(9*x) - 1)/9. - Stefano Spezia, Sep 13 2023
a(n) = A007091(A024049(n)). - Michel Marcus, Jun 16 2024
From Elmo R. Oliveira, Jul 19 2025: (Start)
a(n) = 4*A002275(n).
a(n) = A010785(A017209(n-1)) for n >= 1. (End)

A017221 a(n) = 9*n + 5.

Original entry on oeis.org

5, 14, 23, 32, 41, 50, 59, 68, 77, 86, 95, 104, 113, 122, 131, 140, 149, 158, 167, 176, 185, 194, 203, 212, 221, 230, 239, 248, 257, 266, 275, 284, 293, 302, 311, 320, 329, 338, 347, 356, 365, 374, 383, 392, 401, 410, 419, 428, 437, 446, 455, 464, 473, 482

Offset: 0

N. J. A. Sloane

Numbers whose digital root is 5. - Halfdan Skjerning, Mar 15 2018

R. K. Guy, Unsolved Problems in Number Theory, Springer, 1st edition, 1981. See section D5.

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Sequences of the form (9*n+5)^k: this sequence (k=1), A017222 (k=2), A017223 (k=3), A017224 (k=4), A017225 (k=5), A017226 (k=6), A017227 (k=7), A017228 (k=8), A017229 (k=9), A017230 (k=10), A017231 (k=11).
Cf. A008591, A016789, A017209.
Cf. similar sequences with closed form (2*k-1)*n+k listed in A269044.

[9*n+5: n in [0..60]]; // Vincenzo Librandi, Jul 24 2011

seq(9*w+5, w=0..100); # Matt C. Anderson, May 18 2017

Range[5, 1000, 9] (* Vladimir Joseph Stephan Orlovsky, May 28 2011 *)
9*Range[0,60]+5 (* or *) LinearRecurrence[{2,-1},{5,14},60] (* Harvey P. Dale, Jul 05 2021 *)

forstep(n=5,500,9,print1(n", ")) \\ Charles R Greathouse IV, May 28 2011

[9*n+5 for n in range(51)] # G. C. Greubel, Jan 06 2023

G.f.: (5+4*x)/(1-x)^2. - R. J. Mathar, Mar 20 2018
From G. C. Greubel, Jan 06 2023: (Start)
a(n) = a(n-1) + 9, with a(0) = 5.
E.g.f.: (5 + 9*x)*exp(x). (End)
From Elmo R. Oliveira, Apr 10 2025: (Start)
a(n) = 2*a(n-1) - a(n-2).
a(n) = A016789(3*n+1). (End)

A056991 Numbers with digital root 1, 4, 7 or 9.

Original entry on oeis.org

1, 4, 7, 9, 10, 13, 16, 18, 19, 22, 25, 27, 28, 31, 34, 36, 37, 40, 43, 45, 46, 49, 52, 54, 55, 58, 61, 63, 64, 67, 70, 72, 73, 76, 79, 81, 82, 85, 88, 90, 91, 94, 97, 99, 100, 103, 106, 108, 109, 112, 115, 117, 118, 121, 124, 126, 127, 130, 133, 135, 136, 139, 142

Offset: 1

Eric W. Weisstein

All squares are members (see A070433).
May also be defined as: possible sums of digits of squares. - Zak Seidov, Feb 11 2008
First differences are periodic: 3, 3, 2, 1, 3, 3, 2, 1, 3, 3, 2, 1, 3, 3, 2, 1, 3, 3, 2, 1, 3, 3, 2, 1, 3, 3, 2, 1, ... - Zak Seidov, Feb 11 2008
Minimal n with corresponding sum-of-digits(n^2) are: 1, 2, 4, 3, 8, 7, 13, 24, 17, 43, 67, 63, 134, 83, 167, 264, 314, 313, 707, 1374, 836, 1667, 2236, 3114, 4472, 6833, 8167, 8937, 16667, 21886, 29614, 60663, 41833, 74833, 89437, 94863, 134164, 191833.
a(n) is the set of all m such that 9k+m can be a perfect square (quadratic residues of 9 including the trivial case of 0). - Gary Detlefs, Mar 19 2010
From Klaus Purath, Feb 20 2023: (Start)
The sum of digits of any term belongs to the sequence. Also the products of any terms belong to the sequence.
This is the union of A017173, A017209, A017245 and A008591.
Positive integers of the forms x^2 + (2*m+1)*x*y + (m^2+m-2)*y^2, for integers m.
This sequence is closed under multiplication. (End)

R. J. Mathar, Table of n, a(n) for n = 1..22222
H. I. Okagbue, M. O. Adamu, S. A. Iyase, and A. A. Opanuga, Sequence of Integers Generated by Summing the Digits of their Squares, Indian Journal of Science and Technology, Vol 8(15), DOI: 10.17485/ijst/2015/v8i15/69912, July 2015.
Eric Weisstein's World of Mathematics, Square Number
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A000290, A008591, A017173, A017209, A017245, A056992, A070433.

For complement see A268226.

seq( 3*(n-floor(n/4)) - (3-I^n-(-I)^n-(-1)^n)/2, n=1..63); # Gary Detlefs, Mar 19 2010

LinearRecurrence[{1,0,0,1,-1},{1,4,7,9,10},70] (* Harvey P. Dale, Aug 29 2015 *)

forstep(n=1,1e3,[3,3,2,1],print1(n", ")) \\ Charles R Greathouse IV, Sep 21 2012

From R. J. Mathar, Feb 14 2008: (Start)
O.g.f.: x*(2x+1)*(x^2+x+1)/((-1+x)^2*(x+1)*(x^2+1)).
a(n) = a(n-4) + 9. (End)
a(n) = 3*(n - floor(n/4)) - (3 - i^n - (-i)^n - (-1)^n)/2, where i = sqrt(-1). - Gary Detlefs, Mar 19 2010
a(n) = a(n-1)+a(n-4)-a(n-5). - Wesley Ivan Hurt, May 27 2021
a(n) = 3*n - floor(n/4) - 2*floor((n+3)/4). - Ridouane Oudra, Jan 21 2024
E.g.f.: (cos(x) + (9*x - 1)*cosh(x) - 3*sin(x) + (9*x - 2)*sinh(x))/4. - Stefano Spezia, Feb 21 2024

Edited by N. J. A. Sloane, May 16 2008 at the suggestion of R. J. Mathar

A060816 a(0) = 1; a(n) = (5*3^(n-1) - 1)/2 for n > 0.

Original entry on oeis.org

1, 2, 7, 22, 67, 202, 607, 1822, 5467, 16402, 49207, 147622, 442867, 1328602, 3985807, 11957422, 35872267, 107616802, 322850407, 968551222, 2905653667, 8716961002, 26150883007, 78452649022, 235357947067, 706073841202

Offset: 0

Jason Earls, Apr 29 2001

From Erich Friedman's math magic page 2nd paragraph under "Answers" section.
Let A be the Hessenberg matrix of order n, defined by: A[1,j] = 1, A[i,i] = 2,(i>1),  A[i,i-1] = -1, and A[i,j] = 0 otherwise. Then, for n >= 1, a(n) = (-1)^n*charpoly(A,-1). - Milan Janjic, Jan 26 2010
If n > 0 and H = hex number (A003215), then 9*H(a(n)) - 2 = H(a(n+1)), for example 9*H(2) - 2 = 9*19 - 2 = 169 = H(7). For n > 2, this is a subsequence of A017209, see formula. - Klaus Purath, Mar 31 2021
Consider the Tower of Hanoi puzzle of order n (i.e., with n rings to be moved). Label each ring with a distinct symbol from an alphabet of size n. Construct words by performing moves according to the standard rules of the puzzle, recording the corresponding symbol each time a ring is moved. To ensure finiteness, we forbid returning to any previously encountered state. Additionally, we impose the constraint that the same ring cannot be moved twice in succession. Then, a(n) is the number of distinct words that can be formed under these rules. - Thomas Baruchel, Jul 22 2025

Harry J. Smith, Table of n, a(n) for n = 0..200
Erich Friedman, Math. Magic
Index entries for linear recurrences with constant coefficients, signature (4,-3).

Cf. A005030 (first differences), A244762 (partial sums).

Cf. A003215, A003462, A007051, A008343, A017209, A057198, A134931.

LinearRecurrence[{4,-3},{1,2,7},30] (* Harvey P. Dale, Nov 15 2022 *)

A060816(n)=if(n, 3^n*5\6, 1) \\ M. F. Hasler, Apr 06 2019

The following is a summary of formulas added over the past 18 years.
a(n) = A057198(n) - 1.
a(n) = 3*a(n-1) + 1; with a(0)=1, a(1)=2. - Jason Earls, Apr 29 2001
From Henry Bottomley, May 01 2001: (Start)
For n>0, a(n) = a(n-1)+5*3^(n-2) = a(n-1)+A005030(n-2).
For n>0, a(n) = (5*A003462(n)+1)/3. (End)
From Colin Barker, Apr 24 2012: (Start)
a(n) = 4*a(n-1) - 3*a(n-2) for n > 2.
G.f.: (1-2*x+2*x^2)/((1-x)*(1-3*x)). (End)
a(n+1) = A134931(n) + 1. - Philippe Deléham, Apr 14 2013
For n > 0, A008343(a(n)) = 0. - Dmitry Kamenetsky, Feb 14 2017
For n > 0, a(n) = floor(3^n*5/6). - M. F. Hasler, Apr 06 2019
From Klaus Purath, Mar 31 2021: (Start)
a(n) = A017209(a(n-2)), n > 2.
a(n) = 2 + Sum_{i = 0..n-2} A005030(i).
a(n+1)*a(n+2) = a(n)*a(n+3) + 20*3^n, n > 1.
a(n) = 3^n - A007051(n-1). (End)
E.g.f.: (5*exp(3*x) - 3*exp(x) + 4)/6. - Stefano Spezia, Aug 28 2023

Edited by M. F. Hasler, Apr 06 2019 and by N. J. A. Sloane, Apr 09 2019

A017233 a(n) = 9*n + 6.

Original entry on oeis.org

6, 15, 24, 33, 42, 51, 60, 69, 78, 87, 96, 105, 114, 123, 132, 141, 150, 159, 168, 177, 186, 195, 204, 213, 222, 231, 240, 249, 258, 267, 276, 285, 294, 303, 312, 321, 330, 339, 348, 357, 366, 375, 384, 393, 402, 411, 420, 429, 438, 447, 456, 465, 474, 483

Offset: 0

David J. Horn and Laura Krebs Gordon (lkg615(AT)verizon.net), 1985

General form: (q*n-1)*q, cf. A017233 (q=3), A098502 (q=4). - Vladimir Joseph Stephan Orlovsky, Feb 16 2009

Numbers whose digital root is 6; that is, A010888(a(n)) = 6. (Ball essentially says that Iamblichus (circa 350) announced that a number equal to the sum of three integers 3*n, 3*n - 1, and 3*n - 2 has 6 as what is now called the number's digital root.) - Rick L. Shepherd, Apr 01 2014

W. W. R. Ball, A Short Account of the History of Mathematics, Sterling Publishing Company, Inc., 2001 (Facsimile Edition) [orig. pub. 1912], pages 110-111.

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Tanya Khovanova, Recursive Sequences.
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A008585, A008591, A010888, A016789, A017209, A017221, A017233, A019557, A098502.

[9*n+6: n in [0..60]]; // Vincenzo Librandi, Jul 24 2011

Range[6, 1000, 9] (* Vladimir Joseph Stephan Orlovsky, May 28 2011 *)
LinearRecurrence[{2,-1},{6,15},60] (* Harvey P. Dale, Feb 01 2014 *)

a(n)=9*n+6 \\ Charles R Greathouse IV, Oct 07 2015

G.f.: 3*(2+x)/(x-1)^2. - R. J. Mathar, Mar 20 2018
Sum_{n>=0} (-1)^n/a(n) = sqrt(3)*Pi/27 - log(2)/9. - Amiram Eldar, Dec 12 2021
E.g.f.: 3*exp(x)*(2 + 3*x). - Stefano Spezia, Dec 07 2024
From Elmo R. Oliveira, Apr 10 2025: (Start)
a(n) = 3*A016789(n) = A019557(n+1)/2.
a(n) = 2*a(n-1) - a(n-2). (End)

A083487 Triangle read by rows: T(n,k) = 2nk + n + k (1 <= k <= n).

Original entry on oeis.org

4, 7, 12, 10, 17, 24, 13, 22, 31, 40, 16, 27, 38, 49, 60, 19, 32, 45, 58, 71, 84, 22, 37, 52, 67, 82, 97, 112, 25, 42, 59, 76, 93, 110, 127, 144, 28, 47, 66, 85, 104, 123, 142, 161, 180, 31, 52, 73, 94, 115, 136, 157, 178, 199, 220, 34, 57, 80, 103, 126, 149, 172, 195, 218, 241, 264

Offset: 1

Artemario Tadeu Medeiros da Silva (artemario(AT)uol.com.br), Jun 09 2003

T(n,k) gives number of edges (of unit length) in a k X n grid.

The values 2*T(n,k)+1 = (2*n+1)*(2*k+1) are nonprime and therefore in A047845.

			Triangle begins:
   4;
   7, 12;
  10, 17, 24;
  13, 22, 31, 40;
  16, 27, 38, 49,  60;
  19, 32, 45, 58,  71,  84;
  22, 37, 52, 67,  82,  97, 112;
  25, 42, 59, 76,  93, 110, 127, 144;
  28, 47, 66, 85, 104, 123, 142, 161, 180;

Vincenzo Librandi, Rows n = 1..100, flattened
Alain Kraus, Cours-Arithmétique et algèbre, 2016-2017, Université de Paris VI. See Exercice 6 p. 13.
OEIS Wiki, Odd composites

Cf. A000217, A016777, A016873, A017017, A017209, A017449, A033954.
Cf. A056220, A082652, A083003, A090288, A182868, A186113, A271625.
See A151890 for another version.

[(2*n*k + n + k): k in [1..n], n in [1..11]]; // Vincenzo Librandi, Jun 01 2014

T[n_,k_]:= 2 n k + n + k; Table[T[n, k], {n, 10}, {k, n}]//Flatten (* Vincenzo Librandi, Jun 01 2014 *)

def T(r, c): return 2*r*c + r + c
a = [T(r, c) for r in range(12) for c in range(1, r+1)]
print(a) # Michael S. Branicky, Sep 07 2022

flatten([[2*n*k +n +k for k in range(1,n+1)] for n in range(1,14)]) # G. C. Greubel, Oct 17 2023

From G. C. Greubel, Oct 17 2023: (Start)
T(n, 1) = A016777(n).
T(n, 2) = A016873(n).
T(n, 3) = A017017(n).
T(n, 4) = A017209(n).
T(n, 5) = A017449(n).
T(n, 6) = A186113(n).
T(n, n-1) = A056220(n).
T(n, n-2) = A090288(n-2).
T(n, n-3) = A271625(n-2).
T(n, n) = 4*A000217(n).
T(2*n, n) = A033954(n).
Sum_{k=1..n} T(n, k) = A162254(n).
Sum_{k=1..n} (-1)^(k-1)*T(n, k) = A182868((n+1)/2) if n is odd otherwise A182868(n/2) + 1. (End)

Edited by N. J. A. Sloane, Jul 23 2009

Name edited by Michael S. Branicky, Sep 07 2022

A237930 a(n) = 3^(n+1) + (3^n-1)/2.

Original entry on oeis.org

3, 10, 31, 94, 283, 850, 2551, 7654, 22963, 68890, 206671, 620014, 1860043, 5580130, 16740391, 50221174, 150663523, 451990570, 1355971711, 4067915134, 12203745403, 36611236210, 109833708631, 329501125894, 988503377683, 2965510133050, 8896530399151

Offset: 0

Philippe Deléham, Feb 16 2014

a(n-1) agrees with the graph radius of the n-Sierpinski carpet graph for n = 2 to at least n = 5. See A100774 for the graph diameter of the n-Sierpinski carpet graph.
The inverse binomial transform gives 3, 7, 14, 28, 56, ... i.e., A005009 with a leading 3. - R. J. Mathar, Jan 08 2020
First differences of A108765. The digital root of a(n) for n > 1 is always 4. a(n) is never divisible by 7 or by 12. a(n) == 10 (mod 84) for odd n. a(n) == 31 (mod 84) for even n > 0. Conjecture: This sequence contains no prime factors p == {11, 13, 23, 61 71, 73} (mod 84). - Klaus Purath, Apr 13 2020
This is a subsequence of A017209 for n > 1. See formula. - Klaus Purath, Jul 03 2020

			Ternary....................Decimal
10...............................3
101.............................10
1011............................31
10111...........................94
101111.........................283
1011111........................850
10111111......................2551
101111111.....................7654, etc.

Colin Barker, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Graph Radius.
Eric Weisstein's World of Mathematics, Sierpinski Carpet Graph.
Index entries for linear recurrences with constant coefficients, signature (4,-3).

Cf. A000244, A003462, A005009, A005032 (first differences), A017209, A060816, A100774, A108765 (partial sums), A199109, A329774.

Cf. A024023, A029858, A057198, A116952, A171498.

[3^(n+1) + (3^n-1)/2: n in [0..40]]; // Vincenzo Librandi, Jan 09 2020

(* Start from Eric W. Weisstein, Mar 13 2018 *)
Table[(7 3^n - 1)/2, {n, 0, 20}]
(7 3^Range[0, 20] - 1)/2
LinearRecurrence[{4, -3}, {10, 31}, {0, 20}]
CoefficientList[Series[(3 - 2 x)/((x - 1) (3 x - 1)), {x, 0, 20}], x]
(* End *)

Vec((3 - 2*x) / ((1 - x)*(1 - 3*x)) + O(x^30)) \\ Colin Barker, Nov 27 2019

G.f.: (3-2*x)/((1-x)*(1-3*x)).
a(n) = A000244(n+1) + A003462(n).
a(n) = 3*a(n-1) + 1 for n > 0, a(0)=3. (Note that if a(0) were 1 in this recurrence we would get A003462, if it were 2 we would get A060816. - N. J. A. Sloane, Dec 06 2019)
a(n) = 4*a(n-1) - 3*a(n-2) for n > 1, a(0)=3, a(1)=10.
a(n) = 2*a(n-1) + 3*a(n-2) + 2 for n > 1.
a(n) = A199109(n) - 1.
a(n) = (7*3^n - 1)/2. - Eric W. Weisstein, Mar 13 2018
From Klaus Purath, Apr 13 2020: (Start)
a(n) = A057198(n+1) + A024023(n).
a(n) = A029858(n+2) - A024023(n).
a(n) = A052919(n+1) + A029858(n+1).
a(n) = (A000244(n+1) + A171498(n))/2.
a(n) = 7*A003462(n) + 3.
a(n) = A116952(n) + 2. (End)
a(n) = A017209(7*(3^(n-2)-1)/2 + 3), n > 1. - Klaus Purath, Jul 03 2020
E.g.f.: exp(x)*(7*exp(2*x) - 1)/2. - Stefano Spezia, Aug 28 2023

A228949 Coins left when packing boomerangs into n X n coins.

Original entry on oeis.org

4, 3, 7, 13, 6, 13, 22, 9, 19, 31, 12, 25, 40, 15, 31, 49, 18, 37, 58, 21, 43, 67, 24, 49, 76, 27, 55, 85, 30, 61, 94, 33, 67, 103, 36, 73, 112, 39, 79, 121, 42, 85, 130, 45, 91, 139, 48, 97, 148, 51, 103, 157, 54, 109, 166, 57

Offset: 2

Kival Ngaokrajang, Oct 05 2013

The coins left after packing boomerangs into n X n coins using the same rule as A229593. See illustration in links.

Kival Ngaokrajang, Illustration of initial terms
Index entries for linear recurrences with constant coefficients, signature (0,0,2,0,0,-1).

Cf. A229593 (Boomerangs), A229598 (Voids), A017209 (trisection), A008585 (trisection), A016921 (trisection).

LinearRecurrence[{0,0,2,0,0,-1},{4,3,7,13,6,13},80] (* Harvey P. Dale, Jan 19 2019 *)

G.f. -x^2*(-4-3*x-7*x^2-5*x^3+x^5) / ( (x-1)^2*(1+x+x^2)^2 ). - R. J. Mathar, Oct 16 2013

cp's OEIS Frontend

A144829 Partial products of successive terms of A017209; a(0)=1 .

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A017173 a(n) = 9*n + 1.

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A002278 a(n) = 4*(10^n - 1)/9.

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A017221 a(n) = 9*n + 5.

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A056991 Numbers with digital root 1, 4, 7 or 9.

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A060816 a(0) = 1; a(n) = (5*3^(n-1) - 1)/2 for n > 0.

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A017233 a(n) = 9*n + 6.

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A083487 Triangle read by rows: T(n,k) = 2nk + n + k (1 <= k <= n).