A017185 -id:A017185 - cp's OEIS frontend

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

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

A002276 a(n) = 2*(10^n - 1)/9.

Original entry on oeis.org

0, 2, 22, 222, 2222, 22222, 222222, 2222222, 22222222, 222222222, 2222222222, 22222222222, 222222222222, 2222222222222, 22222222222222, 222222222222222, 2222222222222222, 22222222222222222, 222222222222222222, 2222222222222222222, 22222222222222222222, 222222222222222222222

Offset: 0

N. J. A. Sloane

a(n) is also the total number of holes in a variation of a box fractal as in illustration. - Kival Ngaokrajang, May 23 2014 [As observed by Hans Havermann, this seems to be incorrect: e.g., for n = 2 the illustration shows 28 small holes plus two larger holes. - M. F. Hasler, Oct 05 2020]

Ivan Panchenko, Table of n, a(n) for n = 0..200
Kival Ngaokrajang, Illustration for n = 1..4.
Index entries for linear recurrences with constant coefficients, signature (11,-10).

Cf. A002275, A002277, A002278, A002279, A002280, A002281, A002282, A002283, A178630, A178634.

Cf. A010785, A017185.

LinearRecurrence[{11, -10}, {0, 2}, 50] (* Jinyuan Wang, Feb 27 2020 *)

A002276(n):=2*(10^n - 1)/9$
makelist(A002276(n),n,0,20); /* Martin Ettl, Nov 12 2012 */

a(n)=10^n\9*2 \\ M. F. Hasler, Mar 27 2015

a(n) = A178630(n)/A002283(n). - Reinhard Zumkeller, May 31 2010
From Vincenzo Librandi, Jul 22 2010: (Start)
a(n) = a(n-1) + 2*10^(n-1) with a(0) = 0.
a(n) = 11*a(n-1) - 10*a(n-2) with a(0) = 0, a(1) = 2. (End)
G.f.: 2*x/((1 - x)*(1 - 10*x)). - Ilya Gutkovskiy, Feb 24 2017
E.g.f.: 2*exp(x)*(exp(9*x) - 1)/9. - Stefano Spezia, Sep 13 2023
From Elmo R. Oliveira, Jul 19 2025: (Start)
a(n) = 2*A002275(n).
a(n) = A010785(A017185(n-1)) for n >= 1. (End)

A017209 a(n) = 9*n + 4.

Original entry on oeis.org

4, 13, 22, 31, 40, 49, 58, 67, 76, 85, 94, 103, 112, 121, 130, 139, 148, 157, 166, 175, 184, 193, 202, 211, 220, 229, 238, 247, 256, 265, 274, 283, 292, 301, 310, 319, 328, 337, 346, 355, 364, 373, 382, 391, 400, 409, 418, 427, 436, 445, 454, 463, 472, 481

Offset: 0

N. J. A. Sloane

Numbers whose digital root is 4. - L. Edson Jeffery, Nov 26 2016

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).

Cf. A008591, A010888, A017185, A017197.

[9*n+4: n in [0..60]]; // Vincenzo Librandi, Jul 23 2011

Range[4, 1000, 9] (* Vladimir Joseph Stephan Orlovsky, May 28 2011 *)

forstep(n=4,500,9,print1(n", ")) \\ Charles R Greathouse IV, May 28 2011

G.f.: (4 + 5*x)/(x - 1)^2. - R. J. Mathar, Jul 14 2016
A010888(a(n)) = 4. - L. Edson Jeffery, Nov 26 2016
E.g.f.: exp(x)*(4 + 9*x). - Stefano Spezia, Dec 25 2022

A247676 Odd composite numbers congruent to 2 modulo 9.

Original entry on oeis.org

65, 119, 155, 209, 245, 299, 335, 371, 407, 425, 497, 515, 533, 551, 605, 623, 695, 713, 731, 749, 767, 785, 803, 875, 893, 965, 1001, 1037, 1055, 1073, 1127, 1145, 1199, 1235, 1253, 1271, 1325, 1343, 1379, 1397, 1415, 1469, 1505, 1541, 1577, 1595, 1631, 1649

Offset: 1

Odimar Fabeny, Sep 22 2014

Subsequence of A017185 (9n+2).

Composites == 11 mod 18. - Robert Israel, Sep 24 2014

Odimar Fabeny, Table of n, a(n) for n = 1..10000

Cf. A017185, A247678, A247679, A247681, A247682, A247683.

remove(isprime,[seq(18*k+11,k=1..1000)]); # Robert Israel, Sep 24 2014

Select[18Range[100] + 11, Not[PrimeQ[#]] &] (* Alonso del Arte, Sep 25 2014 *)
Select[Range[11,2000,18],CompositeQ] (* Harvey P. Dale, Oct 29 2023 *)

lista(nn) = {forcomposite(n=1, nn, if ((n % 2) && ((n % 9) == 2), print1(n, ", ")););} \\ Michel Marcus, Sep 22 2014

A154680 Triangle read by rows where T(m,n)=2mn + m + n - 2.

Original entry on oeis.org

2, 5, 10, 8, 15, 22, 11, 20, 29, 38, 14, 25, 36, 47, 58, 17, 30, 43, 56, 69, 82, 20, 35, 50, 65, 80, 95, 110, 23, 40, 57, 74, 91, 108, 125, 142, 26, 45, 64, 83, 102, 121, 140, 159, 178, 29, 50, 71, 92, 113, 134, 155, 176, 197, 218, 32, 55, 78, 101, 124, 147, 170, 193, 216, 239, 262

Offset: 1

Vincenzo Librandi, Jan 18 2009

All terms are in A153052.

First column: A016789; second column: 5*A000027; third column: A016993; fourth column: A017185. - Vincenzo Librandi, Nov 18 2012

			Triangle begins:
2;
5,  10;
8,  15, 22;
11, 20, 29, 38;
14, 25, 36, 47, 58;
17, 30, 43, 56, 69,  82;
20, 35, 50, 65, 80,  95,  110;
23, 40, 57, 74, 91,  108, 125, 142;
26, 45, 64, 83, 102, 121, 140, 159, 178;
29, 50, 71, 92, 113, 134, 155, 176, 197, 218; etc.

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

Cf. A153052, A089038, A016789, A000027, A017185.

[2*n*k+n+k-2: k in [1..n], n in [1..11]]; // Vincenzo Librandi, Nov 18 2012

Flatten[Table[Floor[2 n m + m + n - 2], {n, 1, 16}, {m, n}]] (* Vincenzo Librandi, May 14 2012 *)

A163672 Triangle T(n,m) = 2mn + m + n + 7 read by rows.

Original entry on oeis.org

11, 14, 19, 17, 24, 31, 20, 29, 38, 47, 23, 34, 45, 56, 67, 26, 39, 52, 65, 78, 91, 29, 44, 59, 74, 89, 104, 119, 32, 49, 66, 83, 100, 117, 134, 151, 35, 54, 73, 92, 111, 130, 149, 168, 187, 38, 59, 80, 101, 122, 143, 164, 185, 206, 227, 41, 64, 87, 110, 133, 156, 179

Offset: 1

Vincenzo Librandi, Aug 03 2009

2*T(n,n) - 13 = (2n+1)^2.

The numbers 2*T(m,n)-13 =(2*n+1)*(2*m+1) are not prime. Also: first column: A016789; second column: A016897; third column: A017017; fourth column: A017185. - Vincenzo Librandi, Nov 20 2012

			Triangle begins:
  11;
  14,  19;
  17,  24,  31;
  20,  29,  38,  47;
  23,  34,  45,  56,  67;
  26,  39,  52,  65,  78,  91;
  29,  44,  59,  74,  89, 104, 119;
  32,  49,  66,  83, 100, 117, 134, 151;

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

Cf. A016789, A016897, A017017, A017185, A153049, A163674, A163657.

[2*n*k + n + k + 7: k in [1..n], n in [1..11]]; // Vincenzo Librandi, Nov 20 2012

t[n_,k_]:=2 n*k + n + k + 7; Table[t[n, k], {n, 15}, {k, n}]//Flatten (* Vincenzo Librandi, Nov 20 2012 *)

for(n=1,10, for(m=1,n, print1(2*m*n + m + n + 7, ", "))) \\ G. C. Greubel, Aug 02 2017

T(n,m) = A163674(n,m)-2 = A163657(n,m)-1.

Edited by R. J. Mathar, Oct 12 2009

A181418 a(n) = A000984(n)*A000172(n), which is the term-wise product of the Central binomial coefficients and Franel numbers, respectively.

Original entry on oeis.org

1, 4, 60, 1120, 24220, 567504, 14030016, 360222720, 9513014940, 256758913840, 7051260776560, 196403499277440, 5535202897806400, 157551884911456000, 4522682234563776000, 130783762623673221120, 3806221127760278029980

Offset: 0

Paul D. Hanna, Jan 28 2011

This sequence is s_6 in Cooper's paper. - Jason Kimberley, Nov 25 2012

Diagonal of the rational function R(x,y,z,w)=1/(1-(w*x*y+w*z+x*y+x*z+y+z)). - Gheorghe Coserea, Jul 13 2016

			E.g.f.: A(x) = 1 + 4*x/2! + 60*x^2/(2!*4!) + 1120*x^3/(3!*6!) + 24220*x^4/(4!*8!) + 567504*x^5/(5!*10!) +....
where A(x)^(1/2) = 1 + x + x^2/2!^3 + x^3/3!^3 + x^4/4!^3 +x^5/5!^3 +...

Jason Kimberley, Table of n, a(n) for n = 0..226
S. Cooper, Sporadic sequences, modular forms and new series for 1/pi, Ramanujan J. (2012).
Timothy Huber, Daniel Schultz, and Dongxi Ye, Ramanujan-Sato series for 1/pi, Acta Arith. (2023) Vol. 207, 121-160. See p. 11.

Cf. A000984, A000172, A199813.

Related to diagonal of rational functions: A268545-A268555.

Table[Binomial[2n,n]*Sum[Binomial[n,k]^3,{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Mar 06 2014 *)

{a(n)=binomial(2*n,n)*sum(k=0,n,binomial(n,k)^3)}

{a(n)=(2*n)!*n!*polcoeff(sum(m=0, n, x^m/m!^3+x*O(x^n))^2, n)}

a(n) = C(2n,n) * Sum_{k=0..n} C(n,k)^3.
E.g.f.: Sum_{n>=0} a(n)*x^n/(n!*(2*n)!) = ( Sum_{n>=0} x^n/n!^3 )^2.
From Jason Kimberley, Nov 26 2012: (Start)
1/Pi
  = (2/25)*Sum_{n>=0} a(n)*(9n+2)/50^n. [Cooper, equation (5)]
  = (2/25)*Sum_{n>=0} a(n)*A017185(n)/A165800(n). (End)
G.f.: 4*hypergeom([1/6, 1/3],[1],(27/2)*(1+(1-32*x)^(1/2))*(1-(1-32*x)^(1/2))^2/(3+(1-32*x)^(1/2))^3)^2/(3+(1-32*x)^(1/2)). - Mark van Hoeij, May 07 2013
Recurrence: n^3*a(n) = 2*(2*n-1)*(7*n^2 - 7*n + 2)*a(n-1) + 32*(n-1)*(2*n-3)*(2*n-1)*a(n-2). - Vaclav Kotesovec, Mar 06 2014
a(n) ~ 2^(5*n+1) / (sqrt(3) * (Pi*n)^(3/2)). - Vaclav Kotesovec, Mar 06 2014
0 = (-x^2+28*x^3+128*x^4)*y''' + (-3*x+126*x^2+768*x^3)*y'' + (-1+92*x+864*x^2)*y' + (4+96*x)*y, where y is g.f. - Gheorghe Coserea, Jul 13 2016

A063289 Dimension of the space of weight n cuspidal newforms for Gamma_1( 16 ).

Original entry on oeis.org

-1, 2, 7, 11, 16, 20, 25, 29, 34, 38, 43, 47, 52, 56, 61, 65, 70, 74, 79, 83, 88, 92, 97, 101, 106, 110, 115, 119, 124, 128, 133, 137, 142, 146, 151, 155, 160, 164, 169, 173, 178, 182, 187, 191, 196, 200, 205, 209, 214, 218, 223, 227, 232, 236

Offset: 2

N. J. A. Sloane, Jul 14 2001

It appears that for n > 2 a(n) = floor((9n-22)/2). - Gary Detlefs, Mar 02 2010

William A. Stein, Dimensions of the spaces S_k^{new}(Gamma_1(N)).
William A. Stein, The modular forms database.
Index entries for linear recurrences with constant coefficients, signature (1, 1, -1).

Cf. A063232, A063233, A017185 (bisection), A130880, A332438.

Join[{-1}, Table[9*n/2 + (-1)^n/4 - 45/4, {n, 3, 60}]] (* Amiram Eldar, Jan 12 2024 *)

a(n) = 9*n/2 + (-1)^n/4 - 45/4 for n >= 3, with first differences in A010710. - R. J. Mathar, Dec 06 2010
From M. F. Hasler, Mar 05 2012: (Start)
G.f.: x^2*(-1 + 3*x + 6*x^2 + x^3)/(1 - x - x^2 + x^3).
a(n+2) = a(n)+9 (n>2), a(2n+1) = a(2n)+4 (n>1), a(2n) = a(2n-1)+5 (n>1). (End)
Sum_{n>=3} (-1)^(n+1)/a(n) = cot(2*Pi/9)*Pi/9. - Amiram Eldar, Jan 12 2024
From Amiram Eldar, Nov 22 2024: (Start)
Product_{n>=3} (1 - (-1)^n/a(n)) = 2*sin(Pi/18) + 1 (= A130880 + 1).
Product_{n>=3} (1 + (-1)^n/a(n)) = (1/2) * sec(Pi/9) (= A332438 - 3). (End)

A139610 a(n) = 45*n + 10.

Original entry on oeis.org

10, 55, 100, 145, 190, 235, 280, 325, 370, 415, 460, 505, 550, 595, 640, 685, 730, 775, 820, 865, 910, 955, 1000, 1045, 1090, 1135, 1180, 1225, 1270, 1315, 1360, 1405, 1450, 1495, 1540, 1585, 1630, 1675, 1720, 1765, 1810, 1855, 1900

Offset: 0

Omar E. Pol, Apr 27 2008

Numbers of the 10th column of positive numbers in the square array of nonnegative and polygonal numbers A139600.

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Index to sequences related to polygonal numbers
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A017185, A057145, A062708, A139600.

[5*(9*n + 2): n in [0..60]]; // Vincenzo Librandi, Jul 23 2011

45*Range[0,50]+10 (* or *) LinearRecurrence[{2,-1},{10,55},50] (* Harvey P. Dale, Dec 27 2021 *)

a(n)=45*n+10 \\ Charles R Greathouse IV, Oct 05 2011

a(n) = A057145(n+2,10).
G.f.: 5*(2+7*x)/(x-1)^2. - R. J. Mathar, Jul 28 2016
From Elmo R. Oliveira, Apr 16 2024: (Start)
E.g.f.: 5*exp(x)*(2 + 9*x).
a(n) = 5*A017185(n) = 5*(A062708(n+1) - A062708(n)).
a(n) = 2*a(n-1) - a(n-2) for n >= 2. (End)

A157262 a(n) = 36n^2 - 55n + 21.

Original entry on oeis.org

2, 55, 180, 377, 646, 987, 1400, 1885, 2442, 3071, 3772, 4545, 5390, 6307, 7296, 8357, 9490, 10695, 11972, 13321, 14742, 16235, 17800, 19437, 21146, 22927, 24780, 26705, 28702, 30771, 32912, 35125, 37410, 39767, 42196, 44697, 47270

Offset: 1

Vincenzo Librandi, Feb 26 2009

The identity (10368*n^2 - 15840*n + 6049)^2 - (36*n^2 - 55*n + 21)*(1728*n - 1320)^2 = 1 can be written as A157264(n)^2 - a(n)*A157263(n)^2 = 1. - Vincenzo Librandi, Jan 27 2012

The continued fraction expansion of sqrt(a(n)) is [6n-5; {2, 2, 2, 12n-10}]. For n=1, this collapses to [1; {2}]. - Magus K. Chu, Sep 05 2022

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

Cf. A157263, A157264.

I:=[2, 55, 180]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..40]]; // Vincenzo Librandi, Jan 27 2012

LinearRecurrence[{3,-3,1},{2,55,180},40] (* Vincenzo Librandi, Jan 27 2012 *)
Table[36*n^2-55*n+21, {n,1,30}] (* G. C. Greubel, Feb 04 2018 *)

a(n)=36*n^2-55*n+21 \\ Charles R Greathouse IV, Dec 28 2011

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Vincenzo Librandi, Jan 27 2012
G.f.: x*(-2-49*x-21*x^2)/(x-1)^3. - Vincenzo Librandi, Jan 27 2012
a(n) = A016813(n-1)*A017185(n-1). - Bruno Berselli, Jan 27 2012
E.g.f.: (36*x^2 - 19*x + 21)*exp(x) - 21. - G. C. Greubel, Feb 04 2018

cp's OEIS Frontend

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

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A002276 a(n) = 2*(10^n - 1)/9.

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A017209 a(n) = 9*n + 4.

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A247676 Odd composite numbers congruent to 2 modulo 9.

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A154680 Triangle read by rows where T(m,n)=2*m*n + m + n - 2.

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Magma

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A163672 Triangle T(n,m) = 2mn + m + n + 7 read by rows.

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Magma

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A181418 a(n) = A000984(n)*A000172(n), which is the term-wise product of the Central binomial coefficients and Franel numbers, respectively.

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A154680 Triangle read by rows where T(m,n)=2mn + m + n - 2.

A157262 a(n) = 36n^2 - 55n + 21.