A017173 -id:A017173 - cp's OEIS frontend

A050489 a(n) = C(n)(10n + 1) where C(n) = Catalan numbers (A000108).

1, 11, 42, 155, 574, 2142, 8052, 30459, 115830, 442442, 1696396, 6525246, 25169452, 97319900, 377096040, 1463921595, 5692584870, 22169259090, 86452604700, 337547269290, 1319388204420, 5162382341220, 20217646564440, 79246770753150, 310866899505084

Offset: 0

Barry E. Williams, Dec 27 1999

Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

Andrew Howroyd, Table of n, a(n) for n = 0..200

Column k=10 of A330965.

Cf. A017173, A027810, A000108.

[Catalan(n)*(10*n+1):n in [0..30] ]; // Marius A. Burtea, Jan 05 2020

Table[CatalanNumber[n](10n+1),{n,0,30}] (* Harvey P. Dale, Jul 19 2011 *)

a(n)=binomial(2*n,n)/(n+1)*(10*n+1) \\ Charles R Greathouse IV, Oct 23 2023

-(n+1)*(10*n-9)*a(n) + 2*(10*n+1)*(2*n-1)*a(n-1) = 0. - R. J. Mathar, Dec 03 2014
From Stefano Spezia, Feb 16 2020: (Start)
O.g.f.: 2*(1 + sqrt(1 - 4*x) + 16*x)/((1 + sqrt(1 - 4*x))^2*sqrt(1 - 4*x)).
E.g.f.: exp(2*x)*(I_0(2*x) + 9*I_1(2*x)), where I_n(x) is the modified Bessel function of the first kind.
(End)
G.f.: (9 - 16*x - 9*sqrt(1 - 4*x))/(2*x*sqrt(1 - 4*x)). - Amiram Eldar, Jul 08 2023
From Peter Bala, Aug 23 2025: (Start)
a(n) = binomial(2*n, n) + 9*binomial(2*n, n-1) = A000984(n) + 9*A001791(n).
a(n) ~ 4^n * 10/sqrt(Pi*n). (End)

Corrected and extended by Harvey P. Dale, Jul 19 2011

A083678 Numbers m = d_1 d_2 ... d_k (in base 10) with properties that k is even and d_i + d_{k+1-i} = 10 for all i.

Original entry on oeis.org

19, 28, 37, 46, 55, 64, 73, 82, 91, 1199, 1289, 1379, 1469, 1559, 1649, 1739, 1829, 1919, 2198, 2288, 2378, 2468, 2558, 2648, 2738, 2828, 2918, 3197, 3287, 3377, 3467, 3557, 3647, 3737, 3827, 3917, 4196, 4286, 4376, 4466, 4556, 4646, 4736, 4826, 4916

Offset: 1

Zak Seidov Jun 15 2003

The two-digit terms here occur in many sequences, e.g., A066686, A081926, A017173, A030108, A043457, A052224, A061388, A084364.

			1469 and 6284 are members because 1+9=4+6=10 and 6+4=2+8=10.

Harvey P. Dale, Table of n, a(n) for n = 1..819 (All terms through 999999.)

Cf. A066686, A081926, A017173, A030108, A043457, A052224, A061388, A084364.

ok10Q[n_]:=Module[{idn=IntegerDigits[n]},idn[[1]]+idn[[4]]==idn[[2]]+idn[[3]]==10]; Join[ Select[ Range[10,99],Total[IntegerDigits[#]]==10&],Select[Range[1000,9999],ok10Q]] (* Harvey P. Dale, Oct 14 2023 *)

isok(n) = {digs = digits(n); if (#digs % 2 == 0, for (i = 1, #digs/2, if ((digs[i] + digs[#digs+1-i]) ! = 10, return (0));); return (1);); return (0);} \\ Michel Marcus, Oct 05 2013

A139619 a(n) = 171*n + 19.

Original entry on oeis.org

19, 190, 361, 532, 703, 874, 1045, 1216, 1387, 1558, 1729, 1900, 2071, 2242, 2413, 2584, 2755, 2926, 3097, 3268, 3439, 3610, 3781, 3952, 4123, 4294, 4465, 4636, 4807, 4978, 5149, 5320, 5491, 5662, 5833, 6004, 6175, 6346, 6517, 6688

Offset: 0

Omar E. Pol, May 21 2008

Numbers of the 19th column of positive numbers in the square array of nonnegative and polygonal numbers A139600. Also, numbers of the 19th column in the square array A057145.

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

Cf. A016957, A017329, A057145, A139600, A139606-A139618, A139620.

Cf. A017173, A051682.

[19*(9*n + 1): n in [0..50]]; // Vincenzo Librandi, Jul 23 2011

Range[0, 100]*171 + 19 (* Paolo Xausa, Apr 17 2024 *)

a(n)=171*n+19 \\ Charles R Greathouse IV, Oct 05 2011

From Chai Wah Wu, Apr 14 2017: (Start)
a(n) = 2*a(n-1) - a(n-2) for n > 1.
G.f.: (152*x + 19)/(x - 1)^2. (End)
From Elmo R. Oliveira, Apr 11 2024: (Start)
E.g.f.: 19*exp(x)*(1 + 9*x).
a(n) = 19*A017173(n) = 19*(A051682(n+1) - A051682(n)). (End)

A156145 Number of partitions of 9*n-8 into parts having in decimal representation digital root 1.

Original entry on oeis.org

1, 2, 3, 5, 8, 13, 20, 31, 46, 68, 98, 140, 195, 271, 370, 502, 673, 897, 1183, 1553, 2021, 2618, 3367, 4312, 5489, 6961, 8782, 11039, 13815, 17232, 21409, 26519, 32732, 40288, 49433, 60496, 73824, 89872, 109125, 132204, 159785, 192715, 231921

Offset: 1

Reinhard Zumkeller, Feb 05 2009

A010888(9*n-8) = 1;

record values in A156144: a(n)=A156144(A017173(n-1)).

A163652 Triangle read by rows where T(n,m)=2mn + m + n + 6.

Original entry on oeis.org

10, 13, 18, 16, 23, 30, 19, 28, 37, 46, 22, 33, 44, 55, 66, 25, 38, 51, 64, 77, 90, 28, 43, 58, 73, 88, 103, 118, 31, 48, 65, 82, 99, 116, 133, 150, 34, 53, 72, 91, 110, 129, 148, 167, 186, 37, 58, 79, 100, 121, 142, 163, 184, 205, 226, 40, 63, 86, 109, 132, 155, 178

Offset: 1

Vincenzo Librandi, Aug 02 2009

The numbers 2*T(n,m)-11 = (2*n+1)*(2*m+1) are not prime, and 2*T(n,n) = (2n+1)^2.

First column: A112414, second column: A016885, third column: A017005, fourth column: A017173. - Vincenzo Librandi, Nov 20 2012

			Triangle begins:
  10;
  13, 18;
  16, 23, 30;
  19, 28, 37, 46;
  22, 33, 44, 55,  66;
  25, 38, 51, 64,  77,  90;
  28, 43, 58, 73,  88,  103, 118;
  31, 48, 65, 82,  99,  116, 133, 150;
  34, 53, 72, 91,  110, 129, 148, 167, 186;
  37, 58, 79, 100, 121, 142, 163, 184, 205, 226;
  40, 63, 86, 109, 132, 155, 178, 201, 224, 247, 270;
  etc.

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

Cf. A153045, A154685, A163657, A112414, A016885, A017005, A017173.

[2*n*k + n + k + 6: k in [1..n], n in [1..11]]; // Vincenzo Librandi, Nov 20 2012

t[n_,k_]:=2 n*k + n + k +  6; Table[t[n, k], {n, 15}, {k, n}]//Flatten (* Vincenzo Librandi, Nov 20 2012 *)

T(n,m) = A154685(n,m)+2 = A163657(n,m)-2. [R. J. Mathar, Oct 22 2009]

Comment clarified by R. J. Mathar, Oct 22 2009

A211822 Nonprime numbers with all divisors with additive digital root of 1.

Original entry on oeis.org

1, 361, 703, 1369, 1387, 2071, 2413, 2701, 3097, 3439, 3781, 4033, 4699, 5149, 5329, 5833, 6031, 6697, 6859, 7201, 7363, 7543, 7957, 8227, 9253, 9271, 9937, 10027, 10279, 10963, 11359, 11647, 11881, 11899, 11989, 13213, 13357

Offset: 1

Jaroslav Krizek, Apr 26 2012

Complement of A061237 (prime numbers == 1 (mod 9)) with respect to A211821.

			Number 6859 with divisors 1, 19, 361, 6859 is in sequence because all divisors have additive digital root of 1.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A211821, A024906, A061237, A017173, A211823.

filter:= n -> not isprime(n) and numtheory:-factorset(n) mod 9 = {1}:
filter(1):= true:
select(filter, [seq(i,i=1..20000,9)]); # Robert Israel, May 10 2020

(* First run the program for A211821 *) Select[A211821, Not[PrimeQ[#]] &] (* Alonso del Arte, May 02 2012 *)

A211823 Numbers k such that 9*k+1 are numbers with all divisors with additive digital root = 1.

Original entry on oeis.org

0, 2, 4, 8, 12, 14, 18, 20, 22, 30, 34, 40, 42, 44, 48, 54, 58, 60, 64, 68, 70, 78, 82, 84, 90, 92, 98, 102, 104, 110, 112, 118, 124, 128, 130, 142, 144, 152, 154, 158, 162, 170, 172, 174, 180, 184, 188, 194, 198, 200, 208, 222, 224, 228, 230, 232, 238, 240, 242

Offset: 1

Jaroslav Krizek, Apr 26 2012

Numbers of form 9*k+1 with all divisors with digital root = 1 is in A211821.

Supersequence of A024906 (numbers n such that 9*n+1 is prime).

			Number k = 40 is in sequence because number 9*40 + 1 = 361 is number with all divisors (1, 19, 361) with additive digital root = 1.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A211821, A024906, A061237, A017173, A211822

adrQ[n_]:=NestWhile[Total[IntegerDigits[#]]&,n,#>9&]==1; Select[Range[ 0,250],AllTrue[Divisors[9#+1],adrQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 27 2020 *)

Corrected (230 inserted) by Harvey P. Dale, Aug 27 2020

A326833 Numbers whose sum of digits is a power of 10.

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, 208, 217, 226, 235, 244, 253, 262, 271, 280, 307, 316, 325, 334, 343, 352, 361, 370, 406, 415, 424, 433, 442, 451, 460, 505, 514, 523, 532, 541, 550, 604, 613

Offset: 1

Alois P. Heinz, Oct 20 2019

Alois P. Heinz, Table of n, a(n) for n = 1..43758 (all terms with up to 9 digits)

Subsequence of A326806.

Cf. A007953, A017173, A028838, A052224, A328560.

q:= n-> (m-> m>0 and m=10^ilog[10](m))(add(i, i=convert(n, base, 10))):
select(q, [$1..1000])[];

isok(n) = my(s=sumdigits(n), k); (s==1) || (s==10) || (ispower(s,,&k) && (k==10)); \\ Michel Marcus, Oct 21 2019

A172178 a(n) = 99*n + 1.

Original entry on oeis.org

1, 100, 199, 298, 397, 496, 595, 694, 793, 892, 991, 1090, 1189, 1288, 1387, 1486, 1585, 1684, 1783, 1882, 1981, 2080, 2179, 2278, 2377, 2476, 2575, 2674, 2773, 2872, 2971, 3070, 3169, 3268, 3367, 3466, 3565, 3664, 3763, 3862, 3961, 4060, 4159, 4258

Offset: 0

Mark Dols, Jan 28 2010

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A017173.

[99*n+1: n in [0..50]]; // G. C. Greubel, Apr 26 2022

99*Range[0, 50] + 1 (* or *) LinearRecurrence[{2, -1}, {1, 100}, 50] (* Harvey P. Dale, Apr 09 2012 *)

[99*n+1 for n in (0..50)] # G. C. Greubel, Apr 26 2022

a(n) = a(n-1) + 99 with a(1) = 1.
a(0) = 1, a(1) = 100, a(n) = 2*a(n-1) - a(n-2). - Harvey P. Dale, Apr 09 2012
From G. C. Greubel, Apr 26 2022: (Start)
G.f.: (1 + 98*x)/(1-x)^2.
E.g.f.: (1 + 99*x)*exp(x). (End)

Offset corrected by Harvey P. Dale, Apr 09 2012

A179955 Numbers whose sum of digits is 10 and which contain no 0 digits.

Original entry on oeis.org

19, 28, 37, 46, 55, 64, 73, 82, 91, 118, 127, 136, 145, 154, 163, 172, 181, 217, 226, 235, 244, 253, 262, 271, 316, 325, 334, 343, 352, 361, 415, 424, 433, 442, 451, 514, 523, 532, 541, 613, 622, 631, 712, 721, 811, 1117, 1126, 1135, 1144, 1153, 1162, 1171

Offset: 1

Dominick Cancilla, Aug 03 2010

Subset of A052224.
Finite sequence. Highest member is 1111111111.
Contribution from Zak Seidov, Aug 06 2010, as corrected by D. S. McNeil: There are exactly 511 terms.

			19 is an element of the list because 1+9 = 10.
109 is not an element because it contains a 0.

Alois P. Heinz, Table of n, a(n) for n = 1..511

Cf. A007953, A017173, A052224.

with(combinat):
sort(select(y->y<>10, map(x->parse(cat(x[])), map(p->permute(p)[], partition(10)))))[]; # Alois P. Heinz, Sep 24 2013

Reap[For[n=1; k=1, n <= 2*10^9, n++, id = IntegerDigits[n]; If[Total[id] == 10 && FreeQ[id, 0], Print["a(", k, ") = ", n]; Sow[n]; k++]]][[2, 1]] (* Jean-François Alcover, Jan 08 2016 *)
sd10Q[n_]:=Module[{idn=IntegerDigits[n]},FreeQ[idn,0]&&Total[idn]==10]; Select[Range[1200],sd10Q] (* Harvey P. Dale, Oct 17 2016 *)

cp's OEIS Frontend

A050489 a(n) = C(n)*(10*n + 1) where C(n) = Catalan numbers (A000108).

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A083678 Numbers m = d_1 d_2 ... d_k (in base 10) with properties that k is even and d_i + d_{k+1-i} = 10 for all i.

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A139619 a(n) = 171*n + 19.

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Magma

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A156145 Number of partitions of 9*n-8 into parts having in decimal representation digital root 1.

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A163652 Triangle read by rows where T(n,m)=2*m*n + m + n + 6.

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Magma

Mathematica

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A211822 Nonprime numbers with all divisors with additive digital root of 1.

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Maple

Mathematica

A211823 Numbers k such that 9*k+1 are numbers with all divisors with additive digital root = 1.

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Mathematica

Extensions

A326833 Numbers whose sum of digits is a power of 10.

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A050489 a(n) = C(n)(10n + 1) where C(n) = Catalan numbers (A000108).

A163652 Triangle read by rows where T(n,m)=2mn + m + n + 6.