A017281 -id:A017281 - cp's OEIS frontend

A050490 a(n) = C(n)*(11n+1) where C(n) = Catalan numbers (A000108).

1, 12, 46, 170, 630, 2352, 8844, 33462, 127270, 486200, 1864356, 7171892, 27665596, 106977600, 414538200, 1609344270, 6258307590, 24373220520, 95050101300, 371125269900, 1450670612820, 5676173948640, 22230262964520, 87137141867100, 341824599040860, 1341897206800752

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=11 of A330965.

Cf. A017281, A027810, A000108.

[Catalan(n)*(11*n+1):n in [0..25] ]; // Marius A. Burtea, Jan 05 2020

Table[CatalanNumber[n](11n+1),{n,0,20}] (* Harvey P. Dale, Jul 12 2018 *)

From R. J. Mathar, Feb 13 2015: (Start)
5*(n+1)*a(n) + (-29*n-1)*a(n-1) + 18*(2*n-3)*a(n-2) = 0.
-(n+1)*(11*n-10)*a(n) + 2*(11*n+1)*(2*n-1)*a(n-1) = 0. (End)
G.f.: (5 - 9*x - 5*sqrt(1 - 4*x))/(x*sqrt(1 - 4*x)). - Amiram Eldar, Jul 08 2023

Terms a(21) and beyond from Andrew Howroyd, Jan 05 2020

A279895 a(n) = n(5n + 11)/2.

Original entry on oeis.org

0, 8, 21, 39, 62, 90, 123, 161, 204, 252, 305, 363, 426, 494, 567, 645, 728, 816, 909, 1007, 1110, 1218, 1331, 1449, 1572, 1700, 1833, 1971, 2114, 2262, 2415, 2573, 2736, 2904, 3077, 3255, 3438, 3626, 3819, 4017, 4220, 4428, 4641, 4859, 5082, 5310, 5543, 5781, 6024, 6272, 6525

Offset: 0

Bruno Berselli, Dec 22 2016

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

Cf. A000566, A008589, A017281.
Second bisection of A165720.
The first differences are in A016885.
Cf. similar sequences provided by P(s,m)+s*m, where P(s,m) = ((s-2)*m^2-(s-4)*m)/2 is the m-th s-gonal number: A008585 (s=2), A055999 (s=3), A028347 (s=4), A140091 (s=5), A033537 (s=6), this sequence (s=7), A067725 (s=8).

Magma
```
[n*(5*n+11)/2: n in [0..60]];
```

Table[n (5 n + 11)/2, {n, 0, 60}]
LinearRecurrence[{3,-3,1},{0,8,21},60] (* Harvey P. Dale, Nov 14 2022 *)

PARI
```
vector(60, n, n--; n*(5*n+11)/2)
```
Python
```
[n*(5*n+11)/2 for n in range(60)]
```
Sage
```
[n*(5*n+11)/2 for n in range(60)]
```

O.g.f.: x*(8 - 3*x)/(1 - x)^3.
E.g.f.: x*(16 + 5*x)*exp(x)/2.
a(n+h) - a(n-h) = h*A017281(n+1), with h>=0. A particular case:
a(n) - a(-n) = 11*n = A008593(n).
a(n+h) + a(n-h) = 2*a(n) + A033429(h), with h>=0. A particular case:
a(n) + a(-n) = A033429(n).
a(n) - a(n-2) = A017281(n) for n>1. Also:
40*a(n) + 121 = A017281(n+1)^2.
a(n) = A000566(n) + 7*n, also a(n) = A000566(n) + A008589(n). - Michel Marcus, Dec 22 2016

A346510 a(n) is the number of nontrivial divisors of A346507(n) ending with 1.

Original entry on oeis.org

1, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 4, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 2, 2, 4, 2, 4, 2

Offset: 1

Stefano Spezia, Jul 21 2021

			a(42) = 4 since there are 4 nontrivial divisors of A346507(42) = 2541 ending with 1: 11, 21, 121 and 231.

Cf. A017281, A070824, A346388 (ending with 5), A346389 (ending with 6), A346392, A346507, A346508, A346509.

b={}; For[n=1, n<=500, n++, For[k=1, kMax[b], AppendTo[b, 10n+1]]]]; (* A346507 *) a={}; For[i =1, i<=Length[b], i++, AppendTo[a, Length[Drop[Select[Divisors[Part[b, i]], (Mod[#, 10]==1&)], -1]]-1]]; a

f(n) = sumdiv(n, d, (d>1) && (d(f(x)), [1..5000])) \\ Michel Marcus, Jul 28 2021

from sympy import divisors
def f(n): return sum(d%10 == 1 for d in divisors(n)[1:-1])
def A346507upto(lim): return sorted(set(a*b for a in range(11, lim//11+1, 10) for b in range(a, lim//a+1, 10)))
print(list(map(f, A346507upto(5000)))) # Michael S. Branicky, Jul 31 2021

a(n) = A346392(A346507(n)) - 1.

A354336 a(n) is the integer w such that (L(2n)^2, -L(2n-1)^2, -w) is a primitive solution to the Diophantine equation 2x^3 + 2y^3 + z^3 = 125, where L(n) is the n-th Lucas number (A000032).

Original entry on oeis.org

1, 11, 61, 401, 2731, 18701, 128161, 878411, 6020701, 41266481, 282844651, 1938646061, 13287677761, 91075098251, 624238009981, 4278590971601, 29325898791211, 201002700566861, 1377693005176801, 9442848335670731, 64722245344518301, 443612869075957361

Offset: 0

XU Pingya, Jun 20 2022

Subsequence of A017281.

			2*(L(4)^2)^3 + 2*(-L(3)^2)^3 + (-61)^3 = 2*(49)^3 + 2*(-1)^3 + (-61)^3 = 125, a(2) = 61.

Index entries for linear recurrences with constant coefficients, signature (8,-8,1).

Cf. A000032, A002878, A005248, A017281, A056914, A081015, A092521.

Cf. A337928, A354337.

LucasL[4*Range[22]-3] + 1 - LucasL[2*Range[22]-3]^2

a(n) = (-125 + 2*A005248(n)^6 - 2*A002878(n-1)^6)^(1/3).
a(n) = Lucas(4*n+1) - Lucas(4*n-2) + 3 = A056914(n) - 15*A092521(n-1), for n > 1.
a(n) = Lucas(4*n+1) + 1 - Lucas(2*n-1)^2.
a(n) = 2*A081015(n-1) + 1.
a(n) = 8*a(n-1) - 8*a(n-2) + a(n-3).
G.f.: (1 + 3*x - 19*x^2)/((1 - x)*(1 - 7*x + x^2)). - Stefano Spezia, Jun 22 2022
a(n) = (F(2*n+1) + F(2*n-1))^2 + (F(2*n+1) + F(2*n-1)) * (F(2*n-1) + F(2*n-3)) - (F(2*n-1) + F(2*n-3))^2. - XU Pingya, Jul 17 2024

A385623 Array read by ascending antidiagonals: A(n,k) is the number obtained by concatenation of n with k in that order, with k >= 0.

Original entry on oeis.org

0, 10, 1, 20, 11, 2, 30, 21, 12, 3, 40, 31, 22, 13, 4, 50, 41, 32, 23, 14, 5, 60, 51, 42, 33, 24, 15, 6, 70, 61, 52, 43, 34, 25, 16, 7, 80, 71, 62, 53, 44, 35, 26, 17, 8, 90, 81, 72, 63, 54, 45, 36, 27, 18, 9, 100, 91, 82, 73, 64, 55, 46, 37, 28, 19, 10, 110, 101, 92, 83, 74, 65, 56, 47, 38, 29, 110, 11

Offset: 0

Stefano Spezia, Jul 05 2025

			Array begins as:
   0,  1,  2,  3,  4,  5,  6,  7, ...
  10, 11, 12, 13, 14, 15, 16, 17, ...
  20, 21, 22, 23, 24, 25, 26, 27, ...
  30, 31, 32, 33, 34, 35, 36, 37, ...
  40, 41, 42, 43, 44, 45, 46, 47, ...
  50, 51, 52, 53, 54, 55, 56, 57, ...
  60, 61, 62, 63, 64, 65, 66, 67, ...
  ...

Stefano Spezia, Table of n, a(n) for n = 0..11475 (first 151 antidiagonals of the array, flattened)

Columns for k=0..9 give: A008592, A017281, A017293, A017305, A017317, A017329, A017341, A017353, A017365, A017377.

Cf. A001477 (1st row), A020338 (main diagonal), A055642, A385624 (antidiagonal sums).

A[n_,k_]:=FromDigits[Join[IntegerDigits[n],IntegerDigits[k]]]; Table[A[n,k],{n,0,6},{k,0,7}] (* or *)
A[n_,k_]:=If[k==0,10n,n*10^(Floor[Log10[k]]+1)+k]; Table[A[n-k,k],{n,0,11},{k,0,n}]//Flatten

T(n, k) = fromdigits(concat(digits(n), digits(k))); \\ Michel Marcus, Jul 06 2025

A(n,0) = 10*n and A(n,k) = n*10^(floor(log_10(k)) + 1) + k for k > 0.

A001535 a(n) = (10n+1)*(10n+9).

Original entry on oeis.org

9, 209, 609, 1209, 2009, 3009, 4209, 5609, 7209, 9009, 11009, 13209, 15609, 18209, 21009, 24009, 27209, 30609, 34209, 38009, 42009, 46209, 50609, 55209, 60009, 65009, 70209, 75609, 81209, 87009, 93009, 99209, 105609, 112209, 119009, 126009, 133209, 140609

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. A001622, A017281, A017377.

seq((10*n+1)*(10*n+9),n = 0 .. 100); # Robert Israel, Dec 17 2014

Times@@@Table[10n+{1,9},{n,0,40}] (* or *) LinearRecurrence[{3,-3,1},{9,209,609},40] (* Harvey P. Dale, Oct 15 2014 *)
CoefficientList[Series[(9 + 182 x + 9 x^2) / (1 - x)^3, {x, 0, 50}], x] (* Vincenzo Librandi, Dec 17 2014 *)

a(n)=(10*n+1)*(10*n+9) \\ Charles R Greathouse IV, Jun 16 2017

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(0)=9, a(1)=209, a(2)=609. - Harvey P. Dale, Oct 15 2014
G.f.: (9 + 182*x + 9*x^2)/(1 - x)^3. - Vincenzo Librandi, Dec 17 2014
E.g.f.: (100*x^2 + 200*x + 9)*exp(x). - Robert Israel, Dec 17 2014
From Amiram Eldar, Feb 20 2023: (Start)
a(n) = A017281(n)*A017377(n).
Sum_{n>=0} 1/a(n) = sqrt(5+2*sqrt(5))*Pi/80.
Sum_{n>=0} (-1)^n/a(n) = (sqrt(10+2*sqrt(5)) * log(cot(Pi/20)) + sqrt(10-2*sqrt(5)) * log(cot(3*Pi/20)))/80.
Product_{n>=0} (1 - 1/a(n)) = 2*phi*cos(sqrt(17)*Pi/10), where phi is the golden ratio (A001622).
Product_{n>=0} (1 + 1/a(n)) = 2*phi*cos(sqrt(15)*Pi/10). (End)

A017290 a(n) = (10*n + 1)^10.

Original entry on oeis.org

1, 25937424601, 16679880978201, 819628286980801, 13422659310152401, 119042423827613001, 713342911662882601, 3255243551009881201, 12157665459056928801, 38941611811810745401, 110462212541120451001

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 (11, -55, 165, -330, 462, -462, 330, -165, 55, -11, 1).

Cf. A011557, A017281.

[(10*n+1)^10: n in [0..10]]; // Vincenzo Librandi, Jul 30 2011

A017290:=n->(10*n+1)^10; seq(A017290(n), n=0..10); # Wesley Ivan Hurt, Jan 29 2014

(10Range[0,10]+1)^10  (* Harvey P. Dale, Feb 18 2011 *)

a(n) = A011557(A017281(n)). - Wesley Ivan Hurt, Jan 29 2014

A017291 a(n) = (10*n + 1)^11.

Original entry on oeis.org

1, 285311670611, 350277500542221, 25408476896404831, 550329031716248441, 6071163615208263051, 43513917611435838661, 231122292121701565271, 984770902183611232881, 3543686674874777831491, 11156683466653165551101

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
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), A017281 (10n+1).

[(10*n+1)^11: n in [0..20]]; // Vincenzo Librandi, May 15 2011

(10Range[0,20]+1)^11 (* Harvey P. Dale, May 11 2011 *)

A032585 Lucky numbers ending with digit 1.

Original entry on oeis.org

1, 21, 31, 51, 111, 141, 151, 171, 201, 211, 231, 241, 261, 321, 331, 361, 391, 421, 451, 511, 541, 591, 601, 621, 631, 651, 741, 781, 801, 831, 841, 931, 961, 981, 991, 1011, 1021, 1041, 1101, 1201, 1231, 1251, 1261, 1281, 1291, 1401, 1441, 1471, 1491

Offset: 1

Patrick De Geest, Apr 15 1998

Also, lucky numbers (A000959) which are congruent to 1 mod 5. - R. J. Mathar, Apr 29 2008

Amiram Eldar, Table of n, a(n) for n = 1..10000

Intersection of A000959 and A017281.

A053178 Numbers ending in 1 which are not prime.

Original entry on oeis.org

1, 21, 51, 81, 91, 111, 121, 141, 161, 171, 201, 221, 231, 261, 291, 301, 321, 341, 351, 361, 371, 381, 391, 411, 441, 451, 471, 481, 501, 511, 531, 551, 561, 581, 591, 611, 621, 651, 671, 681, 711, 721, 731, 741, 771, 781, 791, 801, 831, 841, 851, 861, 871

Offset: 1

Enoch Haga, Feb 29 2000

Nonprime numbers of the form k*10+1. - Juri-Stepan Gerasimov, Oct 14 2009

			a(4) = 91 may look prime to some, but is composite.

Cf. A002808, A017281, A153275.

remove(isprime, [10*j+1$j=0..99])[];  # Alois P. Heinz, Jan 21 2021

isok(n) = ((n % 10) == 1) && (! isprime(n)) \\ Michel Marcus, Jul 26 2013

from sympy import isprime
def aupto(lim): return [m for m in range(1, lim+1, 10) if not isprime(m)]
print(aupto(871)) # Michael S. Branicky, Jan 21 2021

cp's OEIS Frontend

A050490 a(n) = C(n)*(11n+1) where C(n) = Catalan numbers (A000108).

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A279895 a(n) = n*(5*n + 11)/2.

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A346510 a(n) is the number of nontrivial divisors of A346507(n) ending with 1.

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A354336 a(n) is the integer w such that (L(2*n)^2, -L(2*n-1)^2, -w) is a primitive solution to the Diophantine equation 2*x^3 + 2*y^3 + z^3 = 125, where L(n) is the n-th Lucas number (A000032).

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A385623 Array read by ascending antidiagonals: A(n,k) is the number obtained by concatenation of n with k in that order, with k >= 0.

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Mathematica

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A001535 a(n) = (10n+1)*(10n+9).

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Maple

Mathematica

PARI

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A017290 a(n) = (10*n + 1)^10.

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Magma

Maple

Mathematica

A279895 a(n) = n(5n + 11)/2.

A354336 a(n) is the integer w such that (L(2n)^2, -L(2n-1)^2, -w) is a primitive solution to the Diophantine equation 2x^3 + 2y^3 + z^3 = 125, where L(n) is the n-th Lucas number (A000032).