A017509 -id:A017509 - cp's OEIS frontend

A017520 a(n) = (11*n + 10)^12.

1000000000000, 7355827511386641, 1152921504606846976, 39959630797262576401, 614787626176508399616, 5688009063105712890625, 37133262473195501387776, 188031682201497672618081, 784716723734800033386496, 2812664781782894485727281

Offset: 0

N. J. A. Sloane

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).

Powers of the form (11*n+10)^m: A017509 (m=1), A017510 (m=2), A017511 (m=3), A017512 (m=4), A017513 (m=5), A017514 (m=6), A017515 (m=7), A017516 (m=8), A017517 (m=9), A017518 (m=10), A017519 (m=11), this sequence (m=12).

List([0..20], n-> (11*n+10)^12); # G. C. Greubel, Oct 29 2019

[(11*n+10)^12: n in [0..20]]; // G. C. Greubel, Oct 29 2019

seq((11*n+10)^12, n=0..20); # G. C. Greubel, Oct 29 2019

(11*Range[0,20]+10)^12 (* Harvey P. Dale, Oct 14 2012 *)

makelist((11*n+10)^12,n,0,30); /* Martin Ettl, Oct 21 2012 */

vector(21, n, (11*n-1)^12) \\ G. C. Greubel, Oct 29 2019

[(11*n+10)^12 for n in (0..20)] # G. C. Greubel, Oct 29 2019

From G. C. Greubel, Oct 29 2019: (Start)
G.f.: (1000000000000 + 7342827511386641*x + 1057373746958820643*x^2 + 25545119783261723711*x^3 + 183137251503172391205*x^4 + 488143704350667868074*x^5 + 528998728358533109886*x^6 + 234662813343627300126*x^7 + 39635845367890711434*x^8 + 2102226021911800565*x^9 + 21798715126193071*x^10 + 8916100448243*x^11 + x^12)/(1-x)^13.
E.g.f.: (1000000000000 + 7354827511386641*x + 569105424792036847*x^2 + 6087155460996032566*x^3 + 19243217071043901221*x^4 + 25018123360727376000*x^5 + 15895943833149490132*x^6 + 5437280856006223356*x^7 + 1053961441036472067*x^8 + 117674853236661875*x^9 + 7405264410708505*x^10 + 241373673336906*x^11 + 3138428376721*x^12)*exp(x). (End)

A267541 Expansion of (2 + 4x + x^2 + x^3 + 2x^4 + x^5)/(1 - x - x^5 + x^6).

Original entry on oeis.org

2, 6, 7, 8, 10, 13, 17, 18, 19, 21, 24, 28, 29, 30, 32, 35, 39, 40, 41, 43, 46, 50, 51, 52, 54, 57, 61, 62, 63, 65, 68, 72, 73, 74, 76, 79, 83, 84, 85, 87, 90, 94, 95, 96, 98, 101, 105, 106, 107, 109, 112, 116, 117, 118, 120, 123, 127, 128, 129, 131, 134, 138, 139, 140

Offset: 0

Bruno Berselli, Jan 16 2016

Also, numbers that are congruent to {2, 6, 7, 8, 10} mod 11.
(m^k+1)/11 is a nonnegative integer when
. m is a member of this sequence and k is an odd multiple of 5 (A017329),
. m is a member of A017509 and k is odd but not multiple of 5 (A045572).
If k is even, (m^k+1)/11 is never an integer.
The product of two terms does not belong to the sequence.

			From the linear recurrence:
(-A267755) ..., -12, -9, -5, -4, -3, -1, 2, 6, 7, 8, 10, 13, ... (A267541)

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

Cf. A002266, A017329, A017509, A267755.

Cf. A088225: numbers congruent to {2,6,7,8} mod 11.

m:=70; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((2+4*x+x^2+x^3+2*x^4+x^5)/(1-x-x^5+x^6)));

gf := (2+4*x+x^2+x^3+2*x^4+x^5)/((1-x)^2*(1+x+x^2+x^3+ x^4)): deg := 64: series(gf,x,deg): seq(coeff(%,x,n), n=0..deg-1); # Peter Luschny, Jan 19 2016

CoefficientList[Series[(2 + 4 x + x^2 + x^3 + 2 x^4 + x^5)/(1 - x - x^5 + x^6), {x, 0, 70}], x]
LinearRecurrence[{1, 0, 0, 0, 1, -1}, {2, 6, 7, 8, 10, 13}, 70]
Select[Range[150], MemberQ[{2, 6, 7, 8, 10}, Mod[#, 11]]&]

Vec((2+4*x+x^2+x^3+2*x^4+x^5)/(1-x-x^5+x^6)+O(x^70))

gf = (2+4*x+x^2+x^3+2*x^4+x^5)/((1-x)^2*(1+x+x^2+x^3+ x^4))
print(taylor(gf, x, 0, 63).list()) # Peter Luschny, Jan 19 2016

G.f.: (2 + 4*x + x^2 + x^3 + 2*x^4 + x^5)/((1 - x)^2*(1 + x + x^2 + x^3 + x^4)).
a(n) = a(n-1) + a(n-5) - a(n-6).
a(-n) = -A267755(n-1).

A001536 a(n) = (11n+1)(11*n+10).

Original entry on oeis.org

10, 252, 736, 1462, 2430, 3640, 5092, 6786, 8722, 10900, 13320, 15982, 18886, 22032, 25420, 29050, 32922, 37036, 41392, 45990, 50830, 55912, 61236, 66802, 72610, 78660, 84952, 91486, 98262, 105280, 112540, 120042, 127786, 135772, 144000, 152470, 161182, 170136

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. A017401, A017509.

Table[(11*n + 1)*(11*n + 10), {n, 0, 40}] (* Amiram Eldar, Feb 20 2023 *)

a(n)=(11*n+1)*(11*n+10) \\ Charles R Greathouse IV, Jun 16 2017

a(n) = 242*n + a(n-1) with a(0)=10. - Vincenzo Librandi, Nov 12 2010
G.f.: -2*(5+111*x+5*x^2)/(x-1)^3. - R. J. Mathar, May 30 2022
From Amiram Eldar, Feb 20 2023: (Start)
a(n) = A017401(n)*A017509(n).
Sum_{n>=0} 1/a(n) = cot(Pi/11)*Pi/99.
Product_{n>=0} (1 - 1/a(n)) = cosec(Pi/11)*cos(sqrt(85)*Pi/22).
Product_{n>=0} (1 + 1/a(n)) = cosec(Pi/11)*cos(sqrt(77)*Pi/22). (End)
From Elmo R. Oliveira, Oct 25 2024: (Start)
E.g.f.: exp(x)*(10 + 121*x*(2 + x)).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

A159551 a(n) = 101*n + 10.

Original entry on oeis.org

10, 111, 212, 313, 414, 515, 616, 717, 818, 919, 1020, 1121, 1222, 1323, 1424, 1525, 1626, 1727, 1828, 1929, 2030, 2131, 2232, 2333, 2434, 2535, 2636, 2737, 2838, 2939, 3040, 3141, 3242, 3343, 3444, 3545, 3646, 3747, 3848, 3949, 4050, 4151, 4252, 4353, 4454, 4555

Offset: 0

Robert G. Wilson v, Apr 14 2009

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

Cf. A017281, A017509.

[101*n+10: n in [0..50]]; // Vincenzo Librandi, Jul 30 2011

f[n_] := FromDigits[ IntegerDigits[n^3 + n^2 + n - 1, n + 1]]; Array[f, 54]

From Elmo R. Oliveira, Apr 03 2025: (Start)
G.f.: (10 + 91*x)/(1-x)^2.
E.g.f.: exp(x)*(10 + 101*x).
a(n) = 2*a(n-1) - a(n-2). (End)

Offset changed from 1 to 0 by Vincenzo Librandi, Jul 30 2011

A108317 Smallest a(n) such that a(n) n's plus a(n) is prime, or 0 if no such a(n) exists.

Original entry on oeis.org

1, 1, 140, 1, 0, 1, 2, 0, 2, 1, 0, 1, 4, 0, 4, 1, 0, 1, 4, 0, 0, 1, 0, 23, 4, 0, 2, 1, 0, 1, 8, 0, 4198, 497, 0, 1, 2, 0, 8, 1, 0, 1, 0, 0, 2, 1, 0, 35, 2, 0, 2, 1, 0, 0, 2, 0, 4, 1, 0, 1, 2, 0, 4, 17, 0, 1, 64, 0, 2, 1, 0, 1, 14, 0, 2, 0, 0, 1

Offset: 1

Ray G. Opao, Jun 30 2005

Some of the larger entries may only correspond to probable primes.
Some or all of the zero values are merely conjectures. - N. J. A. Sloane
a(n)=0 for n = 3m+2 (1<=m) (they are all divisible by 3) or n=11m+10 (1<=m<9) (they are all divisible by 11) and if a(n) is not 0 then n and a(n) are of opposite parity. - Robert G. Wilson v and Rick L. Shepherd, Jul 28 2005
The sequence continues: 0,4490,1,0,13,14,0,0,1,0,349,10,0,86,2539,0,1,4,0,124,1,0,1,4,0,2,1,0,1,2,0,302,1,0,83,2,0,2,5,0,a(120)>5364,2,0,278,5,0,...,. - Robert G. Wilson v, Jul 28 2005
a(79)>14179. - Robert G. Wilson v, Jul 28 2005

			a(13)=4: 4 13s plus 4 = 13131313+4 = 13131317, which is prime.

Cf. A006093 (primes minus 1), A016789 (3n + 2), A017509 (11n + 10).

f[n_] := If[(n > 4 && Mod[n, 3] == 2) || (n > 20 && Mod[n, 11] == 10), k = 0, If[n == 1, k = 1, Block[{id = IntegerDigits[n]}, k = Mod[n, 2] + 1; While[ !PrimeQ[ FromDigits[ Flatten[ Table[id, {k}]]] + k], k += 2]]]; k]; Table[ f[n], {n, 100}] (* only good for n<109 *) (* Robert G. Wilson v, Jun 30 2005 *)

/* for nonzero terms */ a(n) = m=1;pr=n;while(!isprime(pr+m),m++;pr=eval(concat(Str(pr),n)));m \\ Rick L. Shepherd, Jul 26 2005

a(A016789(n)) = a(A017509(n)) = 0 for n >= 1. a(n) = 1 iff n is a term of A006093. - Rick L. Shepherd, Jul 26 2005

a(33) - a(78) from Robert G. Wilson v with guidance from Rick L. Shepherd, Jul 28 2005

cp's OEIS Frontend

A017520 a(n) = (11*n + 10)^12.

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A267541 Expansion of (2 + 4*x + x^2 + x^3 + 2*x^4 + x^5)/(1 - x - x^5 + x^6).

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A001536 a(n) = (11*n+1)*(11*n+10).

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A159551 a(n) = 101*n + 10.

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A108317 Smallest a(n) such that a(n) n's plus a(n) is prime, or 0 if no such a(n) exists.

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Mathematica

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A267541 Expansion of (2 + 4x + x^2 + x^3 + 2x^4 + x^5)/(1 - x - x^5 + x^6).

A001536 a(n) = (11n+1)(11*n+10).