A017533 -id:A017533 - cp's OEIS frontend

A161709 a(n) = 22*n + 1.

1, 23, 45, 67, 89, 111, 133, 155, 177, 199, 221, 243, 265, 287, 309, 331, 353, 375, 397, 419, 441, 463, 485, 507, 529, 551, 573, 595, 617, 639, 661, 683, 705, 727, 749, 771, 793, 815, 837, 859, 881, 903, 925, 947, 969, 991, 1013, 1035, 1057, 1079, 1101, 1123

Offset: 0

Reinhard Zumkeller, Jun 17 2009

Italo Ghersi, Matematica dilettevole e curiosa, p. 139, Hoepli, Milano, 1967. [From Vincenzo Librandi, Dec 02 2009]

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

Cf. A005408, A008604, A016813, A016921, A017281, A017533, A128470, A158057, A161700, A161705, A161714.

List([0..60], n-> 1+22*n ); # G. C. Greubel, Sep 18 2019

[1+22*n: n in [0..60]]; // G. C. Greubel, Sep 18 2019

seq(1+22*n, n=0..60); # G. C. Greubel, Sep 18 2019

22*Range[0,60]+1 (* Harvey P. Dale, Jan 09 2011 *)

vector(60, n, 22*n-21) \\ G. C. Greubel, Sep 18 2019

[1+22*n for n in (0..60)] # G. C. Greubel, Sep 18 2019

From G. C. Greubel, Sep 18 2019: (Start)
a(n) = 2*a(n-1) - a(n-2).
G.f.: (1 + 21*x)/(1-x)^2.
E.g.f.: (1 + 22*x)*exp(x). (End)

A161714 a(n) = 28*n + 1.

Original entry on oeis.org

1, 29, 57, 85, 113, 141, 169, 197, 225, 253, 281, 309, 337, 365, 393, 421, 449, 477, 505, 533, 561, 589, 617, 645, 673, 701, 729, 757, 785, 813, 841, 869, 897, 925, 953, 981, 1009, 1037, 1065, 1093, 1121, 1149, 1177, 1205, 1233, 1261, 1289, 1317, 1345, 1373

Offset: 0

Reinhard Zumkeller, Jun 17 2009

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

Cf. A005408, A016813, A016921, A017281, A017533, A128470, A158057, A161700, A161705, A161709.

List([0..60], n-> 1+28*n); # G. C. Greubel, Sep 18 2019

[28*n + 1: n in [0..60]]; // Vincenzo Librandi, May 04 2011

seq(1+28*n, n=0..60); # G. C. Greubel, Sep 18 2019

Range[1, 1700, 28] (* Vladimir Joseph Stephan Orlovsky, Jun 15 2011 *)
CoefficientList[Series[(1+27x)/(1-x)^2, {x,0,60}], x] (* Michael De Vlieger, Apr 05 2017 *)

vector(60, n, 28*n-27) \\ G. C. Greubel, Sep 18 2019

[1+28*n for n in (0..60)] # G. C. Greubel, Sep 18 2019

G.f.: (1 + 27*x)/(1-x)^2. - Indranil Ghosh, Apr 05 2017

E.g.f.: (1 + 28*x)*exp(x). - G. C. Greubel, Sep 18 2019

A056530 Sequence remaining after third round of Flavius Josephus sieve; remove every fourth term of A047241.

Original entry on oeis.org

1, 3, 7, 13, 15, 19, 25, 27, 31, 37, 39, 43, 49, 51, 55, 61, 63, 67, 73, 75, 79, 85, 87, 91, 97, 99, 103, 109, 111, 115, 121, 123, 127, 133, 135, 139, 145, 147, 151, 157, 159, 163, 169, 171, 175, 181, 183, 187, 193, 195, 199, 205, 207, 211, 217, 219, 223, 229, 231

Offset: 1

Henry Bottomley, Jun 19 2000

Numbers {1, 3, 7} mod 12: A017533, A017557, A017605 interleaved.

Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).
Index entries for sequences related to the Josephus Problem

We have A000027 after 0 rounds of sieving, A005408 after 1 round of sieving, A047241 after 2 rounds, A056530 after 3 rounds, A056531 after 4 rounds, A000960 after all rounds. After n rounds the remaining sequence comprises A002944(n) numbers mod A003418(n+1), i.e. 1/(n+1) of them.

LinearRecurrence[{1,0,1,-1},{1,3,7,13},60] (* Harvey P. Dale, Oct 19 2022 *)

From Chai Wah Wu, Jul 24 2016: (Start)
a(n) = a(n-1) + a(n-3) - a(n-4) for n > 4.
G.f.: x*(5*x^3 + 4*x^2 + 2*x + 1)/(x^4 - x^3 - x + 1). (End)
a(n) = 4*n - (13 + 2*A131713(n))/3. - R. J. Mathar, Jun 22 2020

A103214 a(n) = 24*n + 1.

Original entry on oeis.org

1, 25, 49, 73, 97, 121, 145, 169, 193, 217, 241, 265, 289, 313, 337, 361, 385, 409, 433, 457, 481, 505, 529, 553, 577, 601, 625, 649, 673, 697, 721, 745, 769, 793, 817, 841, 865, 889, 913, 937, 961, 985, 1009, 1033, 1057, 1081, 1105, 1129, 1153, 1177, 1201

Offset: 0

Ralf Stephan, Jan 28 2005

Vincenzo Librandi, Table of n, a(n) for n = 0..3000
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Equals A008606 + 1. Bisection of A017533.

Cf. A255185.

[24*n+1: n in [0..60]]; // Vincenzo Librandi, Jun 15 2011

Range[1, 2000, 24] (* Vladimir Joseph Stephan Orlovsky, Jun 14 2011 *)

a(n)=24*n+1 \\ Charles R Greathouse IV, Jun 14 2011

From Elmo R. Oliveira, Mar 21 2024: (Start)
G.f.: (1+23*x)/(1-x)^2.
E.g.f.: exp(x)*(1 + 24*x).
a(n) = A255185(n+1) - A255185(n).
a(n) = 2*a(n-1) - a(n-2) for n >= 2. (End)

A130154 Triangle read by rows: T(n, k) = 1 + 2(n-k)(k-1) (1 <= k <= n).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 5, 5, 1, 1, 7, 9, 7, 1, 1, 9, 13, 13, 9, 1, 1, 11, 17, 19, 17, 11, 1, 1, 13, 21, 25, 25, 21, 13, 1, 1, 15, 25, 31, 33, 31, 25, 15, 1, 1, 17, 29, 37, 41, 41, 37, 29, 17, 1, 1, 19, 33, 43, 49, 51, 49, 43, 33, 19, 1, 1, 21, 37, 49, 57, 61, 61, 57, 49, 37, 21, 1

Offset: 1

Emeric Deutsch, May 22 2007

Column k, except for the initial k-1 0's, is an arithmetic progression with first term 1 and common difference 2(k-1). Row sums yield A116731. First column of the inverse matrix is A129779.
Studied by Paul Curtz circa 1993.
From Rogério Serôdio, Dec 19 2017: (Start)
T(n, k) gives the number of distinct sums of 2(k-1) elements in {1,1,2,2,...,n-1,n-1}. For example, T(6, 2) = the number of distinct sums of 2 elements in {1,1,2,2,3,3,4,4,5,5}, and because each sum from the smallest 1 + 1 = 2 to the largest 5 + 5 = 10 appears, T(6, 2) = 10 - 1 = 9. [In general: 2*(Sum_{j=1..(k-1)} n-j) - (2*(Sum_{j=1..k-1} j) - 1) = 2*n*(k-1) - 4*(k-1)*k/2 + 1 = 2*(k-1)*(n-k) + 1 = T(n, k). - Wolfdieter Lang, Dec 20 2017]
T(n, k) is the number of lattice points with abscissa x = 2*(k-1) and even ordinate in the closed region bounded by the parabola y = x*(2*(n-1) - x) and the x axis. [That is, (1/2)*y(2*(k-1)) + 1 = T(n, k). - Wolfdieter Lang, Dec 20 2017]
Pascal's triangle (A007318, but with apex in the middle) is formed using the rule South = West + East; the rascal triangle A077028 uses the rule South = (West*East + 1)/North; the present triangle uses a similar rule: South = (West*East + 2)/North. See the formula section for this recurrence. (End)

			The triangle T(n, k) starts:
  n\k  1  2  3  4  5  6  7  8  9 10 ...
  1:   1
  2:   1  1
  3:   1  3  1
  4:   1  5  5  1
  5:   1  7  9  7  1
  6:   1  9 13 13  9  1
  7:   1 11 17 19 17 11  1
  8:   1 13 21 25 25 21 13  1
  9:   1 15 25 31 33 31 25 15  1
 10:   1 17 29 37 41 41 37 29 17  1
 ... reformatted. - _Wolfdieter Lang_, Dec 19 2017

G. C. Greubel, Rows n = 1..100 of triangle, flattened

Cf. A116731, A129779. A007318, A077028.

Column sequences (no leading zeros): A000012, A016813, A016921, A017077, A017281, A017533, A131877, A158057, A161705, A215145.

Flat(List([1..12], n-> List([1..n], k-> 1 + 2*(n-k)*(k-1) ))); # G. C. Greubel, Nov 25 2019

[1 + 2*(n-k)*(k-1): k in [1..n], n in [1..12]]; // G. C. Greubel, Nov 25 2019

T:=proc(n,k) if k<=n then 2*(n-k)*(k-1)+1 else 0 fi end: for n from 1 to 14 do seq(T(n,k),k=1..n) od; # yields sequence in triangular form

Flatten[Table[1+2(n-k)(k-1),{n,0,20},{k,n}]] (* Harvey P. Dale, Jul 13 2013 *)

T(n, k) = 1 + 2*(n-k)*(k-1) \\ Iain Fox, Dec 19 2017

first(n) = my(res = vector(binomial(n+1,2)), i = 1); for(r=1, n, for(k=1, r, res[i] = 1 + 2*(r-k)*(k-1); i++)); res \\ Iain Fox, Dec 19 2017

[[1 + 2*(n-k)*(k-1) for k in (1..n)] for n in (1..12)] # G. C. Greubel, Nov 25 2019

T(n, k) = 1 + 2*(n-k)*(k-1) (1 <= k <= n).
G.f.: G(t,z) = t*z*(3*t*z^2 - z - t*z + 1)/((1-t*z)*(1-z))^2.
Equals = 2 * A077028 - A000012 as infinite lower triangular matrices. - Gary W. Adamson, Oct 23 2007
T(n, 1) = 1 and  T(n, n) = 1 for n >= 1; T(n, k) = (T(n-1, k-1)*T(n-1, k) + 2)/T(n-2, k-1), for n > 2 and 1 < k < n. See a comment above. - Rogério Serôdio, Dec 19 2017
G.f. column k (with leading zeros): (x^k/(1-x)^2)*(1 + (2*k-3)*x), k >= 1. See the g.f. of the triangle G(t,z) above: (d/dt)^k G(t,x)/k!|{t=0}. - _Wolfdieter Lang, Dec 20 2017

Edited by Wolfdieter Lang, Dec 19 2017

A001538 a(n) = (12n+1)(12*n+11).

Original entry on oeis.org

11, 299, 875, 1739, 2891, 4331, 6059, 8075, 10379, 12971, 15851, 19019, 22475, 26219, 30251, 34571, 39179, 44075, 49259, 54731, 60491, 66539, 72875, 79499, 86411, 93611, 101099, 108875, 116939, 125291, 133931, 142859, 152075, 161579, 171371, 181451, 191819

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. A001533, A017533, A017653.

Table[(12n+1)(12n+11),{n,0,40}] (* or *) LinearRecurrence[{3,-3,1},{11,299,875},40] (* Harvey P. Dale, Jul 22 2024 *)

a(n)=(12*n+1)*(12*n+11) \\ Charles R Greathouse IV, Jun 16 2017

a(n) = (9*A001533(n) - 19)/4.
a(n) = 288*n + a(n-1) with a(0)=11. - Vincenzo Librandi, Nov 12 2010
G.f.: -(11 + 266*x + 11*x^2)/(x-1)^3. - R. J. Mathar, Jun 30 2020
From Amiram Eldar, Feb 20 2023: (Start)
a(n) = A017533(n)*A017653(n).
Sum_{n>=0} 1/a(n) = (sqrt(3)+2)*Pi/120.
Sum_{n>=0} (-1)^n/a(n) = (4*log(sqrt(2)+1) + sqrt(3)*log(5+2*sqrt(6)))/(60*sqrt(2)).
Product_{n>=0} (1 - 1/a(n)) = (2*sqrt(2)/(sqrt(3)-1))*cos(sqrt(13/2)*Pi/6).
Product_{n>=0} (1 + 1/a(n)) = 2*sqrt(2+sqrt(3))*cos(Pi/sqrt(6)). (End)
From Elmo R. Oliveira, Oct 25 2024: (Start)
E.g.f.: exp(x)*(11 + 144*x*(2 + x)).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

A157326 a(n) = 10368n^2 + 288n + 1.

Original entry on oeis.org

10657, 42049, 94177, 167041, 260641, 374977, 510049, 665857, 842401, 1039681, 1257697, 1496449, 1755937, 2036161, 2337121, 2658817, 3001249, 3364417, 3748321, 4152961, 4578337, 5024449, 5491297, 5978881, 6487201, 7016257

Offset: 1

Vincenzo Librandi, Feb 27 2009

The identity (10368*n^2 + 288*n + 1)^2 - (36*n^2 + n)*(1728*n + 24)^2 = 1 can be written as a(n)^2 - A157324(n)*A157325(n)^2 = 1 (see also second part of the comment at A157324). - Vincenzo Librandi, Jan 26 2012

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Vincenzo Librandi, X^2-AY^2=1
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A157324, A157325.

I:=[10657, 42049, 94177]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..50]]; // Vincenzo Librandi, Jan 26 2012

LinearRecurrence[{3,-3,1},{10657,42049,94177},50] (* Vincenzo Librandi, Jan 26 2012 *)

for(n=1, 22, print1(10368*n^2 + 288*n + 1", ")); \\ Vincenzo Librandi, Jan 26 2012

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Vincenzo Librandi, Jan 26 2012
G.f.: x*(-x^2 - 10078*x - 10657)/(x-1)^3. - Vincenzo Librandi, Jan 26 2012
a(n) = 2*A017533(6n)^2 - 1. - Bruno Berselli, Jan 29 2012

A321483 a(n) = 7*2^n + (-1)^n.

Original entry on oeis.org

8, 13, 29, 55, 113, 223, 449, 895, 1793, 3583, 7169, 14335, 28673, 57343, 114689, 229375, 458753, 917503, 1835009, 3670015, 7340033, 14680063, 29360129, 58720255, 117440513, 234881023, 469762049, 939524095, 1879048193, 3758096383, 7516192769, 15032385535

Offset: 0

Paul Curtz, Nov 11 2018

Difference table:
   8, 13, 29, 55, 113, 223, 449, ...
   5, 16, 26, 58, 110, 226, 446, 898, ...
  11, 10, 32, 52, 116, 220, 452, 892, 1796, ...
  -1, 22, 20, 64, 104, 232, 440, 904, 1784, 3592, ...
      -2, 44, 40, 128, 208, 464, 880, 1808, 3568, 7184, ...
etc.
Every diagonal is a sequence of the form k*2^m.
a(n) is divisible by
.  5 if n is a term of A004767,
. 11 if n is a term of A016885,
. 13 if n is a term of A017533.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,2).

Cf. A000079, A001045, A005029, A010710, A014551, A062092, A062510, A070366, A175805, A199207, A206372.
Cf. A033999, A005009.
Subsequence of A001651, A160543, A160544.

a[n_] := 7*2^n + (-1)^n ; Array[a, 32, 0] (* Amiram Eldar, Nov 12 2018 *)
CoefficientList[Series[E^-x + 7 E^(2 x), {x, 0, 20}], x]*Table[n!, {n, 0, 20}] (* Stefano Spezia, Nov 12 2018 *)
LinearRecurrence[{1,2},{8,13},40] (* Harvey P. Dale, Mar 18 2022 *)

Vec((8 + 5*x) / ((1 + x)*(1 - 2*x)) + O(x^40)) \\ Colin Barker, Nov 11 2018

O.g.f.: (8 + 5*x) / ((1 + x)*(1 - 2*x)). - Colin Barker, Nov 11 2018
E.g.f.: exp(-x) + 7*exp(2*x). - Stefano Spezia, Nov 12 2018
a(n) = a(n-1) + 2*a(n-2).
a(n) = 2*a(n-1) + 3*(-1)^n for n>0, a(0)=8.
a(2*k) = 7*4^k + 1, a(2*k+1) = 14*4^k - 1.
a(n) = A014551(n) + A014551(n-1) + A014551(n-2).
a(n) = 2^(n+3) - 3*A001045(n).
a(n) mod 9 = A070366(n+3).
a(n) + a(n+1) = 21*2^n.

Two terms corrected, and more terms added by Colin Barker, Nov 11 2018

A049513 Array T by antidiagonals: T(k,n) = kn2^(n-1) + 1, n >= 0, k >= 0.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 5, 1, 1, 4, 9, 13, 1, 1, 5, 13, 25, 33, 1, 1, 6, 17, 37, 65, 81, 1, 1, 7, 21, 49, 97, 161, 193, 1, 1, 8, 25, 61, 129, 241, 385, 449, 1, 1, 9, 29, 73, 161, 321, 577, 897, 1025, 1, 1, 10, 33, 85, 193, 401, 769, 1345, 2049, 2305, 1, 1, 11, 37, 97, 225, 481

Offset: 0

Michael Somos, Sep 25 1999

			Antidiagonals: 1; 1,1; 1,2,1; 1,3,5,1; 1,4,9,13,1; ...

Cf. A000337, A002064, A005183, A016813, A017533, A048474, A049069, A048472.

Essentially the same as A049069.

PARI
```
{T(k, n) = k * n * 2^(n-1) + 1}
```

A005183(n) = T(1, n), A002064(n) = T(2, n), A048474(n) = T(3, n), A000337(n) = T(4, n), A016813(n) = T(n, 2), A017533(n) = T(n, 3).

A382809 a(n) = (6n + 1)(12n + 1)(18*n + 1).

Original entry on oeis.org

1, 1729, 12025, 38665, 89425, 172081, 294409, 464185, 689185, 977185, 1335961, 1773289, 2296945, 2914705, 3634345, 4463641, 5410369, 6482305, 7687225, 9032905, 10527121, 12177649, 13992265, 15978745, 18144865, 20498401, 23047129, 25798825, 28761265, 31942225, 35349481

Offset: 0

Stefano Spezia, Apr 05 2025

a(n) is a Carmichael number if all the three factors (6*n + 1), (12*n + 1), and (18*n + 1) are prime (see Chernick and Ribenboim).

Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 101.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 146.

Jack Chernick, On Fermat's simple theorem, Bulletin of the American Mathematical Society, Vol. 45, No. 4 (1939), pp. 269-274.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A002997, A033502, A046025.

Cf. A002476, A002997, A016921, A017533, A068228, A161705.

LinearRecurrence[{4,-6,4,-1},{1,1729,12025,38665},31]

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n > 3.
G.f.: (1 + 1725*x + 5115*x^2 + 935*x^3)/(1 - x)^4.
E.g.f.: exp(x)*(1 + 1728*x + 4284*x^2 + 1296*x^3).
a(n) = A016921(n) * A017533(n) * A161705(n).
a(n) == 1 (mod 72).

cp's OEIS Frontend

A161709 a(n) = 22*n + 1.

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A161714 a(n) = 28*n + 1.

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A056530 Sequence remaining after third round of Flavius Josephus sieve; remove every fourth term of A047241.

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A103214 a(n) = 24*n + 1.

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A130154 Triangle read by rows: T(n, k) = 1 + 2*(n-k)*(k-1) (1 <= k <= n).

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A001538 a(n) = (12*n+1)*(12*n+11).

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A157326 a(n) = 10368*n^2 + 288*n + 1.

A130154 Triangle read by rows: T(n, k) = 1 + 2(n-k)(k-1) (1 <= k <= n).

A001538 a(n) = (12n+1)(12*n+11).

A157326 a(n) = 10368n^2 + 288n + 1.

A049513 Array T by antidiagonals: T(k,n) = kn2^(n-1) + 1, n >= 0, k >= 0.

A382809 a(n) = (6n + 1)(12n + 1)(18*n + 1).