A008594 -id:A008594 - cp's OEIS frontend

A072065 Define a "piece" to consist of 3 mutually touching pennies welded together to form a triangle; sequence gives side lengths of triangles that can be made from such pieces.

0, 2, 9, 11, 12, 14, 21, 23, 24, 26, 33, 35, 36, 38, 45, 47, 48, 50, 57, 59, 60, 62, 69, 71, 72, 74, 81, 83, 84, 86, 93, 95, 96, 98, 105, 107, 108, 110, 117, 119, 120, 122, 129, 131, 132, 134, 141, 143, 144, 146, 153, 155, 156, 158, 165, 167, 168, 170, 177, 179, 180

Offset: 1

Jim McCann (jmccann(AT)umich.edu), Aug 04 2002

The "piece" in question is also called a "tribone" [Ardila and Stanley]. - N. J. A. Sloane, Feb 27 2014

			A possible side-9 arrangement:
          A
         A A
        B B C
       D B C C
      D D E E F
     G H H E F F
    G G H I I J J
   K L L M I N J O
  K K L M M N N O O

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
F. Ardila and R. P. Stanley, Tilings, arXiv:math/0501170 [math.CO], 2005.
J. H. Conway and J. C. Lagarias, Tiling with Polyominoes and Combinatorial Group Theory, Journal of Combinatorial Theory, Series A 53 (1990), 183-208. [From _N. J. A. Sloane_, Jul 04 2011]
Jim McCann, Triangle Sequence
N. C. Saldanha and C. Tomei, An overview of domino and lozenge tilings, arXiv:math/9801111 [math.CO], 1998.
Torsten Sillke, A Word Problem
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Union of A008594, A017545, A017629 and A017653.

a072065 n = a072065_list !! n
a072065_list = filter ((`elem` [0,2,9,11]) . (`mod` 12)) [0..]
-- Reinhard Zumkeller, Jan 09 2013

f:=r-> {seq(12*i+r,i=0..100)}; t1:= f(0) union f(2) union f(9) union f(11); t2:=sort(convert(t1,list)); # N. J. A. Sloane, Jul 04 2011

Select[Range[0,200],MemberQ[{0,2,9,11},Mod[#,12]]&] (* Harvey P. Dale, Dec 15 2011 *)
LinearRecurrence[{1,0,0,1,-1},{0,2,9,11,12},70] (* Harvey P. Dale, Jan 30 2015 *)

concat(0, Vec(x^2*(2+7*x+2*x^2+x^3)/((1-x)^2*(1+x)*(1+x^2)) + O(x^100))) \\ Colin Barker, Dec 12 2015

A number n is in the sequence iff n == 0, 2, 9 or 11 (mod 12). See Conway-Lagarias or the Sillke link. - Sascha Kurz, Mar 04 2003
a(1)=0, a(2)=2, a(3)=9, a(4)=11, a(5)=12, a(n) = a(n-1)+a(n-4)-a(n-5). - Harvey P. Dale, Jan 30 2015
From Colin Barker, Dec 12 2015: (Start)
a(n) = (3/4+(3*i)/4)*(i^n-i*(-i)^n)-(-1)^n/2+3*(n+1)-5 where i = sqrt(-1).
G.f.: x^2*(2+7*x+2*x^2+x^3) / ((1-x)^2*(1+x)*(1+x^2)). (End)
E.g.f.: (2 + 3*cos(x) + (6*x - 5)*cosh(x) - 3*sin(x) + (6*x - 3)*sinh(x))/2. - Stefano Spezia, May 05 2022
a(n) = (6*n-4-(-1)^n+3*(-1)^((2*n+1-(-1)^n)/4))/2. - Wesley Ivan Hurt, Nov 09 2023

Offset corrected by Reinhard Zumkeller, Jan 09 2013

A169823 Multiples of 60.

Original entry on oeis.org

0, 60, 120, 180, 240, 300, 360, 420, 480, 540, 600, 660, 720, 780, 840, 900, 960, 1020, 1080, 1140, 1200, 1260, 1320, 1380, 1440, 1500, 1560, 1620, 1680, 1740, 1800, 1860, 1920, 1980, 2040, 2100, 2160, 2220, 2280, 2340, 2400, 2460, 2520, 2580, 2640, 2700

Offset: 0

N. J. A. Sloane, May 29 2010

Numbers that are divisible by all of 1, 2, 3, 4, 5, 6.

Erich Friedman, What's Special About This Number? (See the entry for "60")
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A001477, A005843, A008585, A008586, A008587, A008588, A008592, A008594.

Cf. A008597, A008602, A169825, A169827, A249674.

Range[0, 2700, 60] (* Vladimir Joseph Stephan Orlovsky, Jul 12 2011 *)

a(n)=60*n \\ Charles R Greathouse IV, Mar 19 2017

From Elmo R. Oliveira, Apr 16 2024: (Start)
G.f.: 60*x/(x-1)^2.
E.g.f.: 60*x*exp(x).
a(n) = 60*n = 2*a(n-1) - a(n-2) for n >= 2.
a(n) = 2*A249674(n) = 3*A008602(n) = 4*A008597(n) = 5*A008594(n) = 6*A008592(n) = 10*A008588(n) = 12*A008587(n) = 15*A008586(n) = 20*A008585(n) = 30*A005843(n) = 60*A001477(n) = A169827(n)/14 = A169825(n)/7. (End)

A332512 Numbers k such that phi(k) == 0 (mod 12), where phi is the Euler totient function (A000010).

Original entry on oeis.org

13, 21, 26, 28, 35, 36, 37, 39, 42, 45, 52, 56, 57, 61, 63, 65, 70, 72, 73, 74, 76, 77, 78, 84, 90, 91, 93, 95, 97, 99, 104, 105, 108, 109, 111, 112, 114, 117, 119, 122, 124, 126, 129, 130, 133, 135, 140, 143, 144, 146, 147, 148, 152, 153, 154, 155, 156, 157, 161

Offset: 1

Amiram Eldar, Feb 14 2020

nonn

Dence and Pomerance showed that the asymptotic number of the terms below x is ~ x.

			13 is a term since phi(13) = 12 == 0 (mod 12).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Thomas Dence and Carl Pomerance, Euler's function in residue classes, in: K. Alladi, P. D. T. A. Elliott, A. Granville and G. Tenenbaum (eds.), Analytic and Elementary Number Theory, Developments in Mathematics, Vol. 1, Springer, Boston, MA, 1998, pp. 7-20, alternative link.

Cf. A000010, A008594, A332511, A332513, A332514, A332515, A332516.

[k:k in [1..170]| EulerPhi(k) mod 12 eq 0]; // Marius A. Burtea, Feb 14 2020

Select[Range[200], Divisible[EulerPhi[#], 12] &]

A094519 Numbers having at least one pair (x,y) of divisors with x

Original entry on oeis.org

6, 12, 18, 20, 24, 30, 36, 40, 42, 48, 54, 56, 60, 66, 70, 72, 78, 80, 84, 90, 96, 100, 102, 108, 110, 112, 114, 120, 126, 132, 138, 140, 144, 150, 156, 160, 162, 168, 174, 180, 182, 186, 192, 198, 200, 204, 210, 216, 220, 222, 224, 228, 234, 240, 246

Offset: 1

Reinhard Zumkeller, May 06 2004

nonn

If m is in the sequence then so is k*m for k > 0. Furthermore, all terms are even. - David A. Corneth, Aug 31 2019
If (x,y) = (1,m) with m > 1, then oblong numbers m*(m+1) >= 6 belong to this sequence, and each oblong number >= 6 is a primitive term of the subsequence {k*m*(m+1), k >= 1}. Examples: with pair (1,2), we get multiples of 6 (see A008588); with (1,3) we get multiples of 12 (see A008594); with (1,4) we get multiples of 20 (see A008602); with (1,7) we get multiples of 56. - Bernard Schott, Aug 31 2019
The numbers of terms that do not exceed 10^k, for k = 1, 2, ..., are 1, 22, 230, 2317, 23201, 232209, 2322920, 23232166, 232332309, 2323370184, ... . Apparently, the asymptotic density of this sequence exists and equals 0.23233... . - Amiram Eldar, Apr 20 2025

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

Cf. A094518.
Complement of A094520.
A superset of A088723. - R. J. Mathar, Sep 16 2007
Subsequences: A002378 \ {0, 2}, A008588 \ {0}, A008602 \ {0}.

aQ[n_] := AnyTrue[Total /@ Subsets[Divisors[n], {2}], Divisible[n, #] &]; Select[Range[250], aQ] (* Amiram Eldar, Aug 31 2019 *)

is(n) = {my(d = divisors(n)); for(i = 1, #d - 2, for(j = i + 1, #d - 1, if(n % (d[i] + d[j]) == 0, return(1) ) ) ); 0 } \\ David A. Corneth, Aug 31 2019

from itertools import count, islice
from sympy import divisors
def A094519_gen(): # generator of terms
    for n in count(1):
        for i in range(1,len(d:=divisors(n))):
            di = d[i]
            for j in range(i):
                if n % (di+d[j]) == 0:
                    yield n
                    break
            else:
                continue
            break
A094519_list = list(islice(A094519_gen(),20)) # Chai Wah Wu, Dec 26 2021

A094518(a(n)) > 0.

A063664 Numbers whose reciprocal is the sum of two reciprocals of squares.

Original entry on oeis.org

2, 8, 18, 20, 32, 50, 72, 80, 90, 98, 128, 144, 162, 180, 200, 242, 272, 288, 320, 338, 360, 392, 450, 468, 500, 512, 576, 578, 648, 650, 720, 722, 800, 810, 882, 968, 980, 1058, 1088, 1152, 1250, 1280, 1296, 1332, 1352, 1440, 1458, 1568, 1620, 1682, 1800

Offset: 1

Henry Bottomley, Jul 28 2001

nonn

These are numbers which can be written either as b^2*c^2*(b^2+c^2)*d^2 or if (b^2+c^2) is a square then as b^2*c^2*d^2, since 1/(b*(b^2+c^2)*d)^2+1/(c*(b^2+c^2)*d)^2 =1/(b^2*c^2*(b^2+c^2)*d^2) and 1/(b*sqrt(b^2+c^2)*d)^2+1/(c*sqrt(b^2+c^2)*d)^2 = 1/(b^2*c^2*d^2).

			98 is in the sequence since 1/98=1/10^2+1/70^2 (also 1/98=1/14^2+1/14^2).

Either products of terms in A063663 and A000290, or squares of A008594.

Cf. A001481, A000404, A063665, A063669.

from fractions import Fraction
def aupto(lim):
  sqr_recips = [Fraction(1, i*i) for i in range(1, lim+2)]
  ssr = set(f + g for i, f in enumerate(sqr_recips) for g in sqr_recips[i:])
  representable = [f.denominator for f in ssr if f.numerator == 1]
  return sorted(r for r in representable if r <= lim)
print(aupto(1800)) # Michael S. Branicky, Feb 08 2021

Offset changed to 1 by Derek Orr, Jun 23 2015

A076515 Numbers k such that 1 + 3^k + 5^k is prime.

Original entry on oeis.org

0, 12, 36, 48, 72, 120, 605376

Offset: 1

Zak Seidov, Oct 17 2002

Next term, if it exists, is greater than 35000. - Vaclav Kotesovec, Jan 26 2016
No more terms up to 75000 (previous range rechecked). All terms are multiples of 12: if k > 0 is not a multiple of 12, 1 + 3^k + 5^k is divisible by 3, 5 or 7. - Rick L. Shepherd, Aug 06 2017
Next term, if it exists, is greater than 10^6. - Jon Grantham, Jul 29 2023

Cf. A008594, A074507, A268064, A268065, A268067.

[n: n in [0..1000]|IsPrime(3^n+5^n+1)] // Vincenzo Librandi, Jan 22 2011

A076515:=n->`if`(isprime(1+3^n+5^n), n, NULL): seq(A076515(n), n=0..200); # Wesley Ivan Hurt, Aug 06 2017

Do[ If[ PrimeQ[1 + 3^n + 5^n], Print[n]], {n, 0, 3500, 2}]
Select[Range[0,5000],PrimeQ[1+3^#+5^#]&] (* Harvey P. Dale, Mar 09 2012 *)

lista(nn) = for(n=0, nn, if(ispseudoprime(1 + 3^n + 5^n), print1(n, ", "))); \\ Altug Alkan, Jan 25 2016

a(7) from Jon Grantham, Jul 29 2023

A105020 Array read by antidiagonals: row n (n >= 0) contains the numbers m^2 - n^2, m >= n+1.

Original entry on oeis.org

1, 3, 4, 5, 8, 9, 7, 12, 15, 16, 9, 16, 21, 24, 25, 11, 20, 27, 32, 35, 36, 13, 24, 33, 40, 45, 48, 49, 15, 28, 39, 48, 55, 60, 63, 64, 17, 32, 45, 56, 65, 72, 77, 80, 81, 19, 36, 51, 64, 75, 84, 91, 96, 99, 100, 21, 40, 57, 72, 85, 96, 105, 112, 117, 120, 121

Offset: 0

Andrew S. Plewe and Franklin T. Adams-Watters, Jul 11 2005

A "Goldbach Conjecture" for this sequence: when there are n terms between consecutive odd integers (2n+1) and (2n+3) for n > 0, at least one will be the product of 2 primes (not necessarily distinct). Example: n=3 for consecutive odd integers a(7)=7 and a(11)=9 and of the 3 sequence entries a(8)=12, a(9)=15 and a(10)=16 between them, one is the product of 2 primes a(9)=15=3*5. - Michael Hiebl, Jul 15 2007
A024352 gives distinct values in the array, minus the first row (1, 4, 9, 16, etc.). a(n) gives all solutions to the equation x^2 + xy = n, with y mod 2 = 0, x > 0, y >= 0. - Andrew S. Plewe, Oct 19 2007
Alternatively, triangular sequence of coefficients of Dynkin diagram weights for the Cartan groups C_n: t(n,m) = m*(2*n - m). Row sums are A002412. - Roger L. Bagula, Aug 05 2008

			Array begins:
  1  4  9 16 25 36  49  64  81 100 ...
  3  8 15 24 35 48  63  80  99 120 ...
  5 12 21 32 45 60  77  96 117 140 ...
  7 16 27 40 55 72  91 112 135 160 ...
  9 20 33 48 65 84 105 128 153 180 ...
  ...
Triangle begins:
   1;
   3,  4;
   5,  8,  9;
   7, 12, 15, 16;
   9, 16, 21, 24, 25;
  11, 20, 27, 32, 35, 36;
  13, 24, 33, 40, 45, 48, 49;
  15, 28, 39, 48, 55, 60, 63, 64;
  17, 32, 45, 56, 65, 72, 77, 80, 81;
  19, 36, 51, 64, 75, 84, 91, 96, 99, 100;

R. N. Cahn, Semi-Simple Lie Algebras and Their Representations, Dover, NY, 2006, ISBN 0-486-44999-8, p. 139.

G. C. Greubel, Antidiagonals n = 0..50, flattened

Rows give A000290, A005563, A028347, A028560, A028566, A098603, A098847, A098848, A098849, A098850.
Columns give A005408, A008586, A016945, A008590, A017329, A008594, A008598, A008602, A008606, A000567.
Diagonals give A033428, A045944, A067725.
Cf. A000384, A002412, A024352, A105020, A128076.

[(k+1)*(2*n-k+1): k in [0..n], n in [0..15]]; // G. C. Greubel, Mar 15 2023

t[n_, m_]:= (n^2 - m^2); Flatten[Table[t[i, j], {i,12}, {j,i-1,0,-1}]]
(* to view table *) Table[t[i, j], {j,0,6}, {i,j+1,10}]//TableForm (* Robert G. Wilson v, Jul 11 2005 *)
Table[(k+1)*(2*n-k+1), {n,0,15}, {k,0,n}]//Flatten (* Roger L. Bagula, Aug 05 2008 *)

def A105020(n,k): return (k+1)*(2*n-k+1)
flatten([[A105020(n,k) for k in range(n+1)] for n in range(16)]) # G. C. Greubel, Mar 15 2023

a(n) = r^2 - (r^2 + r - m)^2/4, where r = round(sqrt(m)) and m = 2*n+2. - Wesley Ivan Hurt, Sep 04 2021
a(n) = A128076(n+1) * A105020(n+1). - Wesley Ivan Hurt, Jan 07 2022
From G. C. Greubel, Mar 15 2023: (Start)
Sum_{k=0..n} T(n, k) = A002412(n+1).
Sum_{k=0..n} (-1)^k*T(n, k) = (1/2)*((1+(-1)^n)*A000384((n+2)/2) - (1- (-1)^n)*A000384((n+1)/2)). (End)

More terms from Robert G. Wilson v, Jul 11 2005

A274824 Triangle read by rows: T(n,k) = (n-k+1)*sigma(k), n>=1, 1<=k<=n.

Original entry on oeis.org

1, 2, 3, 3, 6, 4, 4, 9, 8, 7, 5, 12, 12, 14, 6, 6, 15, 16, 21, 12, 12, 7, 18, 20, 28, 18, 24, 8, 8, 21, 24, 35, 24, 36, 16, 15, 9, 24, 28, 42, 30, 48, 24, 30, 13, 10, 27, 32, 49, 36, 60, 32, 45, 26, 18, 11, 30, 36, 56, 42, 72, 40, 60, 39, 36, 12, 12, 33, 40, 63, 48, 84, 48, 75, 52, 54, 24, 28, 13, 36, 44, 70, 54, 96, 56, 90, 65, 72, 36, 56, 14

Offset: 1

Omar E. Pol, Oct 02 2016

Theorem: for any sequence S the partial sums of the partial sums are also the antidiagonal sums of the square array in which the n-th row gives n times the sequence S. Therefore they are also the row sums of the triangular array in which the n-th diagonal gives n times the sequence S.
In this case the sequence S is A000203.
The n-th diagonal of this triangle gives n times A000203.
The row sums give A175254 which gives the partial sums of A024916 which gives the partial sums of A000203.
T(n,k) is also the total number of unit cubes that are exactly below the terraces of the k-th level (starting from the top) up the base of the stepped pyramid with n levels described in A245092. This fact is because the mentioned terraces have the same shape as the symmetric representation of sigma(k). For more information see A237593 and A237270.
In the definition of this sequence the value n-k+1 is also the height of the terraces associated to sigma(k) in the mentioned pyramid with n levels, or in other words, the distance between the mentioned terraces and the base of the pyramid.
The sum of the n-th row of triangle equals the volume (also the number of cubes) of the mentioned pyramid with n levels.
For an illustration of the pyramid, see the Links section.
The sum of the n-th row is also 1/4 of the volume of the stepped pyramid described in A244050 with n levels.
Column k lists the positive multiples of sigma(k).
The k-th term in the n-th diagonal is equal to n*sigma(k).
Note that this is also a square array read by antidiagonals upwards: T(i,j) = i*sigma(j), i>=1, j>=1. The first row of the array is A000203. So consider that the pyramid is upside down. The value of "i" is the distance between the base of the pyramid and the terraces associated to sigma(j). The antidiagonal sums give the partial sums of the partial sums of A000203.

			Triangle begins:
1;
2,  3;
3,  6,  4;
4,  9,  8,  7;
5,  12, 12, 14, 6;
6,  15, 16, 21, 12, 12;
7,  18, 20, 28, 18, 24,  8;
8,  21, 24, 35, 24, 36,  16, 15;
9,  24, 28, 42, 30, 48,  24, 30,  13;
10, 27, 32, 49, 36, 60,  32, 45,  26,  18;
11, 30, 36, 56, 42, 72,  40, 60,  39,  36,  12;
12, 33, 40, 63, 48, 84,  48, 75,  52,  54,  24, 28;
13, 36, 44, 70, 54, 96,  56, 90,  65,  72,  36, 56,  14;
14, 39, 48, 77, 60, 108, 64, 105, 78,  90,  48, 84,  28, 24;
15, 42, 52, 84, 66, 120, 72, 120, 91,  108, 60, 112, 42, 48, 24;
16, 45, 56, 91, 72, 132, 80, 135, 104, 126, 72, 140, 56, 72, 48, 31;
...
For n = 16 and k = 10 the sum of the divisors of 10 is 1 + 2 + 5 + 10 = 18, and 16 - 10 + 1 = 7, and 7*18 = 126, so T(16,10) = 126.
On the other hand, the symmetric representation of sigma(10) has two parts of 9 cells, giving a total of 18 cells. In the stepped pyramid described in A245092, with 16 levels, there are 16 - 10 + 1 = 7 cubes exactly below every cell of the symmetric representation of sigma(10) up the base of pyramid; hence the total numbers of cubes exactly below the terraces of the 10th level (starting from the top) up the base of the pyramid is equal to 7*18 = 126. So T(16,10) = 126.
The sum of the 16th row of the triangle is 16 + 45 + 56 + 91 + 72 + 132 + 80 + 135 + 104 + 126 + 72 + 140 + 56 + 72 + 48 + 31 = A175254(16) = 1276, equaling the volume (also the number of cubes) of the stepped pyramid with 16 levels described in A245092 (see Links section).

Indranil Ghosh, Rows 1..100 of triangle, flattened
Omar E. Pol, Illustration of the stepped pyramid with 16 levels

Row sums of triangle give A175254.
Diagonals 1-6: A000203, A074400, A272027, A239050, A274535, A274536.
Column 1 is A000027.
Initial zeros should be omitted in the following sequences related to the columns of triangle:
Columns 2-5: A008585, A008586, A008589, A008588.
Columns 6 and 11: A008594.
Columns 7-9: A008590, A008597, A008595.
Columns 10 and 17: A008600.
Columns 12-13: A135628, A008596.
Columns 14, 15 and 23: A008606.
Columns 16 and 25: A135631.
(There are many other OEIS sequences that are also columns of this triangle.)
Cf. A004736, A024916, A054973, A196020, A236104, A235791, A237591, A237593, A237270, A237271, A244050, A245092, A245093, A262612.

T(n,k) = (n-k+1) * A000203(k).

T(n,k) = A004736(n,k) * A245093(n,k).

A317312 Multiples of 12 and odd numbers interleaved.

Original entry on oeis.org

0, 1, 12, 3, 24, 5, 36, 7, 48, 9, 60, 11, 72, 13, 84, 15, 96, 17, 108, 19, 120, 21, 132, 23, 144, 25, 156, 27, 168, 29, 180, 31, 192, 33, 204, 35, 216, 37, 228, 39, 240, 41, 252, 43, 264, 45, 276, 47, 288, 49, 300, 51, 312, 53, 324, 55, 336, 57, 348, 59, 360, 61, 372, 63, 384, 65, 396, 67, 408, 69

Offset: 0

Omar E. Pol, Jul 25 2018

Partial sums give the generalized 16-gonal numbers (A274978).

a(n) is also the length of the n-th line segment of the rectangular spiral whose vertices are the generalized 16-gonal numbers.

Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Cf. A008594 and A005408 interleaved.
Column 12 of A195151.
Sequences whose partial sums give the generalized k-gonal numbers: A026741 (k=5), A001477 (k=6), zero together with A080512 (k=7), A022998 (k=8), A195140 (k=9), zero together with A165998 (k=10), A195159 (k=11), A195161 (k=12), A195312 (k=13), A195817 (k=14), A317311 (k=15).
Cf. A274978.

{0}~Join~Riffle[2 Range@ # - 1, 12 Range@ #] &@ 35 (* or *)
CoefficientList[Series[x (1 + 12 x + x^2)/((1 - x)^2*(1 + x)^2), {x, 0, 69}], x] (* or *)
LinearRecurrence[{0, 2, 0, -1}, {0, 1, 12, 3}, 70] (* Michael De Vlieger, Jul 26 2018 *)

a(2n) = 12*n, a(2n+1) = 2*n + 1.
From Michael De Vlieger, Jul 26 2018: (Start)
G.f.: x*(1 + 12*x + x^2) / ((1 - x)^2*(1 + x)^2).
a(n) = 2*a(n-2) - a(n-4) for n>3. (End)
Multiplicative with a(2^e) = 3*2^(e+1), and a(p^e) = p^e for an odd prime p. - Amiram Eldar, Oct 14 2023
Dirichlet g.f.: zeta(s-1) * (1 + 5*2^(1-s)). - Amiram Eldar, Oct 25 2023
a(n) = (7 + 5*(-1)^n)*n/2. - Aaron J Grech, Aug 20 2024

A082285 a(n) = 16*n + 13.

Original entry on oeis.org

13, 29, 45, 61, 77, 93, 109, 125, 141, 157, 173, 189, 205, 221, 237, 253, 269, 285, 301, 317, 333, 349, 365, 381, 397, 413, 429, 445, 461, 477, 493, 509, 525, 541, 557, 573, 589, 605, 621, 637, 653, 669, 685, 701, 717, 733, 749, 765, 781, 797, 813, 829, 845

Offset: 0

Cino Hilliard, May 10 2003

Solutions to (7^x + 11^x) mod 17 = 13.

a(n-2), n>=2, gives the second column in triangle A238476 related to the Collatz problem. - Wolfdieter Lang, Mar 12 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Tanya Khovanova, Recursive Sequences.
Leo Tavares, Illustration: Bounded Star Crosses.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A004770, A008594, A008598, A016813, A098502, A106839, A119413.

[[ n : n in [1..1000] | n mod 16 eq 13]]; // Vincenzo Librandi, Oct 10 2011

Range[13, 1000, 16] (* Vladimir Joseph Stephan Orlovsky, May 31 2011 *)
LinearRecurrence[{2,-1},{13,29},60] (* Harvey P. Dale, Jan 28 2023 *)

\\ solutions to 7^x+11^x == 13 mod 17
anpbn(n) = { for(x=1,n, if((7^x+11^x-13)%17==0,print1(x" "))) }

a(n) = 16*n + 13.
a(n) = 32*n - a(n-1) + 10; a(0)=13. - Vincenzo Librandi, Oct 10 2011
From Stefano Spezia, Dec 27 2019: (Start)
O.g.f.: (13 + 3*x)/(1 - x)^2.
E.g.f.: exp(x)*(13 + 16*x). (End)
a(n) = A008594(n+1) + A016813(n+1) - 4. - Leo Tavares, Sep 22 2022
From Elmo R. Oliveira, Apr 12 2025: (Start)
a(n) = 2*a(n-1) - a(n-2).
a(n) = A004770(2*n+2). (End)

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