A079773 -id:A079773 - cp's OEIS frontend

A122853 Numbers k such that (3^k + 5^k)/8 = A074606(k)/8 is a prime.

3, 5, 7, 17, 19, 109, 509, 661, 709, 1231, 12889, 13043, 26723, 43963, 44789

Offset: 1

Alexander Adamchuk, Sep 14 2006

(3^k + 5^k)/8 = A074606(k)/8 = A081186(k)/4.
Corresponding primes of the form (3^k + 5^k)/2^3 are listed in {A121938(n)} = {A079773(a(n))} = {19, 421, 10039, 95383574161, 2384331073699, ...}.
No other terms less than 100000. - Robert Price, Apr 28 2012

Cf. A074606, A081186, A121824, A121877, A005058, A005059, A121938, A109347, A079773.

Do[f=5^n+3^n;If[PrimeQ[f/2^3],Print[{n,f/2^3}]],{n,1,1231}]

select(n->isprime((3^n + 5^n)/8), vector(2000,i,i)) \\ Charles R Greathouse IV, Feb 13 2011

a(11)-a(15) from Robert Price, Apr 28 2012

A238801 Triangle T(n,k), read by rows, given by T(n,k) = C(n+1, k+1)*(1-(k mod 2)).

Original entry on oeis.org

1, 2, 0, 3, 0, 1, 4, 0, 4, 0, 5, 0, 10, 0, 1, 6, 0, 20, 0, 6, 0, 7, 0, 35, 0, 21, 0, 1, 8, 0, 56, 0, 56, 0, 8, 0, 9, 0, 84, 0, 126, 0, 36, 0, 1, 10, 0, 120, 0, 252, 0, 120, 0, 10, 0, 11, 0, 165, 0, 462, 0, 330, 0, 55, 0, 1, 12, 0, 220, 0, 792, 0, 792, 0, 220, 0, 12, 0

Offset: 0

Philippe Deléham, Mar 05 2014

Row sums are powers of 2.

			Triangle begins:
1;
2, 0;
3, 0, 1;
4, 0, 4, 0;
5, 0, 10, 0, 1;
6, 0, 20, 0, 6, 0;
7, 0, 35, 0, 21, 0, 1;
8, 0, 56, 0, 56, 0, 8, 0;
9, 0, 84, 0, 126, 0, 36, 0, 1;
10, 0, 120, 0, 252, 0, 120, 0, 10, 0; etc.

G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened

Cf. Columns: A000027, A000292, A000389, A000580, A000582, A001288, A010966, A010968, A010970, A010972, A010974, A010976, A010980, A010982, A010984, A010986, A010988, A010990, A010992, A010994, A010996, A010998, A011000, A017713, A017715, A017717, A017719, A017721, A017723, A017725, A017727, A017729, A017731, A017733, A017735, A017737, A017739, A017741, A017743, A017745, A017747, A017749, A017751, A017753, A017755, A017757, A017759, A017761, A017763.

Cf. A095704, A178616.

Table[Binomial[n + 1, k + 1]*(1 - Mod[k , 2]), {n, 0, 10}, {k, 0, n}] // Flatten (* G. C. Greubel, Nov 22 2017 *)

T(n,k) = binomial(n+1, k+1)*(1-(k % 2));
tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n,k), ", ")); print); \\ Michel Marcus, Nov 23 2017

G.f.: 1/((1+(y-1)*x)*(1-(y+1)*x)).
T(n,k) = 2*T(n-1,k) + T(n-2,k-2) - T(n-2,k), T(0,0) = 1, T(1,0) = 2, T(1,1) = 0, T(n,k) = 0 if k<0 or if k>n.
Sum_{k=0..n} T(n,k)*x^k = A000027(n+1), A000079(n), A015518(n+1), A003683(n+1), A079773(n+1), A051958(n+1), A080920(n+1), A053455(n), A160958(n+1) for x = 0, 1, 2, 3, 4, 5, 6, 7, 8 respectively.

A080424 a(n) = 3a(n-1) + 18a(n-2), a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 3, 27, 135, 891, 5103, 31347, 185895, 1121931, 6711903, 40330467, 241805655, 1451365371, 8706597903, 52244370387, 313451873415, 1880754287211, 11284396583103, 67706766919107, 406239439253175, 2437440122303451, 14624630273467503, 87747813021864627, 526486783988008935

Offset: 0

Paul Barry, Feb 24 2003

The ratio a(n+1)/a(n) converges to 6 as n approaches infinity. - Felix P. Muga II, Mar 10 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
F. P. Muga II, Extending the Golden Ratio and the Binet-de Moivre Formula, March 2014; Preprint on ResearchGate.
Index entries for linear recurrences with constant coefficients, signature (3,18).

Cf. A001045, A015441, A051958, A079773.

[(6^n-(-3)^n)/9: n in [0..30]]; // Vincenzo Librandi, Aug 13 2011

a[n_]:=(6^n - (-3)^n)/9; Array[a, 22, 0] (* Robert G. Wilson v, Aug 13 2011 *)
LinearRecurrence[{3,18}, {0,1}, 31] (* G. C. Greubel, Dec 22 2023 *)

a(n)=(6^n-(-3)^n)/9 \\ Charles R Greathouse IV, Jun 10 2011

[3^(n-1)*lucas_number1(n,1,-2) for n in range(31)] # G. C. Greubel, Dec 22 2023

G.f.: x/((1+3*x)*(1-6*x)).
a(n) = (6^n - (-3)^n)/9.
a(n+1) = 6*a(n) + (-3)^n. - Paul Curtz, Jun 07 2011
a(n) = 3^(n-1)*A001045(n). - R. J. Mathar, Mar 08 2021

A080920 a(n) = 2a(n-1) + 35a(n-2), a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 2, 39, 148, 1661, 8502, 75139, 447848, 3525561, 22725802, 168846239, 1133095548, 8175809461, 56009963102, 398173257339, 2756695223248, 19449454453361, 135383241720402, 951497389308439, 6641408238830948

Offset: 0

Paul Barry, Feb 24 2003

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (2,35).

Cf. A079773, A051958, A015441.

Join[{a=0,b=1},Table[c=2*b+35*a;a=b;b=c,{n,60}]] (* Vladimir Joseph Stephan Orlovsky, Feb 01 2011 *)
CoefficientList[Series[1 / ((1 + 5 x) (1 - 7 x)), {x, 0, 20}], x] (* Vincenzo Librandi, Aug 05 2013 *)
LinearRecurrence[{2,35},{0,1},30] (* Harvey P. Dale, Aug 24 2017 *)

a(n) = 7^n/12 - (-5)^n/12.
a(n) = Sum{k=1..n, binomial(n, 2k-1)*6^(2(k-1))}.
G.f.: 1/((1+5x)(1-7x)).
a(n+1) = Sum_{k = 0..n} A238801(n,k)*6^k. - Philippe Deléham, Mar 07 2014

A080921 a(n) = 2a(n-1) + 48a(n-2), a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 2, 52, 200, 2896, 15392, 169792, 1078400, 10306816, 72376832, 639480832, 4753049600, 40201179136, 308548739072, 2546754076672, 19903847628800, 162051890937856, 1279488468058112, 10337467701133312, 82090381869056000

Offset: 0

Paul Barry, Feb 24 2003

Essentially the same as A053455: a(n) = A053455(n-1), n>=1.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (2,48).

Cf. A079773, A051958, A015441, A080920.

CoefficientList[Series[x / ((1 + 6 x) (1 - 8 x)), {x, 0, 20}], x] (* Vincenzo Librandi, Aug 05 2013 *)
LinearRecurrence[{2,48},{0,1},30] (* Harvey P. Dale, Jan 20 2016 *)

a(n) = (8^n - (-6)^n)/14.
a(n) = Sum{k=1..n, binomial(n, 2k-1) * 7^(2(k-1)) }
G.f.: x/((1+6*x)*(1-8*x)).
a(n) = A053455(n-1), n>=1. [R. J. Mathar, Sep 18 2008]

A121938 Primes of the form (3^k + 5^k)/2^3 = A074606(k)/8.

Original entry on oeis.org

19, 421, 10039, 95383574161, 2384331073699, 1925929944387235853055979210606894889560480247048440342330377620014353281101

Offset: 1

Zak Seidov, Sep 10 2006

Corresponding numbers k such that (3^k + 5^k)/8 is prime are listed in A122853. All these numbers are primes. - Alexander Adamchuk, Sep 14 2006

The next term is too large to include. - Alexander Adamchuk, Sep 14 2006

Cf. A074706, A122853, A081186, A079773, A121824, A121877, A005058, A005059, A121938, A109347.

Do[f=5^n+3^n;If[PrimeQ[f/2^3],Print[{n,f/2^3}]],{n,1,1231}] (* Alexander Adamchuk, Sep 14 2006 *)

a(n) = (A122853(n)^3 + A122853(n)^5)/8. a(n) = A074606[A122853(n)]/8 = A081186[A122853(n)]/4. a(n) = A079773[A122853(n)]. - Alexander Adamchuk, Sep 14 2006

More terms from Alexander Adamchuk, Sep 14 2006

A292847 a(n) is the smallest odd prime of the form ((1 + sqrt(2n))^k - (1 - sqrt(2n))^k)/(2sqrt(2n)).

Original entry on oeis.org

5, 7, 101, 11, 13, 269, 17, 19, 509, 23, 709, 821, 29, 31, 46957, 55399, 37, 168846239, 41, 43, 9177868096974864412935432937651459122761, 47, 485329129, 2789, 53, 3229, 3461, 59, 61, 1563353111, 139237612541, 67, 5021, 71, 73, 484639, 6221, 79, 6869, 83, 7549

Offset: 1

XU Pingya, Sep 24 2017

nonn

			For k = {1, 2, 3, 4, 5}, ((1 + sqrt(6))^k - (1 - sqrt(6))^k)/(2*sqrt(6)) = {1, 2, 9, 28, 101}. 101 is odd prime, so a(3) = 101.

Cf. A000129, A002605, A015518, A063727, A002532, A083099, A015519, A003683, A002534, A083102, A015520, A091914, A079773, A161007, A099134.

g[n_, k_] := ((1 + Sqrt[n])^k - (1 - Sqrt[n])^k)/(2Sqrt[n]);
Table[k = 3; While[! PrimeQ[Expand@g[2n, k]], k++]; Expand@g[2n, k], {n, 41}]

g(n,k) = ([0,1;2*n-1,2]^k*[0;1])[1,1]
a(n) = for(k=3,oo,if(ispseudoprime(g(n,k)),return(g(n,k)))) \\ Jason Yuen, Apr 12 2025

When 2*n + 3 = p is prime, a(n) = p.

A383637 Expansion of 1/((1-x) * (1+3x) (1-5*x)).

Original entry on oeis.org

1, 3, 22, 90, 511, 2373, 12412, 60420, 307021, 1520343, 7646002, 38097150, 190884331, 953225913, 4769716792, 23837822280, 119221396441, 596010127083, 2980341200782, 14900834307810, 74506786627351, 372526087871853, 1862653975153972, 9313199268385740, 46566208164081061

Offset: 0

Seiichi Manyama, May 03 2025

Index entries for linear recurrences with constant coefficients, signature (3,13,-15).

Cf. A079773, A120612.

PARI
```
a(n) = (5^(n+2)+(-3)^(n+2)-2)/32;
```

a(n) = Sum_{k=0..floor(n/2)} 16^k * binomial(n+2,2*k+2).
a(n) = (5^(n+2) + (-3)^(n+2) - 2)/32 = (A120612(n+2) - 1)/16.
a(n) = 3*a(n-1) + 13*a(n-2) - 15*a(n-3).
a(n) = Sum_{k=0..n} 4^k * (-3)^(n-k) * binomial(n+2,k+2) * Stirling2(k+2,2).
a(n) = Sum_{k=0..n} (-4)^k * 5^(n-k) * binomial(n+2,k+2) * Stirling2(k+2,2).

cp's OEIS Frontend

A122853 Numbers k such that (3^k + 5^k)/8 = A074606(k)/8 is a prime.

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A238801 Triangle T(n,k), read by rows, given by T(n,k) = C(n+1, k+1)*(1-(k mod 2)).

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A080424 a(n) = 3*a(n-1) + 18*a(n-2), a(0)=0, a(1)=1.

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A080920 a(n) = 2a(n-1) + 35a(n-2), a(0)=0, a(1)=1.

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A080921 a(n) = 2*a(n-1) + 48*a(n-2), a(0)=0, a(1)=1.

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A121938 Primes of the form (3^k + 5^k)/2^3 = A074606(k)/8.

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A292847 a(n) is the smallest odd prime of the form ((1 + sqrt(2*n))^k - (1 - sqrt(2*n))^k)/(2*sqrt(2*n)).

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A383637 Expansion of 1/((1-x) * (1+3*x) * (1-5*x)).

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A080424 a(n) = 3a(n-1) + 18a(n-2), a(0)=0, a(1)=1.

A080921 a(n) = 2a(n-1) + 48a(n-2), a(0)=0, a(1)=1.

A292847 a(n) is the smallest odd prime of the form ((1 + sqrt(2n))^k - (1 - sqrt(2n))^k)/(2sqrt(2n)).

A383637 Expansion of 1/((1-x) * (1+3x) (1-5*x)).