A085279 -id:A085279 - cp's OEIS frontend

A206238 T(n,k)=Number of (n+1)X(k+1) 0..3 arrays with every 2X3 or 3X2 subblock having exactly two equal edges, and new values 0..3 introduced in row major order.

15, 60, 60, 310, 256, 310, 1640, 1136, 1136, 1640, 8910, 5728, 4456, 5728, 8910, 51066, 31652, 27168, 27168, 31652, 51066, 294546, 170728, 133392, 283728, 133392, 170728, 294546, 1710184, 943584, 607008, 1236432, 1236432, 607008, 943584, 1710184

Offset: 1

R. H. Hardin Feb 05 2012

Table starts
......15......60......310......1640........8910........51066........294546
......60.....256.....1136......5728.......31652.......170728........943584
.....310....1136.....4456.....27168......133392.......607008.......3503136
....1640....5728....27168....283728.....1236432......9042600......95322432
....8910...31652...133392...1236432....10915392....118573968....1122086640
...51066..170728...607008...9042600...118573968...1448239080...22535636736
..294546..943584..3503136..95322432..1122086640..22535636736..649065145152
.1710184.5175034.17206032.419146392.10022726928.303011941944.8026428934128

			Some solutions for n=4 k=3
..0..0..1..0....0..0..1..1....0..0..1..1....0..1..2..0....0..0..1..1
..0..1..0..0....0..2..3..3....2..2..3..1....3..2..2..0....2..2..0..1
..2..0..0..1....2..3..3..2....1..2..2..3....2..2..1..2....3..2..2..3
..0..0..2..3....3..3..0..3....0..1..2..2....2..1..2..2....2..1..2..2
..0..1..3..3....0..0..3..3....0..0..3..2....3..2..2..3....2..2..0..2

R. H. Hardin, Table of n, a(n) for n = 1..544

Empirical for column k:
k=1: a(n) = 8*a(n-1) -11*a(n-2) +36*a(n-3) -303*a(n-4) +232*a(n-5) +147*a(n-6) +756*a(n-7) for n>8
k=2: a(n) = 3*a(n-1) +20*a(n-2) -14*a(n-3) -133*a(n-4) +95*a(n-5) +123*a(n-6) +9*a(n-7) -102*a(n-8) for n>10
k=3: a(n) = a(n-1) +129*a(n-3) -129*a(n-4) for n>7
k=4: a(n) = a(n-1) +339*a(n-3) -339*a(n-4) for n>8
k=5: a(n) = a(n-1) +921*a(n-3) -921*a(n-4) for n>9
k=6: a(n) = a(n-1) +2571*a(n-3) -2571*a(n-4) for n>10
k=7: a(n) = a(n-1) +7329*a(n-3) -7329*a(n-4) for n>11
k=8: a(n) = a(n-1) +21219*a(n-3) -21219*a(n-4) for n>12
k=9: a(n) = a(n-1) +62121*a(n-3) -62121*a(n-4) for n>13
k=10: a(n) = a(n-1) +183291*a(n-3) -183291*a(n-4) for n>14
k=11: a(n) = a(n-1) +543729*a(n-3) -543729*a(n-4) for n>15
apparently a(n) = a(n-1) +3*A085279(k+1)*a(n-3) -3*A085279(k+1)*a(n-4) for k>2 and n>k+4

A254028 a(n) = 2^(n+1) + 3^n + 3.

Original entry on oeis.org

6, 10, 20, 46, 116, 310, 860, 2446, 7076, 20710, 61100, 181246, 539636, 1610710, 4815740, 14414446, 43177796, 129402310, 387944780, 1163310046, 3488881556, 10464547510, 31389448220, 94159956046, 282463090916, 847355718310

Offset: 0

Luciano Ancora, Jan 22 2015

This is the sequence of third terms of "second partial sums of m-th powers".

Colin Barker, Table of n, a(n) for n = 0..1000
Luciano Ancora, Demonstration of formulas
Index entries for linear recurrences with constant coefficients, signature (6,-11,6).

Cf. A085279, A094374, A099754.

Table[2^(n+1)+3^n+3,{n,0,30}] (* or *) LinearRecurrence[{6,-11,6},{6,10,20},30] (* Harvey P. Dale, Mar 27 2025 *)

a(n)=2<Charles R Greathouse IV, Jan 23 2015

Vec(-2*(13*x^2-13*x+3)/((x-1)*(2*x-1)*(3*x-1)) + O(x^100)) \\ Colin Barker, Jan 24 2015

G.f.: -2*(13*x^2-13*x+3) / ((x-1)*(2*x-1)*(3*x-1)). - Colin Barker, Jan 24 2015
a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3). - Colin Barker, Jan 24 2015
a(n) = A085279(n+1) = 2*( A099754(n)+1 ) = 2*( A094374(n)-2 ). [Bruno Berselli, Jan 26 2015]

A085280 Expansion of (1-4x+x^2)/((1-x)(1-3x)(1-4x)).

Original entry on oeis.org

1, 4, 14, 48, 166, 584, 2094, 7648, 28406, 107064, 408574, 1575248, 6123846, 23963944, 94261454, 372262848, 1474702486, 5855763224, 23293912734, 92788230448, 369990660326, 1476475856904, 5895443074414, 23550391238048

Offset: 0

Paul Barry, Jun 25 2003

Binomial transform of A085279.

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Index entries for linear recurrences with constant coefficients, signature (8,-19,12).

[3^n+4^n/3-1/3: n in [0..30]]; // Vincenzo Librandi, May 26 2013

Table[(3^n + 4^n/3 - 1/3), {n, 0, 30}] (* Vincenzo Librandi, May 26 2013 *)
CoefficientList[Series[(1-4x+x^2)/((1-x)(1-3x)(1-4x)),{x,0,30}],x] (* Harvey P. Dale, Jan 02 2021 *)

a(n) = 3^n+4^n/3-1/3.

A369490 a(n) = 3^(n+1) + 2*(-2)^(n+1).

Original entry on oeis.org

-1, 17, 11, 113, 179, 857, 1931, 7073, 18659, 61097, 173051, 539633, 1577939, 4815737, 14283371, 43177793, 128878019, 387944777, 1161212891, 3488881553, 10456158899, 31389448217, 94126401611, 282463090913, 847221500579, 2542000046057

Offset: 0

Philippe Deléham, Jan 24 2024

Index entries for linear recurrences with constant coefficients, signature (1, 6).

Cf. A000079, A000244, A003063, A015441, A085279.

LinearRecurrence[{1,6},{-1,17},26] (* James C. McMahon, Jan 30 2024 *)

def A369490(n): return 3**(n+1)+(1<Chai Wah Wu, Feb 25 2024

a(n) = a(n-1) + 6*a(n-2); a(0) = -1, a(1) = 17.
G.f.: (18*x-1)/((1+2*x)*(1-3*x)).
a(2*n) = A003063(2*n+2).
a(2*n+1) = A085279(2*n+3).
a(n) = 18*A015441(n) - A015441(n+1).

A123924 Numbers k such that 2^(k+1) + 3^k is prime.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 9, 11, 12, 15, 17, 22, 32, 33, 35, 36, 46, 47, 59, 63, 80, 101, 154, 159, 173, 221, 225, 236, 250, 281, 347, 789, 992, 1607, 1631, 1983, 2072, 3616, 3702, 5076, 5957, 6335, 8771, 10203, 10984, 12203, 12350, 13660, 14891

Offset: 1

Alexander Adamchuk, Nov 20 2006

nonn

Also numbers k such that A123601(k) = A085279(k+1) = 2^(k+1) + 3^k. There are only 4 known primes of form the 2^k + 3^k, {2, 5, 13, 97} = A082101, corresponding to k = {0, 1, 2, 4}.

Cf. A082101 (primes of form 2^k + 3^k), A085279, A123601 (smallest prime of the form p^n + q^n + r^n, where p,q,r are primes).

Do[f=2^(n+1)+3^n;If[PrimeQ[f],Print[{n,f}]],{n,0,347}]
Select[Range[0,6400],PrimeQ[2^(#+1)+3^#]&] (* Harvey P. Dale, Mar 04 2019 *)

is(n)=ispseudoprime(2^(n+1)+3^n) \\ Charles R Greathouse IV, Jun 13 2017

More terms from Stefan Steinerberger, May 12 2007
a(44) from Jinyuan Wang, Aug 02 2021
a(45)-a(50) from Michael S. Branicky, Aug 05 2021

A247313 a(n) = 5*a(n-1) - 2^n for n>0, a(0)=1.

Original entry on oeis.org

1, 3, 11, 47, 219, 1063, 5251, 26127, 130379, 651383, 3255891, 16277407, 81382939, 406906503, 2034516131, 10172547887, 50862673899, 254313238423, 1271565929971, 6357829125567, 31789144579259, 158945720799143, 794728599801411, 3973642990618447

Offset: 0

Vincenzo Librandi, Sep 12 2014

James Boswell Instituut, Sequences, 2006, p. 19 (recurrence 1.d).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,-10).

Cf. A000079, A000351, A074600, A085279.

Magma
```
[(2^(n+1)+5^n)/3: n in [0..30]];
```

RecurrenceTable[{a[0] == 1, a[n] == 5 a[n - 1] - 2^n}, a, {n, 0, 30}] (* or *) Table[(2^(n + 1) + 5^n)/3, {n, 0, 30}]

Vec((1-4*x)/((1-2*x)*(1-5*x)) + O(x^50)) \\ Michel Marcus, Sep 13 2014

G.f.: (1-4*x)/((1-2*x)*(1-5*x)).
a(n) = ( 2^(n+1) + 5^n )/3.
a(n) = 7*a(n-1) - 10*a(n-2) for n>1.

cp's OEIS Frontend

A206238 T(n,k)=Number of (n+1)X(k+1) 0..3 arrays with every 2X3 or 3X2 subblock having exactly two equal edges, and new values 0..3 introduced in row major order.

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A254028 a(n) = 2^(n+1) + 3^n + 3.

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A085280 Expansion of (1-4x+x^2)/((1-x)(1-3x)(1-4x)).

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A369490 a(n) = 3^(n+1) + 2*(-2)^(n+1).

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A123924 Numbers k such that 2^(k+1) + 3^k is prime.

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A247313 a(n) = 5*a(n-1) - 2^n for n>0, a(0)=1.

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