A056450 -id:A056450 - cp's OEIS frontend

A165152 a(n) = (3*12^n - 8^n)/2.

1, 14, 184, 2336, 29056, 356864, 4347904, 52699136, 636583936, 7672561664, 92339175424, 1110217588736, 13339790934016, 160214930161664, 1923678673567744, 23092940175835136, 277185650854199296, 3326790760203812864, 39925992722073124864, 479147941461896462336

Offset: 0

Klaus Brockhaus, Sep 15 2009

Binomial transform of A165151. Sixth binomial transform of A016129. Tenth binomial transform of A056450.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (20,-96).

Cf. A165151, A016129, A056450.

Magma
```
[ (3*12^n-8^n)/2: n in [0..17] ];
```

A165152:=n->(3*12^n - 8^n)/2; seq(A165152(n), n=0..17); # Wesley Ivan Hurt, Jan 29 2014

Table[(3*12^n - 8^n)/2, {n, 0, 17}] (* Wesley Ivan Hurt, Jan 29 2014 *)

a(n) = 20*a(n-1) - 96*a(n-2) for n > 1; a(0) = 1, a(1) = 14.
G.f.: (1 - 6*x)/((1 - 8*x)*(1 - 12*x)).
E.g.f.: exp(8*x)*(3*exp(4*x) - 1)/2. - Stefano Spezia, Mar 30 2023

A001446 a(n) = (4^n + 4^[ n/2 ] )/2.

Original entry on oeis.org

10, 34, 136, 520, 2080, 8224, 32896, 131200, 524800, 2097664, 8390656, 33556480, 134225920, 536879104, 2147516416, 8589967360, 34359869440, 137439084544, 549756338176, 2199023779840, 8796095119360

Offset: 2

N. J. A. Sloane

James Spahlinger, Table of n, a(n) for n = 2..1001
Index entries for linear recurrences with constant coefficients, signature (4,4,-16).

Equals 2^(2[n/2]-1) * (A056450(n)+1). Cf. A001445.

G.f.: x^2*(10-6x-40x^2)/((1-4x)*(1-4x^2)).

a(n) = 4*a(n-1) + 4*a(n-2) - 16*a(n-3) for n > 4. - Chai Wah Wu, Sep 10 2020

A251861 Number of non-palindromic words (length n>0) over the alphabet of 26 letters.

Original entry on oeis.org

0, 650, 16900, 456300, 11863800, 308898200, 8031353200, 208826607600, 5429491797600, 141167083772000, 3670344178072000, 95428956352766400, 2481152865171926400, 64509974695265340800, 1677259342076898860800, 43608742899220046995200, 1133827315379721221875200, 29479510200008489360729600, 766467265200220723378969600, 19928148895209267985244544000

Offset: 1

Mikk Heidemaa, Dec 10 2014

Example: the acronyms 'OEIS' and 'SIEO' are two distinct non-palindromic words of length 4 among all possible such 456300 words (over 26 letters of the Latin alphabet).

			For n=2, the a(2)=650 solutions are {ab,ac,...,az,...,yz}, but not, e.g., 'aa' or 'zz'.

Indranil Ghosh, Table of n, a(n) for n = 1..705
Index entries for linear recurrences with constant coefficients, signature (26,26,-676).

Analogs for other numbers of elements: (1) A000004, (2) A233411, (3) A242278, (4) A242026, (5) A240437.

Cf. A056450.

seq(26^n - 26^ceil(n/2), n = 1 .. 50); # Robert Israel, Dec 11 2014

f[n_, b_] := b^n - b^Ceiling[n/2]; Array[ f[#, 26] &, 50] (* Robert G. Wilson v, Dec 10 2014 *)
Table[2^(n/2-1)*13^(n/2)*((-1)^n*(Sqrt[26]-1)-Sqrt[26]-1)+26^n, {n, 50}]

a(n)=26^n-26^ceil(n/2) \\ Charles R Greathouse IV, Dec 10 2014

a(n) = 2^(n/2-1)*13^(n/2)*((-1)^n*(sqrt(26)-1)-sqrt(26)-1)+26^n.
a(n) = 26^n - 26^ceiling(n/2).
G.f.: 650*x^2/((1 - 26*x)*(1 - 26*x^2)).
a(n+3) = 26*a(n+2) + 26*a(n+1) - 676*a(n). - Robert Israel, Dec 11 2014

A232774 Triangle T(n,k), read by rows, given by T(n,0)=1, T(n,1)=2^(n+1)-n-2, T(n,n)=(-1)^(n-1) for n > 0, T(n,k)=T(n-1,k)-T(n-1,k-1) for 1 < k < n.

Original entry on oeis.org

1, 1, 1, 1, 4, -1, 1, 11, -5, 1, 1, 26, -16, 6, -1, 1, 57, -42, 22, -7, 1, 1, 120, -99, 64, -29, 8, -1, 1, 247, -219, 163, -93, 37, -9, 1, 1, 502, -466, 382, -256, 130, -46, 10, -1, 1, 1013, -968, 848, -638, 386, -176, 56, -11, 1, 2036, -1981, 1816, -1486, 1024

Offset: 0

Philippe Deléham, Nov 30 2013

Row sums are A000079(n) = 2^n.
Diagonal sums are A024493(n+1) = A130781(n).
Sum_{k=0..n} T(n,k)*x^k = -A003063(n+2), A159964(n), A000012(n), A000079(n), A001045(n+2), A056450(n), (-1)^(n+1)*A232015(n+1) for x = -2, -1, 0, 1, 2, 3, 4 respectively.

			Triangle begins:
  1;
  1,    1;
  1,    4,   -1;
  1,   11,   -5,   1;
  1,   26,  -16,   6,   -1;
  1,   57,  -42,  22,   -7,   1;
  1,  120,  -99,  64,  -29,   8,   -1;
  1,  247, -219, 163,  -93,  37,   -9,  1;
  1,  502, -466, 382, -256, 130,  -46, 10,  -1;
  1, 1013, -968, 848, -638, 386, -176, 56, -11, 1;

Columns: A000012, A000295, -A002662, A002663, -A002664, A035038, -A035039, A035040, -A035041, A035042.
Cf. A000079, A055248, A024493, A130781, A000302.
Cf. also A003063, A159964, A000012, A001045, A056450, A232015.

G.f.: Sum_{n>=0, k=0..n} T(n,k)*y^k*x^n=(1+2*(y-1)*x)/((1-2*x)*(1+(y-1)*x)).
|T(2*n,n)| = 4^n = A000302(n).
T(n,k) = (-1)^(k-1) * (Sum_{i=0..n-k} (2^(i+1)-1) * binomial(n-i-1,k-1)) for 0 < k <= n and T(n,0) = 1 for n >= 0. - Werner Schulte, Mar 22 2019

cp's OEIS Frontend

A165152 a(n) = (3*12^n - 8^n)/2.

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A001446 a(n) = (4^n + 4^[ n/2 ] )/2.

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A251861 Number of non-palindromic words (length n>0) over the alphabet of 26 letters.

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A232774 Triangle T(n,k), read by rows, given by T(n,0)=1, T(n,1)=2^(n+1)-n-2, T(n,n)=(-1)^(n-1) for n > 0, T(n,k)=T(n-1,k)-T(n-1,k-1) for 1 < k < n.

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