A081337 -id:A081337 - cp's OEIS frontend

A081336 a(n) = (7^n + 3^n)/2.

1, 5, 29, 185, 1241, 8525, 59189, 412865, 2885681, 20186645, 141267149, 988751945, 6920909321, 48445302365, 339113927909, 2373787929425, 16616486808161, 116315321563685, 814206992665469, 5699448173817305, 39896134892198201

Offset: 0

Paul Barry, Mar 18 2003

Binomial transform of A081336.

5th binomial transform of (1,0,4,0,16,0,64,...).

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Yassine Otmani, The 2-Pascal Triangle and a Related Riordan Array, J. Int. Seq. (2025) Vol. 28, Issue 3, Art. No. 25.3.5. See p. 12.
Index entries for linear recurrences with constant coefficients, signature (10,-21).

Cf. A074608, A081337.

[(7^n+3^n)/2: n in [0..25]]; // Vincenzo Librandi, Aug 08 2013

CoefficientList[Series[(1 - 5 x) / ((1 - 3 x) (1 - 7 x)), {x, 0, 20}], x] (* Vincenzo Librandi, Aug 08 2013 *)
LinearRecurrence[{10,-21},{1,5},30] (* Harvey P. Dale, Dec 07 2014 *)

a(n)=(7^n+3^n)/2 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = 10*a(n-1) - 21*a(n-2), a(0)=1, a(1)=5.
G.f.: (1-5*x)/((1-3*x)*(1-7*x)).
E.g.f.: exp(5*x) * cosh(2*x).
a(n) = A074608(n) / 2. - Michel Marcus, Oct 07 2015
a(n) = Sum_{k=0..n} A027907(n,2k)*4^k . - J. Conrad, Aug 24 2016

A081338 a(n) = (9^n + 5^n)/2.

Original entry on oeis.org

1, 7, 53, 427, 3593, 31087, 273533, 2430547, 21718673, 194686807, 1748275013, 15714943867, 141336838553, 1271543265727, 11441447985293, 102960824836387, 926586388371233, 8338972319559847, 75049224997132373, 675435395579660107

Offset: 0

Paul Barry, Mar 18 2003

Binomial transform of A081337. 7th binomial transform of (1,0,4,0,16,0,64,....).

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (14,-45).

[(9^n+5^n)/2: n in [0..25]]; // Vincenzo Librandi, Aug 08 2013

CoefficientList[Series[(1 - 7 x) / ((1 - 5 x) (1 - 9 x)),{x, 0, 20}], x] (* Vincenzo Librandi, Aug 08 2013 *)

a(n) = 14*a(n-1)-45*a(n-2), a(0)=1, a(1)=7.
G.f.: (1-7*x)/((1-5*x)*(1-9*x)).
E.g.f.: exp(7*x) * cosh(2*x).

A098657 Expansion of (1-x-4x^2)/((1-2x)(1-8x^2)).

Original entry on oeis.org

1, 1, 6, 4, 40, 16, 288, 64, 2176, 256, 16896, 1024, 133120, 4096, 1056768, 16384, 8421376, 65536, 67239936, 262144, 537395200, 1048576, 4297064448, 4194304, 34368126976, 16777216, 274911461376, 67108864, 2199157473280, 268435456, 17592722915328, 1073741824

Offset: 0

Paul Barry, Sep 19 2004

Let A=[1,2,1;2,0,-2;1,-2,1] the 3 X 3 symmetric Krawtchouk matrix. Then a(n) is the 1,1 element of A^n.

P. Feinsilver and J. Kocik, Krawtchouk matrices from classical and quantum walks, Contemporary Mathematics, 287 2001, pp. 83-96.

Index entries for linear recurrences with constant coefficients, signature (2,8,-16).

Cf. A081337, A098655, A098656.

a(n) = 2^((3*n-4)/2)*(1+(-1)^n)+2^(n-1).
a(n) = 2*a(n-1) + 8*a(n-2) - 16*a(n-3).
a(2n) = A081337(n) = (8^n+4^n)/2 and a(2n+1) = 4^n. - Peter Kagey, Jul 14 2023

cp's OEIS Frontend

A081336 a(n) = (7^n + 3^n)/2.

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A081338 a(n) = (9^n + 5^n)/2.

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Magma

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A098657 Expansion of (1-x-4x^2)/((1-2x)(1-8x^2)).

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