A025551 -id:A025551 - cp's OEIS frontend

A081342 a(n) = (8^n + 2^n)/2.

1, 5, 34, 260, 2056, 16400, 131104, 1048640, 8388736, 67109120, 536871424, 4294968320, 34359740416, 274877911040, 2199023263744, 17592186060800, 140737488388096, 1125899906908160, 9007199254872064, 72057594038190080, 576460752303947776, 4611686018428436480

Offset: 0

Paul Barry, Mar 18 2003

Binomial transform of A034494.

5th binomial transform of {1, 0, 9, 0, 81, 0, 729, 0, ...}.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (10,-16).

Cf. A074603, A025551, A081343.

List([0..30], n-> (8^n + 2^n)/2); # G. C. Greubel, Jan 08 2020

[(8^n+2^n)/2: n in [0..30]]; // Vincenzo Librandi, Jun 13 2011

seq( (8^n + 2^n)/2, n=0..30); # G. C. Greubel, Jan 08 2020

Table[(8^n + 2^n)/2, {n, 0, 30}] (* Vladimir Joseph Stephan Orlovsky, Jun 12 2011 *)

a(n)=(8^n+2^n)/2 \\ Charles R Greathouse IV, Sep 28 2015

[(8^n + 2^n)/2 for n in (0..30)] # G. C. Greubel, Jan 08 2020

a(n) = (8^n + 2^n)/2.
a(n) = 10*a(n-1) - 16*a(n-2), a(0)=1, a(1)=5.
G.f.: (1-5*x)/((1-2*x)*(1-8*x)).
E.g.f.: exp(5*x)*cosh(3*x).
a(n) = ((5+sqrt(9))^n + (5-sqrt(9))^n)/2. - Al Hakanson (hawkuu(AT)gmail.com), Dec 08 2008
a(n) = A074603(n)/2. - Michel Marcus, Jan 09 2020

A081343 a(n) = (10^n + 4^n)/2.

Original entry on oeis.org

1, 7, 58, 532, 5128, 50512, 502048, 5008192, 50032768, 500131072, 5000524288, 50002097152, 500008388608, 5000033554432, 50000134217728, 500000536870912, 5000002147483648, 50000008589934592, 500000034359738368

Offset: 0

Paul Barry, Mar 18 2003

Binomial transform of A025551. 7th binomial transform of {1, 0, 9, 0, 81, 0, 729, 0, ...}.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (14,-40).

Cf. A081342.

List([0..25], n-> (10^n + 4^n)/2); # G. C. Greubel, Jan 07 2020

[(10^n+4^n)/2: n in [0..25]]; // Vincenzo Librandi, Aug 08 2013

seq( (10^n + 4^n)/2, n=0..25); # G. C. Greubel, Jan 07 2020

CoefficientList[Series[(1-7x)/((1-4x)(1-10x)), {x, 0, 20}], x] (* Vincenzo Librandi, Aug 08 2013 *)

a(n)=(10^n+4^n)/2 \\ Charles R Greathouse IV, Oct 07 2015

[(10^n + 4^n)/2 for n in (0..25)] # G. C. Greubel, Jan 07 2020

a(n) = 14*a(n-1) -40*a(n-2), a(0)=1, a(1)=7.
G.f.: (1-7*x)/((1-4*x)*(1-10*x)).
E.g.f.: exp(7*x) * cosh(3*x).
a(n) = ((7+sqrt(9))^n + (7-sqrt(9))^n)/2. - Al Hakanson (hawkuu(AT)gmail.com), Dec 08 2008

A073216 The terms of A055235 (sums of two powers of 3) divided by 2.

Original entry on oeis.org

1, 2, 3, 5, 6, 9, 14, 15, 18, 27, 41, 42, 45, 54, 81, 122, 123, 126, 135, 162, 243, 365, 366, 369, 378, 405, 486, 729, 1094, 1095, 1098, 1107, 1134, 1215, 1458, 2187, 3281, 3282, 3285, 3294, 3321, 3402, 3645, 4374, 6561, 9842, 9843, 9846, 9855, 9882, 9963, 10206, 10935, 13122, 19683

Offset: 0

Jeremy Gardiner, Jul 21 2002

n such that 3 is the largest power of 3 dividing binomial(3n,n). - Benoit Cloitre, Jan 01 2004
Equals A023745 + 1.
This sequence is A007051 together with its (successive) multiples by (powers of) 3. - R. K. Guy, Oct 08 2011

			T(2,0) = 5 = (3^2 + 3^0) / 2.
Triangle begins:
     1;
     2,    3;
     5,    6,    9;
    14,   15,   18,   27;
    41,   42,   45,   54,   81;
   122,  123,  126,  135,  162,  243;
   365,  366,  369,  378,  405,  486,  729;
  1094, 1095, 1098, 1107, 1134, 1215, 1458, 2187;
  ...

C. Armana, Coefficients of Drinfeld modular forms and Hecke operators, Journal of Number Theory 131 (2011), 1435-1460.

Cf. A000244 (main diagonal), A055235, A007051 (first column), A023745.

T(2n,n) gives A025551.

from math import isqrt
def A073216(n): return 3**(a:=(k:=isqrt(m:=n<<1))+(m>k*(k+1))-1)+3**(n-1-(a*(a+1)>>1))>>1 # Chai Wah Wu, Apr 08 2025

T(n,m) = (3^n + 3^m) / 2, n = 0, 1, 2, 3 ..., m = 0, 1, 2, 3, ... n.

Edited by Jeremy Gardiner, Oct 08 2011

Offset changed by Alois P. Heinz, Apr 08 2025

A074610 a(n) = 3^n + 9^n.

Original entry on oeis.org

2, 12, 90, 756, 6642, 59292, 532170, 4785156, 43053282, 387440172, 3486843450, 31381236756, 282430067922, 2541867422652, 22876797237930, 205891146443556, 1853020231898562, 16677181828806732, 150094635684419610

Offset: 0

Robert G. Wilson v, Aug 25 2002

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (12,-27).

Cf. A000051, A007689, A025551, A034472, A034474, A034491, A052539.

Cf. A062394, A062395, A062396, A063376, A063481, A074600..A074624.

[3^n+9^n: n in [0..30]]; // G. C. Greubel, Jan 13 2024

Table[3^n + 9^n, {n, 0, 25}]
LinearRecurrence[{12,-27},{2,12},20] (* Harvey P. Dale, May 18 2023 *)

[3^n+9^n for n in range(31)] # G. C. Greubel, Jan 13 2024

From Mohammad K. Azarian, Jan 11 2009: (Start)
G.f.: 1/(1-3*x) + 1/(1-9*x).
E.g.f.: exp(3*x) + exp(9*x). (End)
a(n) = 12*a(n-1) - 27*a(n-2) with a(0)=2, a(1)=12. - Vincenzo Librandi, Jul 21 2010
a(n) = 2*A025551(n). - G. C. Greubel, Jan 13 2024

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A081342 a(n) = (8^n + 2^n)/2.

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A081343 a(n) = (10^n + 4^n)/2.

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A073216 The terms of A055235 (sums of two powers of 3) divided by 2.

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A074610 a(n) = 3^n + 9^n.

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