A074506 -id:A074506 - cp's OEIS frontend

A074605 a(n) = 3^n + 4^n.

2, 7, 25, 91, 337, 1267, 4825, 18571, 72097, 281827, 1107625, 4371451, 17308657, 68703187, 273218425, 1088090731, 4338014017, 17309009347, 69106897225, 276040168411, 1102998412177, 4408506864307, 17623567104025

Offset: 0

Robert G. Wilson v, Aug 25 2002

x^n + y^n = (x+y)*a(n-1) - (x*y)*a(n-2). - Vincenzo Librandi, Jul 19 2010

L. B. W. Jolley, "Summation of Series", Dover Publications, 1961, p. 14.

B. Berselli, Table of n, a(n) for n = 0..1000. - _Bruno Berselli_, Jul 20 2010
Index entries for linear recurrences with constant coefficients, signature (7,-12).

Cf. A000051, A007689, A034472, A034474, A034491, A052539, A062394.

Cf. A062395, A062396, A063376, A063481, A074506, A074600..A074624.

[3^n+4^n: n in [0..40]]; // G. C. Greubel, Jan 16 2024

Mathematica
```
Table[3^n + 4^n, {n, 0, 25}]
```

a(n)=3^n+4^n \\ Charles R Greathouse IV, Oct 07 2015

[3^n+4^n for n in range(41)] # G. C. Greubel, Jan 16 2024

a(n) = A074506(n) - 1.
2 + 7*x + 25*x^2 + 91*x^3  + ... n terms = (1 - (4*x)^n)/(1 - 4*x) + (1 - (3*x)^n)/(1 - 3*x). [Jolley] - Gary W. Adamson, Dec 20 2006
From Mohammad K. Azarian, Jan 11 2009: (Start)
G.f.: 1/(1-3*x) + 1/(1-4*x).
E.g.f.: exp(3*x) + exp(4*x). (End)
a(n) = 3*a(n-1) + 4^(n-1). - Bruno Berselli, Jul 20 2010
a(n) = 7*a(n-1) - 12*a(n-2) with a(0)=2, a(1)=7. - Vincenzo Librandi, Jul 19 2010

A323767 A(n,k) = Sum_{j=0..floor(n/2)} binomial(n-j,j)^k, square array A(n,k) read by antidiagonals, for n >= 0, k >= 0.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 3, 3, 1, 1, 2, 5, 5, 3, 1, 1, 2, 9, 11, 8, 4, 1, 1, 2, 17, 29, 26, 13, 4, 1, 1, 2, 33, 83, 92, 63, 21, 5, 1, 1, 2, 65, 245, 338, 343, 153, 34, 5, 1, 1, 2, 129, 731, 1268, 1923, 1281, 376, 55, 6

Offset: 0

Seiichi Manyama, Jan 27 2019

			Square array begins:
   1,  1,   1,    1,     1,      1,       1, ...
   1,  1,   1,    1,     1,      1,       1, ...
   2,  2,   2,    2,     2,      2,       2, ...
   2,  3,   5,    9,    17,     33,      65, ...
   3,  5,  11,   29,    83,    245,     731, ...
   3,  8,  26,   92,   338,   1268,    4826, ...
   4, 13,  63,  343,  1923,  10903,   62283, ...
   4, 21, 153, 1281, 11553, 108801, 1050753, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns 0-5 give A004526(n+2), A000045(n+1), A051286, A181545, A181546, A181547.
Rows 0-5 give A000012, A000012, A007395, A000051, A168607, A074506.
Main diagonal gives A323769.
Cf. A011973,

f := Sum[Power[Binomial[#1 - i, i], #2], {i, 0, #1/2}] &;a = Flatten[Reverse[DeleteCases[Table[Table[f[m - n, n], {n, 0, 20}], {m, 0, 20}], 0, Infinity], 2]] (* Elijah Beregovsky, Nov 24 2020 *)

A057351 Numbers k that divide 4^k + 3^k + 1.

Original entry on oeis.org

1, 2, 26, 338, 2054, 4394, 13282, 26702, 46306, 57122, 59566, 162266, 221446, 347126, 404293, 742586, 760318, 1301638, 2031406, 2109458, 2857114, 4512638, 4752982, 8340826, 8634886, 9653618, 10066654, 12819014, 19012162, 21263026, 25885054, 27422954, 34901542

Offset: 1

Robert G. Wilson v, Sep 22 2000

Michael S. Branicky, Table of n, a(n) for n = 1..82

Cf. A074506.

Select[ Range[ 10^6 ], Mod[ PowerMod[ 4, #, # ] + PowerMod[ 3, #, # ] + 1, # ] == 0 & ]

a(18) from Hugo Pfoertner, Jan 05 2022

a(19)-a(33) from Michael S. Branicky, Jan 06 2022

cp's OEIS Frontend

A074605 a(n) = 3^n + 4^n.

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A323767 A(n,k) = Sum_{j=0..floor(n/2)} binomial(n-j,j)^k, square array A(n,k) read by antidiagonals, for n >= 0, k >= 0.

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A057351 Numbers k that divide 4^k + 3^k + 1.

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