A001550 -id:A001550 - cp's OEIS frontend

A001576 a(n) = 1^n + 2^n + 4^n.

3, 7, 21, 73, 273, 1057, 4161, 16513, 65793, 262657, 1049601, 4196353, 16781313, 67117057, 268451841, 1073774593, 4295032833, 17180000257, 68719738881, 274878431233, 1099512676353, 4398048608257, 17592190238721, 70368752566273, 281474993487873, 1125899940397057

Offset: 0

N. J. A. Sloane

Equals A135576, except for the first term. - Omar E. Pol, Nov 18 2008
Conjecture: For n > 1, if a(n) = 1^n + 2^n + 4^n is a prime number then n is of the form 3^h. For example, for h=1, n=3, a(n) = 1^3 + 2^3 + 4^3 = 73 (prime); for h=2, n=9, a(n) = 1^9 + 2^9 + 4^9 = 262657 (prime); for h=3, n=27, a(n) is not prime. - Vincenzo Librandi, Aug 03 2010
The previous conjecture was proved by Golomb in 1978. See A051154. - T. D. Noe, Aug 15 2010
Another more elementary proof can be found in Liu link. - Bernard Schott, Mar 08 2019
Fills in one quarter section of the figurate form of the Sierpinski square curve. See illustration in links and A141725. - John Elias, Mar 29 2023

T. D. Noe, Table of n, a(n) for n = 0..200
John Elias, Illustration of Initial Terms: 1/4 Sierpinski Square Curve
Andy Liu, West German Mathematical Olympiad 1982 - Second round, Problem 4, Crux Mathematicorum, p. 105, Vol. 12, May. 86.
Index entries for linear recurrences with constant coefficients, signature (7,-14,8).

Subsequence of A002061.
Cf. A001550, A034513, A001579, A074501-A074580, A135576, A135577.
See also comments in A051154.
Cf. A006095, A022166.

Mathematica
```
Table[1^n + 2^n + 4^n, {n, 0, 24}]
```

a(n)=1+2^n+4^n \\ Charles R Greathouse IV, Jun 10 2011

[sigma(4,n)for n in range(0,23)] # Zerinvary Lajos, Jun 04 2009

a(n) = 6*a(n-1) - 8*a(n-2) + 3.
O.g.f.: -1/(-1+x) - 1/(-1+2*x) - 1/(-1+4*x) = ( -3+14*x-14*x^2 ) / ( (x-1)*(2*x-1)*(4*x-1) ). - R. J. Mathar, Feb 29 2008
E.g.f.: e^x + e^(2*x) + e^(4*x). - Mohammad K. Azarian, Dec 26 2008
a(n) = A024088(n)/A000225(n). - Reinhard Zumkeller, Feb 15 2009
Exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + 7*x + 35*x^2 + 155*x^3 + ... is the o.g.f. for the 2nd subdiagonal of triangle A022166, essentially A006095. - Peter Bala, Apr 07 2015

A074501 a(n) = 1^n + 2^n + 5^n.

Original entry on oeis.org

3, 8, 30, 134, 642, 3158, 15690, 78254, 390882, 1953638, 9766650, 48830174, 244144722, 1220711318, 6103532010, 30517610894, 152587956162, 762939584198, 3814697527770, 19073486852414, 95367432689202, 476837160300278

Offset: 0

Robert G. Wilson v, Aug 23 2002

Jason Bard, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8,-17,10).

Equals A074600(n) + 1.

Cf. A001550, A001576, A034513, A001579, A074502..A074580.

Mathematica
```
Table[1^n + 2^n + 5^n, {n, 0, 24}]
```

a(n)=1+2^n+5^n \\ Charles R Greathouse IV, Jun 10 2011

From Mohammad K. Azarian, Dec 26 2008: (Start)
G.f.: 1/(1-x) + 1/(1-2*x) + 1/(1-5*x).
E.g.f.: e^x + e^(2*x) + e^(5*x). (End)
a(n) = 7*a(n-1) - 10*a(n-2) + 4 with a(0)=3, a(1)=8. - Vincenzo Librandi, Jul 21 2010

A007689 a(n) = 2^n + 3^n.

Original entry on oeis.org

2, 5, 13, 35, 97, 275, 793, 2315, 6817, 20195, 60073, 179195, 535537, 1602515, 4799353, 14381675, 43112257, 129271235, 387682633, 1162785755, 3487832977, 10462450355, 31385253913, 94151567435, 282446313697, 847322163875

Offset: 0

N. J. A. Sloane, Robert G. Wilson v

L. B. W. Jolley, Summation of Series, Dover Publications, 1961, p. 14.
D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 1, p. 92.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..200
I. Amburg, K. Dasaratha, L. Flapan, T. Garrity, C. Lee, C. Mihailak, N. Neumann-Chun, S. Peluse, and M. Stoffregen, Stern Sequences for a Family of Multidimensional Continued Fractions: TRIP-Stern Sequences, arXiv:1509.05239 [math.CO], 2015-2017.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 169
Index entries for linear recurrences with constant coefficients, signature (5,-6).

For odd-indexed members divided by 5 see A096951.
Binomial transform of A000051.
Cf. A000079, A000244, A001550, A063376, A063481.
Cf. A074600 - A074624, A082101 (primes).

a007689 n = a000079 n + a000244 n  -- Reinhard Zumkeller, Apr 28 2013

[2^n+3^n: n in [0..30]]; // G. C. Greubel, Mar 11 2023

A007689:=n->2^n + 3^n: seq(A007689(n), n=0..50); # Wesley Ivan Hurt, Jan 24 2017

Table[2^n + 3^n, {n, 0, 25}]
a=2;Numerator[Table[a=2*a-((a+1)/2),{n,0,7!}]] (*10 times (or more) faster for large numbers.*) (* Vladimir Joseph Stephan Orlovsky, Apr 19 2010 *)
LinearRecurrence[{5,-6},{2,5},30] (* nearly 20 times faster than the above program for large numbers. *) (* Harvey P. Dale, Oct 20 2013 *)

a(n)=2^n+3^n \\ Charles R Greathouse IV, Jun 15 2011

[lucas_number2(n,5,6)for n in range(0,27)] # Zerinvary Lajos, Jul 08 2008

E.g.f.: exp(2*x)*(1+exp(x)).
G.f.: (2-5*x)/((1-2*x)*(1-3*x)).
a(n) = 5*a(n-1) - 6*a(n-2).
Sum_{j=0..n-1} a(j) = (1/2)*(3^n - 1) + (2^n - 1). [Jolley] - Gary W. Adamson, Dec 20 2006
Equals double binomial transform of [2, 1, 1, 1, ...]. - Gary W. Adamson, Apr 23 2008
If p[i] = Fibonacci(2i-5) and if A is the Hessenberg matrix of order n defined by: A[i,j]=p[j-i+1], (i<=j), A[i,j]=-1, (i=j+1), and A[i,j]=0 otherwise. Then, for n>=1, a(n-1)= det A. - Milan Janjic, May 08 2010
a(n) = 2*a(n-1) + 3^(n-1), with a(0)=2. - Vincenzo Librandi, Nov 18 2010
a(n) = A001550(n) - 1 = A000079(n) + A000244(n). - Reinhard Zumkeller, Mar 01 2012

Additional comments from Michael Somos, Jun 10 2000

A034513 a(n) = 1^n + 3^n + 9^n.

Original entry on oeis.org

3, 13, 91, 757, 6643, 59293, 532171, 4785157, 43053283, 387440173, 3486843451, 31381236757, 282430067923, 2541867422653, 22876797237931, 205891146443557, 1853020231898563, 16677181828806733, 150094635684419611

Offset: 0

N. J. A. Sloane

Also the sum of n-th powers of the divisors of 9.

T. D. Noe, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (13,-39,27).

Cf. A001550, A001576, A001579, A074501-A074580.

Table[1^n + 3^n + 9^n, {n, 0, 20}]
LinearRecurrence[{13,-39,27},{3,13,91},20] (* Harvey P. Dale, Apr 13 2012 *)

a(n)=1+3^n+9^n \\ Charles R Greathouse IV, Jun 10 2011

[sigma(9,n)for n in range(0,19)] # Zerinvary Lajos, Jun 04 2009

G.f.: 1/(1-x)+1/(1-3*x)+1/(1-9*x). E.g.f.: e^x+e^(3*x)+e^(9*x). - Mohammad K. Azarian, Dec 26 2008

a(n) = 13*a(n-1) - 39*a(n-2) + 27*a(n-3), a(0)=3, a(1)=13, a(2)=91. - Harvey P. Dale, Apr 13 2012

A074580 a(n) = 7^n + 8^n + 9^n.

Original entry on oeis.org

3, 24, 194, 1584, 13058, 108624, 911234, 7703664, 65588738, 561991824, 4843001474, 41948320944, 364990300418, 3188510652624, 27953062038914, 245823065693424, 2167728096132098, 19161612027339024, 169737447404391554

Offset: 0

Robert G. Wilson v, Aug 23 2002

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Index entries for linear recurrences with constant coefficients, signature (24,-191,504).

Cf. A001550, A001576, A034513, A001579, A074501-A074579.

[7^n + 8^n + 9^n: n in [0..20]]; // Vincenzo Librandi, May 20 2011

Table[7^n + 8^n + 9^n, {n, 0, 18}]
LinearRecurrence[{24,-191,504},{3,24,194},20] (* Harvey P. Dale, Jun 09 2019 *)

a(n)=7^n+8^n+9^n \\ Charles R Greathouse IV, Jun 10 2011

From Mohammad K. Azarian, Dec 26 2008: (Start)
G.f.: 1/(1-7*x) + 1/(1-8*x) + 1/(1-9*x).
E.g.f.: e^(7*x) + e^(8*x) + e^(9*x). (End)

A001579 a(n) = 3^n + 5^n + 6^n.

Original entry on oeis.org

3, 14, 70, 368, 2002, 11144, 63010, 360248, 2076802, 12050504, 70290850, 411802328, 2421454402, 14282991464, 84472462690, 500716911608, 2973740844802, 17689728038024, 105375041354530, 628434388600088

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Henri W. Gould, The Girard-Waring power sum formulas for symmetric functions and Fibonacci sequences, The Fibonacci Quarterly, 37(2):135-140, 1999.
Index entries for linear recurrences with constant coefficients, signature (14,-63,90).

Cf. A001550, A001576, A034513, A074501-A074580.

[3^n + 5^n + 6^n: n in [0..20]]; // Vincenzo Librandi, May 20 2011

Table[3^n + 5^n + 6^n, {n, 0, 20}]
LinearRecurrence[{14,-63,90},{3,14,70},20] (* Harvey P. Dale, Jun 17 2021 *)

a(n)=3^n+5^n+6^n \\ Charles R Greathouse IV, Jun 10 2011

def a(n): return 3**n + 5**n + 6**n
print([a(n) for n in range(20)]) # Michael S. Branicky, Mar 14 2021

From Mohammad K. Azarian, Dec 26 2008: (Start)
G.f.: 1/(1-3*x) + 1/(1-5*x) + 1/(1-6*x).
E.g.f.: e^(3*x) + e^(5*x) + e^(6*x). (End)

A103438 Square array T(m,n) read by antidiagonals: Sum_{k=1..n} k^m.

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 0, 1, 3, 3, 0, 1, 5, 6, 4, 0, 1, 9, 14, 10, 5, 0, 1, 17, 36, 30, 15, 6, 0, 1, 33, 98, 100, 55, 21, 7, 0, 1, 65, 276, 354, 225, 91, 28, 8, 0, 1, 129, 794, 1300, 979, 441, 140, 36, 9, 0, 1, 257, 2316, 4890, 4425, 2275, 784, 204, 45, 10

Offset: 0

Ralf Stephan, Feb 11 2005

For the o.g.f.s of the column sequences for this array, see A196837 and the link given there. - Wolfdieter Lang, Oct 15 2011
T(m,n)/n is the m-th moment of the discrete uniform distribution on {1,2,...,n}. - Geoffrey Critzer, Dec 31 2018
T(1,n) divides T(m,n) for odd m. - Franz Vrabec, Dec 23 2020

			Square array begins:
  0, 1,  2,   3,    4,     5,     6,      7,      8,      9, ... A001477;
  0, 1,  3,   6,   10,    15,    21,     28,     36,     45, ... A000217;
  0, 1,  5,  14,   30,    55,    91,    140,    204,    285, ... A000330;
  0, 1,  9,  36,  100,   225,   441,    784,   1296,   2025, ... A000537;
  0, 1, 17,  98,  354,   979,  2275,   4676,   8772,  15333, ... A000538;
  0, 1, 33, 276, 1300,  4425, 12201,  29008,  61776, 120825, ... A000539;
  0, 1, 65, 794, 4890, 20515, 67171, 184820, 446964, 978405, ... A000540;
Antidiagonal triangle begins as:
  0;
  0, 1;
  0, 1,  2;
  0, 1,  3,  3;
  0, 1,  5,  6,  4;
  0, 1,  9, 14, 10,  5;
  0, 1, 17, 36, 30, 15, 6;

J. Faulhaber, Academia Algebrae, Darinnen die miraculosische inventiones zu den höchsten Cossen weiters continuirt und profitirt werden, Augspurg, bey Johann Ulrich Schönigs, 1631.

G. C. Greubel, Antidiagonals n = 0..50, flattened
José L. Cereceda, Sums of powers of integers and hyperharmonic numbers, arXiv:2005.03407 [math.NT], 2020.
T. A. Gulliver, Divisibility of sums of powers of odd integers, Int. Math. For. 5 (2010) 3059-3066.
T. A. Gulliver, Sums of Powers of Integers Divisible by Three, Int. J. Contemp. Math. Sciences, Vol. 7, 2012, no. 38, pp. 1895-1901. - From _N. J. A. Sloane_, Dec 22 2012
V. J. W. Guo and J. Zeng, A q-analogue of Faulhaber's formula for sums of powers, arXiv:math/0501441 [math.CO], 2005.
H. Helfgott and I. M. Gessel, Enumeration of tilings of diamonds and hexagons with defects, arXiv:math/9810143 [math.CO], 1998.
T. Kim, q-analogues of the sums of powers of consecutive integers, arXiv:math/0502113 [math.NT], 2005.
D. E. Knuth, Johann Faulhaber and sums of powers, Math. Comp. 61 (1993), no. 203, 277-294.
Eric Weisstein's World of Mathematics, Discrete Uniform Distribution.
Wikipedia, Faulhaber's formula

Rows include A000027, A000217, A000330, A000537, A000538, A000539, A000540, A000541, A000542, A007487, A023002.
Columns include A000051, A001550, A001551, A001552, A001553, A001554, A001555, A001556, A001557.
Diagonals include A076015 and A031971.
Antidiagonal sums are in A103439.
Antidiagonals are the rows of triangle A192001.

T:= func< n,k | n eq 0 select k else (&+[j^n: j in [0..k]]) >;
[T(n-k,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Dec 22 2021

seq(print(seq(Zeta(0,-k,1)-Zeta(0,-k,n+1),n=0..9)),k=0..6);
# (Produces the square array from the example.) Peter Luschny, Nov 16 2008
# alternative
A103438 := proc(m,n)
    (bernoulli(m+1,n+1)-bernoulli(m+1))/(m+1) ;
    if m = 0 then
        %-1 ;
    else
        % ;
    end if;
end proc: # R. J. Mathar, May 10 2013
# simpler:
A103438 := proc(m,n)
    (bernoulli(m+1,n+1)-bernoulli(m+1,1))/(m+1) ;
end proc: # Peter Luschny, Mar 20 2024

T[m_, n_]:= HarmonicNumber[m, -n]; Flatten[Table[T[m-n, n], {m, 0, 11}, {n, m, 0, -1}]] (* Jean-François Alcover, May 11 2012 *)

PARI
```
T(m,n)=sum(k=0,n,k^m)
```

from itertools import count, islice
from math import comb
from fractions import Fraction
from sympy import bernoulli
def A103438_T(m,n): return sum(k**m for k in range(1,n+1)) if n<=m else int(sum(comb(m+1,i)*(bernoulli(i) if i!=1 else Fraction(1,2))*n**(m-i+1) for i in range(m+1))/(m+1))
def A103438_gen(): # generator of terms
    for m in count(0):
        for n in range(m+1):
            yield A103438_T(m-n,n)
A103438_list = list(islice(A103438_gen(),100)) # Chai Wah Wu, Oct 23 2024

def T(n,k): return (bernoulli_polynomial(k+1, n+1) - bernoulli_polynomial(1, n+1)) /(n+1)
flatten([[T(n-k,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Dec 22 2021

E.g.f.: e^x*(e^(x*y)-1)/(e^x-1).
T(m, n) = Zeta(-n, 1) - Zeta(-n, m + 1), for m >= 0 and n >= 0, where Zeta(z, v) is the Hurwitz zeta function. - Peter Luschny, Nov 16 2008
T(m, n) = HarmonicNumber(m, -n). - Jean-François Alcover, May 11 2012
T(m, n) = (Bernoulli(m + 1, n + 1) - Bernoulli(m + 1, 1)) / (m + 1). - Peter Luschny, Mar 20 2024
T(m, n) = Sum_{k=0...m-n} B(k)*(-1)^k*binomial(m-n,k)*n^(m-n-k+1)/(m-n-k+1), where B(k) = Bernoulli number A027641(k) / A027642(k). - Robert B Fowler, Aug 20 2024
T(m, n) = Sum_{i=1..n} J_m(i)*floor(n/i), where J_m is the m-th Jordan totient function. - Ridouane Oudra, Jul 19 2025

A144048 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is Euler transform of (j->j^k).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 5, 6, 5, 1, 1, 9, 14, 13, 7, 1, 1, 17, 36, 40, 24, 11, 1, 1, 33, 98, 136, 101, 48, 15, 1, 1, 65, 276, 490, 477, 266, 86, 22, 1, 1, 129, 794, 1828, 2411, 1703, 649, 160, 30, 1, 1, 257, 2316, 6970, 12729, 11940, 5746, 1593, 282, 42, 1, 1, 513

Offset: 0

Alois P. Heinz, Sep 08 2008

In general, column k > 0 is asymptotic to (Gamma(k+2)*Zeta(k+2))^((1-2*Zeta(-k)) /(2*k+4)) * exp((k+2)/(k+1) * (Gamma(k+2)*Zeta(k+2))^(1/(k+2)) * n^((k+1)/(k+2)) + Zeta'(-k)) / (sqrt(2*Pi*(k+2)) * n^((k+3-2*Zeta(-k))/(2*k+4))). - Vaclav Kotesovec, Mar 01 2015

			Square array begins:
  1,  1,   1,   1,    1,     1, ...
  1,  1,   1,   1,    1,     1, ...
  2,  3,   5,   9,   17,    33, ...
  3,  6,  14,  36,   98,   276, ...
  5, 13,  40, 136,  490,  1828, ...
  7, 24, 101, 477, 2411, 12729, ...

Alois P. Heinz, Antidiagonals = 0..99, flattened
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 21.
N. J. A. Sloane, Transforms

Columns k=0-9 give: A000041, A000219, A023871, A023872, A023873, A023874, A023875, A023876, A023877, A023878.
Rows give: 0-1: A000012, 2: A000051, A094373, 3: A001550, 4: A283456, 5: A283457.
Main diagonal gives A252782.
Cf. A283272.

with(numtheory): etr:= proc(p) local b; b:= proc(n) option remember; `if`(n=0,1, add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n) end end: A:= (n,k)-> etr(j->j^k)(n); seq(seq(A(n,d-n), n=0..d), d=0..13);

etr[p_] := Module[{ b}, b[n_] := b[n] = If[n == 0, 1, Sum[Sum[d*p[d], {d, Divisors[j]}]*b[n - j], {j, 1, n}]/n]; b]; A[n_, k_] := etr[Function[j, j^k]][n]; Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 13}] // Flatten (* Jean-François Alcover, Dec 27 2013, translated from Maple *)

G.f. of column k: Product_{j>=1} 1/(1-x^j)^(j^k).

A074527 a(n) = 2^n + 3^n + 5^n.

Original entry on oeis.org

3, 10, 38, 160, 722, 3400, 16418, 80440, 397442, 1973320, 9825698, 49007320, 244676162, 1222305640, 6108314978, 30531959800, 152631002882, 763068724360, 3815084948258, 19074649113880, 95370919473602, 476847620653480

Offset: 0

Robert G. Wilson v, Aug 23 2002

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (10,-31,30).

Cf. A001550, A001576, A034513, A001579, A074501-A074580.

[2^n + 3^n + 5^n: n in [0..25]]; // Vincenzo Librandi, Jun 11 2011

Table[2^n + 3^n + 5^n, {n, 0, 21}]
LinearRecurrence[{10,-31,30},{3,10,38},30] (* Harvey P. Dale, Feb 05 2022 *)

a(n)=2^n+3^n+5^n \\ Charles R Greathouse IV, Oct 07 2015

From Mohammad K. Azarian, Dec 26 2008: (Start)
G.f.: 1/(1-2*x) + 1/(1-3*x) + 1/(1-5*x).
E.g.f.: exp(2*x) + exp(3*x) + exp(5*x). (End)
a(n) = 10*a(n-1) - 31*a(n-2) + 30*a(n-3). - Wesley Ivan Hurt, May 26 2024

A083321 a(n) = (-1)^n + (-2)^n - (-3)^n.

Original entry on oeis.org

1, 0, -4, 18, -64, 210, -664, 2058, -6304, 19170, -58024, 175098, -527344, 1586130, -4766584, 14316138, -42981184, 129009090, -387158344, 1161737178, -3485735824, 10458256050, -31376865304, 94134790218, -282412759264, 847255055010, -2541798719464, 7625463267258

Offset: 0

Paul Barry, Apr 25 2003

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-6,-11,-6).

Cf. A001550 (1^n + 2^n + 3^n).

[(-1)^n+(-2)^n-(-3)^n: n in [0..30]]; // Vincenzo Librandi, Sep 05 2011

a[n_]:=(-1)^n*(1+2^n-3^n); (* Vladimir Joseph Stephan Orlovsky, Dec 05 2008, corrected by M. F. Hasler, Apr 20 2020 *)
LinearRecurrence[{-6,-11,-6},{1,0,-4},30] (* Harvey P. Dale, Mar 05 2022 *)

apply( A083321(n)=(-1)^n*(1+2^n-3^n), [0..33]) \\ M. F. Hasler, Apr 19 2020

G.f.: (1+6*x+7*x^2)/((1+x)*(1+2*x)*(1+3*x)).
E.g.f.: exp(-x)+exp(-2*x)-exp(-3*x).
a(n) = (-1)^(n-1)*(3^n - 2^n - 1) for n >= 0. - M. F. Hasler, Apr 19 2020

cp's OEIS Frontend

A001576 a(n) = 1^n + 2^n + 4^n.

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A074501 a(n) = 1^n + 2^n + 5^n.

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A007689 a(n) = 2^n + 3^n.

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A034513 a(n) = 1^n + 3^n + 9^n.

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PARI

Sage

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A074580 a(n) = 7^n + 8^n + 9^n.

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Magma

Mathematica

PARI

Formula

A001579 a(n) = 3^n + 5^n + 6^n.

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Programs

Magma

Mathematica

PARI

Python

Formula

A103438 Square array T(m,n) read by antidiagonals: Sum_{k=1..n} k^m.

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