A001579 -id:A001579 - 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

A001550 a(n) = 1^n + 2^n + 3^n.

Original entry on oeis.org

3, 6, 14, 36, 98, 276, 794, 2316, 6818, 20196, 60074, 179196, 535538, 1602516, 4799354, 14381676, 43112258, 129271236, 387682634, 1162785756, 3487832978, 10462450356, 31385253914, 94151567436, 282446313698, 847322163876

Offset: 0

N. J. A. Sloane

a(n)*(-1)^n, n>=0, gives the z-sequence for the Sheffer triangle A049458 ((signed) 3-restricted Stirling1 numbers), which is the inverse triangle of A143495 with offset [0,0] (3-restricted Stirling2 numbers). See the W. Lang link under A006232 for a- and z-sequences for Sheffer matrices. The a-sequence for each (signed) r-restricted Stirling1 Sheffer triangle is A027641/A027642 (Bernoulli numbers). - Wolfdieter Lang, Oct 10 2011

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 813.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
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
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 363
C. J. Pita Ruiz V., Some Number Arrays Related to Pascal and Lucas Triangles, J. Int. Seq. 16 (2013) #13.5.7.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
Kai Wang, Girard-Waring Type Formula For A Generalized Fibonacci Sequence, Fibonacci Quarterly (2020) Vol. 58, No. 5, 229-235.
Index entries for linear recurrences with constant coefficients, signature (6,-11,6).

Cf. A000051, A000079, A000244, A007689, A034472.
Cf. A001576, A001579, A034513, A074501 - A074580.
Column 3 of array A103438.

a001550 n = sum $ map (^ n) [1..3]  -- Reinhard Zumkeller, Mar 01 2012

[1^n + 2^n + 3^n : n in [0..30]]; // Wesley Ivan Hurt, Jun 25 2020

A001550:=-(3-12*z+11*z^2)/(z-1)/(3*z-1)/(2*z-1); # Simon Plouffe in his 1992 dissertation.

Table[1^n + 2^n + 3^n, {n, 0, 30}]
CoefficientList[Series[(3-12x+11x^2)/(1-6x+11x^2-6x^3),{x,0,30}],x] (* or *) LinearRecurrence[{6,-11,6},{3,6,14},31] (* Harvey P. Dale, Apr 30 2011 *)
Total[Range[3]^#]&/@Range[0,30] (* Harvey P. Dale, Sep 23 2019 *)

a(n)=1+2^n+3^n \\ Charles R Greathouse IV, Jun 10 2011

def A001550(n): return 3**n+(1<Chai Wah Wu, Nov 01 2024

From Michael Somos: (Start)
G.f.: (3 -12*x +11*x^2)/(1 -6*x +11*x^2 -6*x^3).
a(n) = 5*a(n-1) - 6*a(n-2) + 2. (End)
E.g.f.: exp(x) + exp(2*x) + exp(3*x). - Mohammad K. Azarian, Dec 26 2008
a(0)=3, a(1)=6, a(2)=14, a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3). - Harvey P. Dale, Apr 30 2011
a(n) = A007689(n) + 1. - Reinhard Zumkeller, Mar 01 2012
From Kai Wang, May 18 2020: (Start)
a(n) = 3*A000392(n+3) - 12*A000392(n+2) + 11*A000392(n+1).
A000392(n) = (3*a(n+1) - 12*a(n) + 10*a(n-1))/2. (End)

Additional terms from Michael Somos

Attribute "conjectured" removed from Simon Plouffe's g.f. by R. J. Mathar, Mar 11 2009

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

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)

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

A074507 a(n) = 1^n + 3^n + 5^n.

Original entry on oeis.org

3, 9, 35, 153, 707, 3369, 16355, 80313, 397187, 1972809, 9824675, 49005273, 244672067, 1222297449, 6108298595, 30531927033, 152630937347, 763068593289, 3815084686115, 19074648589593, 95370918425027, 476847618556329

Offset: 0

Robert G. Wilson v, Aug 23 2002

Harvey P. Dale, Table of n, a(n) for n = 0..1000
T. A. Gulliver, Divisibility of sums of powers of odd integers, Int. Math. For. 5 (2010) 3059-3066, eq. (6).
D. Suprijanto, I. W. Suwarno, Observation on Sums of Powers of Integers Divisible by 3k-1, Applied Mathematical Sciences, Vol. 8, 2014, no. 45, 2211 - 2217.
Index entries for linear recurrences with constant coefficients, signature (9,-23,15).

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

Table[1^n + 3^n + 5^n, {n, 0, 22}]
LinearRecurrence[{9,-23,15},{3,9,35},30] (* Harvey P. Dale, Mar 02 2022 *)

a(n) = 1 + 3^n + 5^n; \\ Michel Marcus, Aug 07 2017

a(n) = 8*a(n-1) - 15*a(n-2) + 8.

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

A074528 a(n) = 2^n + 3^n + 6^n.

Original entry on oeis.org

3, 11, 49, 251, 1393, 8051, 47449, 282251, 1686433, 10097891, 60526249, 362976251, 2177317873, 13062296531, 78368963449, 470199366251, 2821153019713, 16926788715971, 101560344351049, 609360902796251

Offset: 0

Robert G. Wilson v, Aug 23 2002

From Álvar Ibeas, Mar 24 2015: (Start)
Number of isomorphism classes of 3-fold coverings of a connected graph with circuit rank n+1 [Kwak and Lee].
Number of orbits of the conjugacy action of Sym(3) on Sym(3)^(n+1) [Kwak and Lee, 2001].
(End)

J. H. Kwak and J. Lee, Enumeration of graph coverings, surface branched coverings and related group theory, in Combinatorial and Computational Mathematics (Pohang, 2000), ed. S. Hong et al., World Scientific, Singapore 2001, pp. 97-161. [Added by N. J. A. Sloane, Nov 12 2009]

Hakan Icoz, Table of n, a(n) for n = 0..1000 (terms 0..200 from Vincenzo Librandi)
M. W. Hero and J. F. Willenbring, Stable Hilbert series as related to the measurement of quantum entanglement, Discrete Math., 309 (2010), 6508-6514.
J. H. Kwak and J. Lee, Isomorphism classes of graph bundles. Can. J. Math., 42(4), 1990, pp. 747-761.
Index entries for linear recurrences with constant coefficients, signature (11,-36,36).

Cf. A001550, A001576, A034513, A001579, A074501-A074580.
A246985 is essentially identical.
Third row of A160449, shifted.

[2^n + 3^n + 6^n: n in [0..25]]; // Vincenzo Librandi, Jun 11 2011

Table[2^n + 3^n + 6^n, {n, 0, 20}]
LinearRecurrence[{11,-36,36},{3,11,49},30] (* Harvey P. Dale, May 02 2016 *)

a(n)=2^n+3^n+6^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/(1-6*x).
E.g.f.: exp(2*x) + exp(3*x) + exp(6*x). (End)
a(n) = 11*a(n-1) - 36*a(n-2) + 36*a(n-3). - Wesley Ivan Hurt, Aug 21 2020

A074506 a(n) = 1^n + 3^n + 4^n.

Original entry on oeis.org

3, 8, 26, 92, 338, 1268, 4826, 18572, 72098, 281828, 1107626, 4371452, 17308658, 68703188, 273218426, 1088090732, 4338014018, 17309009348, 69106897226, 276040168412, 1102998412178, 4408506864308, 17623567104026

Offset: 0

Robert G. Wilson v, Aug 23 2002

Bruno Berselli, Table of n, a(n) for n = 0..1000.
Index entries for linear recurrences with constant coefficients, signature (8,-19,12).

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

Equals A074605(n) + 1.

Table[1^n + 3^n + 4^n, {n, 0, 22}]
LinearRecurrence[{8,-19,12},{3,8,26},30] (* Harvey P. Dale, May 12 2025 *)

a(n) = 7*a(n-1) - 12*a(n-2) + 6 with a(0)=3, a(1)=8. - Vincenzo Librandi, Jul 19 2010
a(n) = 8*a(n-1) - 19*a(n-2) + 12*a(n-3). - R. J. Mathar, Jul 18 2010
From Mohammad K. Azarian, Dec 26 2008: (Start)
G.f.: 1/(1-x) + 1/(1-3*x) + 1/(1-4*x).
E.g.f.: e^x + e^(3*x) + e^(4*x). (End)

A074526 a(n) = 2^n + 3^n + 4^n.

Original entry on oeis.org

3, 9, 29, 99, 353, 1299, 4889, 18699, 72353, 282339, 1108649, 4373499, 17312753, 68711379, 273234809, 1088123499, 4338079553, 17309140419, 69107159369, 276040692699, 1102999460753, 4408508961459, 17623571298329

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 (9,-26,24).

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

[2^n + 3^n + 4^n: n in [0..25]]; // Vincenzo Librandi, Jun 11 2011

Table[2^n + 3^n + 4^n, {n, 0, 23}]
LinearRecurrence[{9,-26,24},{3,9,29},30] (* Harvey P. Dale, Jun 14 2022 *)

From Mohammad K. Azarian, Dec 26 2008: (Start)
G.f.: 1/(1-2*x)+1/(1-3*x)+1/(1-4*x).
E.g.f.: exp(2*x)+exp(3*x)+exp(4*x). (End)

cp's OEIS Frontend

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

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A001550 a(n) = 1^n + 2^n + 3^n.

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

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

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

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

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A074507 a(n) = 1^n + 3^n + 5^n.

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