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Search: a001579 -id:a001579
Displaying 1-10 of 83 results found. page 1 2 3 4 5 6 7 8 9
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A001550 a(n) = 1^n + 2^n + 3^n.
(Formerly M2580 N1020) 		+10
101
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
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
REFERENCES
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).
LINKS
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).
FORMULA
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)
MAPLE
A001550:=-(3-12*z+11*z^2)/(z-1)/(3*z-1)/(2*z-1); # Simon Plouffe in his 1992 dissertation.
MATHEMATICA
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 *)
PROG
(PARI) a(n)=1+2^n+3^n \\ Charles R Greathouse IV, Jun 10 2011
(Haskell) a001550 n = sum $ map (^ n) [1..3] -- Reinhard Zumkeller, Mar 01 2012
(Magma) [1^n + 2^n + 3^n : n in [0..30]]; // Wesley Ivan Hurt, Jun 25 2020
CROSSREFS
Cf. A000051, A000079, A000244, A007689, A034472.
Cf. A001576, A001579, A034513, A074501 - A074580.
Column 3 of array A103438.
KEYWORD
nonn,easy,nice
AUTHOR
N. J. A. Sloane
EXTENSIONS
Additional terms from Michael Somos
Attribute "conjectured" removed from Simon Plouffe's g.f. by R. J. Mathar, Mar 11 2009
STATUS
approved
A001576 a(n) = 1^n + 2^n + 4^n. +10
100
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
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
LINKS
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).
FORMULA
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
MATHEMATICA
Table[1^n + 2^n + 4^n, {n, 0, 24}]
PROG
(Sage) [sigma(4, n)for n in range(0, 23)] # Zerinvary Lajos, Jun 04 2009
(PARI) a(n)=1+2^n+4^n \\ Charles R Greathouse IV, Jun 10 2011
CROSSREFS
Subsequence of A002061.
Cf. A001550, A034513, A001579, A074501-A074580, A135576, A135577.
See also comments in A051154.
Cf. A006095, A022166.
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane
STATUS
approved
A074501 a(n) = 1^n + 2^n + 5^n. +10
99
3, 8, 30, 134, 642, 3158, 15690, 78254, 390882, 1953638, 9766650, 48830174, 244144722, 1220711318, 6103532010, 30517610894, 152587956162, 762939584198, 3814697527770, 19073486852414, 95367432689202, 476837160300278 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
LINKS
Table of n, a(n) for n=0..21.
Index entries for linear recurrences with constant coefficients, signature (8,-17,10).
FORMULA
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
MATHEMATICA
Table[1^n + 2^n + 5^n, {n, 0, 24}]
PROG
(PARI) a(n)=1+2^n+5^n \\ Charles R Greathouse IV, Jun 10 2011
CROSSREFS
Equals A074600(n) + 1.
Cf. A001550, A001576, A034513, A001579, A074502..A074580.
KEYWORD
easy,nonn
AUTHOR
Robert G. Wilson v, Aug 23 2002
STATUS
approved
A034513 a(n) = 1^n + 3^n + 9^n. +10
88
3, 13, 91, 757, 6643, 59293, 532171, 4785157, 43053283, 387440173, 3486843451, 31381236757, 282430067923, 2541867422653, 22876797237931, 205891146443557, 1853020231898563, 16677181828806733, 150094635684419611 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
Also the sum of n-th powers of the divisors of 9.
LINKS
T. D. Noe, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (13,-39,27).
FORMULA
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
MATHEMATICA
Table[1^n + 3^n + 9^n, {n, 0, 20}]
LinearRecurrence[{13, -39, 27}, {3, 13, 91}, 20] (* Harvey P. Dale, Apr 13 2012 *)
PROG
(Sage) [sigma(9, n)for n in range(0, 19)] # Zerinvary Lajos, Jun 04 2009
(PARI) a(n)=1+3^n+9^n \\ Charles R Greathouse IV, Jun 10 2011
CROSSREFS
Cf. A001550, A001576, A001579, A074501-A074580.
KEYWORD
easy,nonn
AUTHOR
N. J. A. Sloane
STATUS
approved
A074580 a(n) = 7^n + 8^n + 9^n. +10
85
3, 24, 194, 1584, 13058, 108624, 911234, 7703664, 65588738, 561991824, 4843001474, 41948320944, 364990300418, 3188510652624, 27953062038914, 245823065693424, 2167728096132098, 19161612027339024, 169737447404391554 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..300
Index entries for linear recurrences with constant coefficients, signature (24,-191,504).
FORMULA
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)
MATHEMATICA
Table[7^n + 8^n + 9^n, {n, 0, 18}]
LinearRecurrence[{24, -191, 504}, {3, 24, 194}, 20] (* Harvey P. Dale, Jun 09 2019 *)
PROG
(Magma) [7^n + 8^n + 9^n: n in [0..20]]; // Vincenzo Librandi, May 20 2011
(PARI) a(n)=7^n+8^n+9^n \\ Charles R Greathouse IV, Jun 10 2011
CROSSREFS
Cf. A001550, A001576, A034513, A001579, A074501-A074579.
KEYWORD
easy,nonn
AUTHOR
Robert G. Wilson v, Aug 23 2002
STATUS
approved
A074527 a(n) = 2^n + 3^n + 5^n. +10
9
3, 10, 38, 160, 722, 3400, 16418, 80440, 397442, 1973320, 9825698, 49007320, 244676162, 1222305640, 6108314978, 30531959800, 152631002882, 763068724360, 3815084948258, 19074649113880, 95370919473602, 476847620653480 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (10,-31,30).
FORMULA
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
MATHEMATICA
Table[2^n + 3^n + 5^n, {n, 0, 21}]
LinearRecurrence[{10, -31, 30}, {3, 10, 38}, 30] (* Harvey P. Dale, Feb 05 2022 *)
PROG
(Magma) [2^n + 3^n + 5^n: n in [0..25]]; // Vincenzo Librandi, Jun 11 2011
(PARI) a(n)=2^n+3^n+5^n \\ Charles R Greathouse IV, Oct 07 2015
CROSSREFS
Cf. A001550, A001576, A034513, A001579, A074501-A074580.
KEYWORD
easy,nonn
AUTHOR
Robert G. Wilson v, Aug 23 2002
STATUS
approved
A074507 a(n) = 1^n + 3^n + 5^n. +10
6
3, 9, 35, 153, 707, 3369, 16355, 80313, 397187, 1972809, 9824675, 49005273, 244672067, 1222297449, 6108298595, 30531927033, 152630937347, 763068593289, 3815084686115, 19074648589593, 95370918425027, 476847618556329 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
LINKS
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).
FORMULA
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]
MATHEMATICA
Table[1^n + 3^n + 5^n, {n, 0, 22}]
LinearRecurrence[{9, -23, 15}, {3, 9, 35}, 30] (* Harvey P. Dale, Mar 02 2022 *)
PROG
(PARI) a(n) = 1 + 3^n + 5^n; \\ Michel Marcus, Aug 07 2017
CROSSREFS
Cf. A001550, A001576, A034513, A001579, A074501-A074580.
KEYWORD
easy,nonn
AUTHOR
Robert G. Wilson v, Aug 23 2002
STATUS
approved
A074528 a(n) = 2^n + 3^n + 6^n. +10
5
3, 11, 49, 251, 1393, 8051, 47449, 282251, 1686433, 10097891, 60526249, 362976251, 2177317873, 13062296531, 78368963449, 470199366251, 2821153019713, 16926788715971, 101560344351049, 609360902796251 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
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)
REFERENCES
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]
LINKS
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).
FORMULA
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
MATHEMATICA
Table[2^n + 3^n + 6^n, {n, 0, 20}]
LinearRecurrence[{11, -36, 36}, {3, 11, 49}, 30] (* Harvey P. Dale, May 02 2016 *)
PROG
(Magma) [2^n + 3^n + 6^n: n in [0..25]]; // Vincenzo Librandi, Jun 11 2011
(PARI) a(n)=2^n+3^n+6^n \\ Charles R Greathouse IV, Oct 07 2015
CROSSREFS
Cf. A001550, A001576, A034513, A001579, A074501-A074580.
A246985 is essentially identical.
Third row of A160449, shifted.
KEYWORD
easy,nonn
AUTHOR
Robert G. Wilson v, Aug 23 2002
STATUS
approved
A074506 a(n) = 1^n + 3^n + 4^n. +10
4
3, 8, 26, 92, 338, 1268, 4826, 18572, 72098, 281828, 1107626, 4371452, 17308658, 68703188, 273218426, 1088090732, 4338014018, 17309009348, 69106897226, 276040168412, 1102998412178, 4408506864308, 17623567104026 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
LINKS
Bruno Berselli, Table of n, a(n) for n = 0..1000.
Index entries for linear recurrences with constant coefficients, signature (8,-19,12).
FORMULA
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)
MATHEMATICA
Table[1^n + 3^n + 4^n, {n, 0, 22}]
CROSSREFS
Cf. A001550, A001576, A034513, A001579, A074501..A074580.
Equals A074605(n) + 1.
KEYWORD
easy,nonn
AUTHOR
Robert G. Wilson v, Aug 23 2002
STATUS
approved
A074526 a(n) = 2^n + 3^n + 4^n. +10
4
3, 9, 29, 99, 353, 1299, 4889, 18699, 72353, 282339, 1108649, 4373499, 17312753, 68711379, 273234809, 1088123499, 4338079553, 17309140419, 69107159369, 276040692699, 1102999460753, 4408508961459, 17623571298329 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (9,-26,24).
FORMULA
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)
MATHEMATICA
Table[2^n + 3^n + 4^n, {n, 0, 23}]
LinearRecurrence[{9, -26, 24}, {3, 9, 29}, 30] (* Harvey P. Dale, Jun 14 2022 *)
PROG
(Magma) [2^n + 3^n + 4^n: n in [0..25]]; // Vincenzo Librandi, Jun 11 2011
CROSSREFS
Cf. A001550, A001576, A034513, A001579, A074501 - A074580.
KEYWORD
easy,nonn
AUTHOR
Robert G. Wilson v, Aug 23 2002
STATUS
approved
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Last modified August 29 15:03 EDT 2024. Contains 375517 sequences. (Running on oeis4.)