A001550 -id:A001550 - cp's OEIS frontend

A074515 a(n) = 1^n + 4^n + 9^n.

3, 14, 98, 794, 6818, 60074, 535538, 4799354, 43112258, 387682634, 3487832978, 31385253914, 282446313698, 2541932937194, 22877060890418, 205892205836474, 1853024483819138, 16677198879535754, 150094704016475858

Offset: 0

Robert G. Wilson v, Aug 23 2002

Index entries for linear recurrences with constant coefficients, signature (14,-49,36).

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

Cf. A052539.

Table[1^n + 4^n + 9^n, {n, 0, 20}]
LinearRecurrence[{14,-49,36},{3,14,98},30] (* Harvey P. Dale, Aug 06 2013 *)

def a(n): return 1 + 4**n + 9**n
print([a(n) for n in range(19)]) # Michael S. Branicky, Mar 14 2021

From Mohammad K. Azarian, Dec 26 2008: (Start)
G.f.: 1/(1-x) + 1/(1-4*x) + 1/(1-9*x).
E.g.f.: e^x + e^(4*x) + e^(9*x). (End)
a(n) = 13*a(n-1) - 36*a(n-2) + 24 with a(0)=3, a(1)=14. - Vincenzo Librandi, Jul 21 2010

A074535 a(n) = 2^n + 4^n + 8^n.

Original entry on oeis.org

3, 14, 84, 584, 4368, 33824, 266304, 2113664, 16843008, 134480384, 1074791424, 8594130944, 68736258048, 549822930944, 4398314962944, 35185445863424, 281479271743488, 2251816993685504, 18014467229220864, 144115462954287104

Offset: 0

Robert G. Wilson v, Aug 23 2002

Number of monic irreducible polynomials of degree 1 in GF(2^n)[x,y,z]. - Max Alekseyev, Jan 23 2006

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (14,-56,64).

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

[2^n + 4^n + 8^n: n in [0..25]]; // Vincenzo Librandi, Jun 11 2011

Mathematica
```
Table[2^n + 4^n + 8^n, {n, 0, 20}]
```

def a(n): return 2**n + 4**n + 8**n
print([a(n) for n in range(20)]) # Michael S. Branicky, Mar 14 2021

G.f.: 1/(1-2*x)+1/(1-4*x)+1/(1-8*x). E.g.f.: exp(2*x)+exp(4*x)+exp(8*x). [Mohammad K. Azarian, Dec 26 2008]

Let A=[1, 1, 1;2, 0, -2;1, -1, 1], the 3 X 3 Krawtchouk matrix. Then a(n)=trace((A*A')^n). - Paul Barry, Sep 18 2004

A074579 a(n) = 6^n + 8^n + 9^n.

Original entry on oeis.org

3, 23, 181, 1457, 11953, 99593, 840241, 7160057, 61503553, 531715913, 4620992401, 40333791257, 353325795553, 3104682336233, 27353203130161, 241545689168057, 2137316275469953, 18945908172796553, 168210593763149521

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 (23,-174,432).

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

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

Table[6^n + 8^n + 9^n, {n, 0, 20}]
LinearRecurrence[{23,-174,432},{3,23,181},30] (* Harvey P. Dale, Sep 20 2016 *)

From Mohammad K. Azarian, Dec 26 2008: (Start)
G.f.: 1/(1-6*x) + 1/(1-8*x) + 1/(1-9*x).
E.g.f.: e^(6*x) + e^(8*x) + e^(9*x). (End)
a(n) = 23*a(n-1)-174*a(n-2)+432*a(n-3). - Wesley Ivan Hurt, Apr 17 2022

A192001 Triangle with sums of nonnegative integer powers of positive first n integers in the columns.

Original entry on oeis.org

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

Offset: 1

Wolfdieter Lang, Jun 25 2011

This is the Abramowitz-Stegun table on p. 813, call it s(m,n), with an extra column n=0 with values n added, and read by antidiagonals. a(n,m) = s(n-m,m), n+1 >= m >= 0.
O.g.f. for column no. m >= 0: (x^(m+1)/(1-x)^(m+2))*E(m;x) with the row polynomials E(m;x) = Sum_{p=0..m} A173018(m,p)*x^p of the Eulerian number triangle (proof via the Worpitzky identity). See the Graham et al. reference p. 253-8 for Eulerian numbers, and the Worpitzky identity (6.37) on p. 255.
E.g.f. for diagonals (starting with k=0 for the main diagonal): g(k,x) = exp(x)*(exp((k+1)*x)-1)/(1-exp(x)).
  Compare with (7.77) on p. 353 of the Graham et al. reference.
O.g.f. for diagonals (starting with k=0 for the main diagonal): G(k,z) =(Psi(1/z+1)-Psi(1/z-k-1))/z - 1.
  with the digamma function Psi(z):=(log(Gamma(z)))'.
  Compare with Graham et al., p. 352, eq.(7.76), where H_z=Psi(z+1)+gamma, with the Euler-Mascheroni constant gamma.
The diagonal sequences are, for k=0..9: A000012, A000051, A001550-A001557.
The negative k-diagonal, -a(k+m+1,m), yields the Sheffer z-sequence Shz(k+1;m) for the Sheffer arrays |S1|(k+1) defined in a comment to A094646.
See also A196837 with a W. Lang link, where the o.g.f.s for the diagonals, numbered with k >= 1, are given as G(k,x) = Sum_{m=0..k} (k-m)*S1(k+1,k+1-m)*x^m / Product_{j=1..k} (1-j*x), with S1 the Stirling numbers of the first kind, A048994. - Wolfdieter Lang, Nov 01 2011

			The triangle a(n,m) begins:
n\m    0   1   2    3    4     5     6     7    8   9  10 ...
n=1:   1
n=2:   2   1
n=3:   3   3   1
n=4:   4   6   5    1
n=5:   5  10  14    9    1
n=6:   6  15  30   36   17     1
n=7:   7  21  55  100   98    33     1
n=8:   8  28  91  225  354   276    65     1
n=9:   9  36 140  441  979  1300   794   129    1
n=10: 10  45 204  784 2275  4425  4890  2316  257   1
n=11: 11  55 285 1296 4676 12201 20515 18700 6818 513   1
...  Reformatted and extended by _Wolfdieter Lang_, Jan 12 2013
a(4,2)= 5 = s(2,2) = 1^2 + 2^2.
Recurrence: 55 = a(7,2) = (7-2)*a(6,1) - (a(2,1) + a(3,1) + a(4,1) + a(5,1)) = 5*15 - (1 + 3 + 6 + 10) = 55. - _Wolfdieter Lang_, Jan 12 2013
The first column, m=0 holds the integers 1,2,3,..., equal to the sums of 0th powers of the n first integers. The second column is 1, 1+2, 1+2+3, ... = A000217. The third column are the sums of squares, 1^2, 1^2+2^2, 1^2+2^2+3^3, ... = A000330, etc. - _M. F. Hasler_, Jan 13 2013

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964. Tenth printing, Wiley, 2002 (also electronically available, see the link), p. 813.
Mohammad K. Azarian, Problem 1218, Pi Mu Epsilon Journal, Vol. 13, No. 2, Spring 2010, p. 116. Solution published in Vol. 13, No. 3, Fall 2010, pp. 183-185.
Ronald L. Graham, Donald E. Knuth and Oren Patashnik, Concrete Mathematics, Addison-Wesley, 1991.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [scanned copy], p. 813.

Cf. A103438 (omitting the first column of zeros, reversed).

Flatten[ Table[ HarmonicNumber[-m + n, -m], {n, 1, 10}, {m, 0, n - 1}]] (* Jean-François Alcover, Sep 26 2011 *)

A192001(n,m) = sum(k=1,n-m,k^m) \\ - M. F. Hasler, Jan 13 2013

a(n,m) = s(n-m,m), n-1 >= m >= 0, n >= 1, else 0, with s(n,m) := Sum_{k=1..n} k^m.
O.g.f. column no. m: see a comment above.
O.g.f.s and e.g.f.s for diagonals k >= 0: see a comment above.
Recurrence known by Ibn al-Haytham (see a comment and link under A000537, and comments by Bruno Berselli under the A-numbers of the first column sequences):
  a(n,m) = (n-m)*a(n-1,m-1) - Sum_{j=m..n-2} a(j,m-1), n >= 1, n-1 >= m >= 1. a(n,0) = n. - Wolfdieter Lang, Jan 12 2013

A364683 a(n) is the least k such that 1 + 2^k + 3^k is divisible by n, or -1 if there is no such k.

Original entry on oeis.org

0, 1, 0, 3, -1, 1, 2, -1, 3, -1, 9, 3, -1, 2, -1, -1, 9, 3, -1, -1, -1, 9, 5, -1, -1, -1, 9, -1, -1, -1, 16, -1, 9, 9, -1, 3, 12, -1, -1, -1, 18, -1, -1, 9, -1, 5, -1, -1, 4, -1, 9, -1, -1, 9, -1, -1, -1, -1, -1, -1, -1, 16, -1, -1, -1, 9, -1, 9, 5, -1, -1, -1, 19, 12, -1, -1, -1, -1, 33, -1, 27

Offset: 1

Robert Israel, Aug 02 2023

If a(n) = -1, then a(m) = -1 for all multiples of n.

			a(7) = 2 because 1 + 2^2 + 3^2 = 14 is divisible by 7 while 1 + 2^0 + 3^0 = 3 and 1 + 2^1 + 3^1 = 6 are not.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A001550.

f:= proc(n) local k;
 for k from 0 to numtheory:-phi(n) + max(padic:-ordp(n,2), padic:-ordp(n,3)) do
   if 1 + 2&^k + 3&^k mod n = 0 then return k fi
 od;
 -1
end proc:
map(f, [$1..100]);

A056645 Numbers n such that n | 3^n + 2^n + 1^n.

Original entry on oeis.org

1, 2, 3, 9, 14, 27, 81, 98, 99, 153, 243, 621, 686, 729, 1199, 2187, 4018, 4802, 5049, 6561, 8019, 12393, 19683, 20541, 31509, 33614, 59049, 70389, 82377, 113974, 150903, 177147, 226557, 235298, 299619, 308014, 340443, 408969, 423929, 531441

Offset: 1

Robert G. Wilson v, Aug 25 2000

nonn

Cf. A001550.

Do[ If[ Mod[ PowerMod[ 3, n, n ] + PowerMod[ 2, n, n ] + 1, n ] == 0, Print[ n ] ], {n, 1, 10^6} ]

A074508 a(n) = 1^n + 3^n + 6^n.

Original entry on oeis.org

3, 10, 46, 244, 1378, 8020, 47386, 282124, 1686178, 10097380, 60525226, 362974204, 2177313778, 13062288340, 78368947066, 470199333484, 2821152954178, 16926788584900, 101560344088906, 609360902271964

Offset: 0

Robert G. Wilson v, Aug 23 2002

Index entries for linear recurrences with constant coefficients, signature (10,-27,18).

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

Mathematica
```
Table[1^n + 3^n + 6^n, {n, 0, 20}]
```

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

a(n) = 9*a(n-1) - 18*a(n-2) + 10, n>1. [Gary Detlefs, Jun 21 2010]

A074511 a(n) = 1^n + 4^n + 5^n.

Original entry on oeis.org

3, 10, 42, 190, 882, 4150, 19722, 94510, 456162, 2215270, 10814202, 53022430, 260917842, 1287811990, 6371951082, 31591319950, 156882857922, 780119322310, 3883416742362, 19348364235070, 96466943268402, 481235204714230, 2401777977060042, 11991297699255790

Offset: 0

Robert G. Wilson v, Aug 23 2002

Index entries for linear recurrences with constant coefficients, signature (10,-29,20).

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

Mathematica
```
Table[1^n + 4^n + 5^n, {n, 0, 21}]
```

From Mohammad K. Azarian, Dec 26 2008: (Start)
G.f.: 1/(1-x) + 1/(1-4*x) + 1/(1-5*x).
E.g.f.: e^x + e^(4*x) + e^(5*x). (End)
a(n) = 9*a(n-1) - 20*a(n-2) + 12 with a(0)=3, a(1)=10. - Vincenzo Librandi, Jul 21 2010

A074516 a(n) = 1^n + 5^n + 6^n.

Original entry on oeis.org

3, 12, 62, 342, 1922, 10902, 62282, 358062, 2070242, 12030822, 70231802, 411625182, 2420922962, 14281397142, 84467679722, 500702562702, 2973697798082, 17689598897862, 105374653934042, 628433226338622

Offset: 0

Robert G. Wilson v, Aug 23 2002

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (12,-41,30).

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

[5^n+6^n+1^n: n in [0..30]]; // Vincenzo Librandi, Mar 24 2013

A074516:=n->1^n+5^n+6^n: seq(A074516(n), n=0..30); # Wesley Ivan Hurt, Jan 27 2017

Table[1^n + 5^n + 6^n, {n, 0, 20}]
LinearRecurrence[{12,-41,30},{3,12,62},20] (* Harvey P. Dale, Apr 06 2018 *)

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

a(n) = 11*a(n-1) - 30*a(n-2) + 20, n>1. [Gary Detlefs, Jun 21 2010]

A074520 1^n + 6^n + 7^n.

Original entry on oeis.org

3, 14, 86, 560, 3698, 24584, 164306, 1103480, 7444418, 50431304, 342941426, 2340123800, 16018069538, 109949704424, 756587236946, 5217746494520, 36054040477058, 249557173431944, 1729973554578866, 12008254925383640

Offset: 0

Robert G. Wilson v, Aug 23 2002

Index entries for linear recurrences with constant coefficients, signature (14,-55,42).

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

A074520:=n->1^n + 6^n + 7^n; seq(A074520(n), n=0..100); # Wesley Ivan Hurt, Nov 11 2013

Mathematica
```
Table[1 + 6^n + 7^n, {n, 0, 20}]
```

G.f.:1/(1-x)+1/(1-6*x)+1/(1-7*x). E.g.f.: e^x+e^(6*x)+e^(7*x). [Mohammad K. Azarian, Dec 26 2008]
a(n) = 13*a(n-1) - 42*a(n-2) + 30, n>1. [Gary Detlefs, Jun 21 2010]
a(n) = A074619(n) + 1. - Michel Marcus, Nov 11 2013

cp's OEIS Frontend

A074515 a(n) = 1^n + 4^n + 9^n.

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A074535 a(n) = 2^n + 4^n + 8^n.

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A074579 a(n) = 6^n + 8^n + 9^n.

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A192001 Triangle with sums of nonnegative integer powers of positive first n integers in the columns.

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A364683 a(n) is the least k such that 1 + 2^k + 3^k is divisible by n, or -1 if there is no such k.

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Maple

A056645 Numbers n such that n | 3^n + 2^n + 1^n.

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A074508 a(n) = 1^n + 3^n + 6^n.

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A074511 a(n) = 1^n + 4^n + 5^n.

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