A000351 -id:A000351 - cp's OEIS frontend

A036291 a(n) = n*5^n.

0, 5, 50, 375, 2500, 15625, 93750, 546875, 3125000, 17578125, 97656250, 537109375, 2929687500, 15869140625, 85449218750, 457763671875, 2441406250000, 12969970703125, 68664550781250, 362396240234375, 1907348632812500, 10013580322265625, 52452087402343750

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (10,-25).

Cf. A000351, A053464.

[n*5^n: n in [0..20]]; // Vincenzo Librandi, Sep 09 2014

g:=1/(1-5*z): gser:=series(g, z=0, 43): seq(coeff(gser, z, n)*n, n=0..31); # Zerinvary Lajos, Jan 09 2009

Table[n 5^n, {n, 0, 20}] (* Vincenzo Librandi, Sep 09 2014 *)

G.f.: 5*x/(1 - 5*x)^2. - Vincenzo Librandi, Sep 09 2014
From Amiram Eldar, Jul 20 2020: (Start)
Sum_{n>=1} 1/a(n) = log(5/4).
Sum_{n>=1} (-1)^(n+1)/a(n) = log(6/5). (End)
From Elmo R. Oliveira, Sep 09 2024: (Start)
E.g.f.: 5*x*exp(5*x).
a(n) = n*A000351(n) = 5*A053464(n).
a(n) = 10*a(n-1) - 25*a(n-2) for n > 1. (End)

A047851 a(n) = A047848(3,n).

Original entry on oeis.org

1, 2, 8, 44, 260, 1556, 9332, 55988, 335924, 2015540, 12093236, 72559412, 435356468, 2612138804, 15672832820, 94036996916, 564221981492, 3385331888948, 20311991333684, 121871948002100, 731231688012596, 4387390128075572, 26324340768453428, 157946044610720564, 947676267664323380

Offset: 0

Clark Kimberling

n-th difference of a(n), a(n-1), ..., a(0) is A000351(n-1) for n >= 1.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,-6).

Cf. A000351, A047848.

[(6^n + 4)/5: n in [0..40]]; // G. C. Greubel, Jan 11 2025

a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=6*a[n-1]+1 od: seq(a[n]+1, n=0..20); # Zerinvary Lajos, Mar 20 2008

(6^Range[0,40] +4)/5 (* G. C. Greubel, Jan 11 2025 *)

def A047851(n): return (pow(6,n) + 4)//5
print([A047851(n) for n in range(41)]) # G. C. Greubel, Jan 11 2025

a(n) = (6^n + 4)/5. - Ralf Stephan, Feb 14 2004
From Philippe Deléham, Oct 05 2009: (Start)
a(0) = 1, a(1) = 2, a(n) = 7*a(n-1) - 6*a(n-2) for n > 1.
G.f.: (1 - 5*x)/(1 - 7*x + 6*x^2). (End)
a(n) = 6*a(n-1) - 4 (with a(0)=1). - Vincenzo Librandi, Aug 06 2010
E.g.f.: exp(x)*(exp(5*x) + 4)/5. - Elmo R. Oliveira, Aug 29 2024

a(21)-a(24) from Elmo R. Oliveira, Aug 29 2024

A135158 a(n) = 5^n - 3^n - 2^n.

Original entry on oeis.org

-1, 0, 12, 90, 528, 2850, 14832, 75810, 383808, 1932930, 9705552, 48648930, 243605088, 1219100610, 6098716272, 30503196450, 152544778368, 762810181890, 3814309582992, 19072323542370, 95363943807648, 476826695752770, 2384154405761712, 11920834803510690, 59604362329076928, 298022376554789250

Offset: 0

Omar E. Pol, Nov 21 2007

Essentially the same as A130072. - Zak Seidov, Oct 03 2011

			a(4) = 528 because 5^4 = 625, 3^4 = 81, 2^4 = 16 and 625 - 81 - 16 = 528.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-31,30).

Cf. A000079, A000244, A000351, A001047, A130072.

[5^n-3^n-2^n: n in [0..50]]; // Vincenzo Librandi, Dec 15 2010

A135158:=n->5^n-3^n-2^n; seq(A135158(n), n=0..30); # Wesley Ivan Hurt, Feb 26 2014

lst={};Do[p=5^n-3^n-2^n;AppendTo[lst, p], {n, 0, 7^2}];lst (* Vladimir Joseph Stephan Orlovsky, Aug 19 2008 *)
LinearRecurrence[{10,-31,30}, {-1, 0, 12}, 25] (* or *) Table[5^n - 3^n - 2^n, {n,0,25}] (* G. C. Greubel, Sep 30 2016 *)

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

G.f.: ( 1+19*x^2-10*x ) / ( (3*x-1)*(2*x-1)*(5*x-1) ).
a(n) = 10*a(n-1) - 31*a(n-2) + 30*a(n-3). - Zak Seidov, Oct 03 2011
E.g.f.: exp(5*x) - exp(3*x) - exp(2*x). - G. C. Greubel, Sep 30 2016

More terms from Vincenzo Librandi, Dec 15 2010

A135159 a(n) = 5^n - 3^n + 2^n.

Original entry on oeis.org

1, 4, 20, 106, 560, 2914, 14960, 76066, 384320, 1933954, 9707600, 48653026, 243613280, 1219116994, 6098749040, 30503261986, 152544909440, 762810444034, 3814310107280, 19072324590946, 95363945904800, 476826699947074, 2384154414150320, 11920834820287906, 59604362362631360, 298022376621898114

Offset: 0

Omar E. Pol, Nov 21 2007

Constants are the prime numbers in decreasing order.

			a(4)=560 because 5^4=625, 3^4=81, 2^4=16 and 625-81+16=560.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-31,30).

Cf. A000079, A000244, A000351, A007689.

[5^n-3^n+2^n: n in [0..50]]; // Vincenzo Librandi, Dec 15 2010

lst={};Do[p=5^n-3^n+2^n;AppendTo[lst, p], {n, 0, 7^2}];lst (* Vladimir Joseph Stephan Orlovsky, Aug 19 2008 *)
CoefficientList[Series[1/(1 - 5 x) - 1/(1 - 3 x) + 1/(1 - 2 x), {x, 0, 40}], x] (* Vincenzo Librandi, May 22 2014 *)
Table[5^n - 3^n + 2^n, {n,0,25}] (* or *) LinearRecurrence[{10, -31, 30}, {1, 4, 20}, 25] (* G. C. Greubel, Sep 30 2016 *)

a(n) = 5^n - 3^n + 2^n.
From Mohammad K. Azarian, Jan 16 2009: (Start)
G.f.: 1/(1-5*x) - 1/(1-3*x) + 1/(1-2*x).
E.g.f.: exp(5*x) - exp(3*x) + exp(2*x). (End)
a(n) = 10*a(n-1) - 31*a(n-2) + 30*a(n-3). - G. C. Greubel, Sep 30 2016

More terms from Vincenzo Librandi, Dec 15 2010

A135160 a(n) = 5^n + 3^n - 2^n.

Original entry on oeis.org

1, 6, 30, 144, 690, 3336, 16290, 80184, 396930, 1972296, 9823650, 49003224, 244667970, 1222289256, 6108282210, 30531894264, 152630871810, 763068462216, 3815084423970, 19074648065304, 95370917376450, 476847616459176, 2384217167880930, 11921023089868344, 59604927188149890, 298024071132008136

Offset: 0

Omar E. Pol, Nov 21 2007

			a(4)=690 because 5^4=625, 3^4=81, 2^4=16 and we can write 625 + 81 - 16 = 690.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-31,30).

Cf. A000079, A000244, A000351, A001047.

[5^n+3^n-2^n: n in [0..50]] // Vincenzo Librandi, Dec 15 2010

lst={};Do[p=5^n+3^n-2^n;AppendTo[lst, p], {n, 0, 7^2}];lst (* Vladimir Joseph Stephan Orlovsky, Aug 19 2008 *)
Table[5^n+3^n-2^n,{n,0,30}] (* or *) LinearRecurrence[{10,-31,30},{1,6,30},30] (* Harvey P. Dale, Mar 10 2013 *)

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

a(n) = 5^n + 3^n - 2^n.
From Mohammad K. Azarian, Jan 16 2009: (Start)
G.f.: 1/(1-5*x) + 1/(1-3*x) - 1/(1-2*x).
E.g.f.: e^(5*x) + e^(3*x) - e^(2*x). (End)
a(0)=1, a(1)=6, a(2)=30, a(n) = 10*a(n-1) - 31*a(n-2) + 30*a(n-3). - Harvey P. Dale, Mar 10 2013

More terms from Vincenzo Librandi, Dec 15 2010

A145232 a(n) = Fibonacci(5^n).

Original entry on oeis.org

1, 5, 75025, 59425114757512643212875125, 18526362353047317310282957646406309593963452838196423660508102562977229905562196608078556292556795045922591488273554788881298750625

Offset: 0

Artur Jasinski, Oct 05 2008

Seiichi Manyama, Table of n, a(n) for n = 0..5
Robert Frontczak, Problem B-1341, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 62, No. 1 (2024), p. 84.
Thomas Koshy and Zhenguang Gao, Polynomial Extensions of a Diminnie Delight, Fibonacci Quart. 55 (2017), no. 1, 13-20.
Achilleas Sinefakopoulos, Solution to Problem 1909, Crux Mathematicorum, 20 (1994), 295-296.

Cf. A000045.

Cf. (k^n)-th Fibonacci number: A058635 (k=2), A045529 (k=3), A145231 (k=4), this sequence (k=5), A145233 (k=6), A145234 (k=7), A250487 (k=8), A250488 (k=9), A250489 (k=10).

a := proc(n) option remember; if n = 0 then 1 else 25*a(n-1)^5 - 25*a(n-1)^3 + 5*a(n-1) end if; end:
seq(a(n), n = 0..5); # Peter Bala, Nov 24 2022

G = (1 + Sqrt[5])/2; Table[Expand[(G^(5^n) - (1 - G)^(5^n))/Sqrt[5]], {n, 1, 6}]
Table[Round[N[(2/Sqrt[5])*Cosh[5^n*ArcCosh[Sqrt[5]/2]], 1000]], {n, 1, 4}]
Fibonacci[5^Range[0,4]] (* Harvey P. Dale, Nov 29 2018 *)

a(n) = (G^(5^n) - (1 - G)^(5^n))/sqrt(5) where G = (1 + sqrt(5))/2.
a(n) = (2/sqrt(5))*cosh((2*k+1)^n*arccosh(sqrt(5)/2)).
a(n) = (2/sqrt(5))*cosh(5^n*arccosh(sqrt(5)/2)).
a(n) = (5^n)*A128935(n). - R. J. Mathar, Nov 04 2010
a(n) = A000045(A000351(n)). - Michel Marcus, Nov 07 2013
a(n+1) = 25*a(n)^5 - 25*a(n)^3 + 5*a(n) with a(0) = 1. - Peter Bala, Nov 24 2022
a(n) = 5^n * Product_{k=0..n-1} (5*a(k)^4 - 5*a(k)^2 + 1) (Frontczak, 2024). - Amiram Eldar, Feb 29 2024

A165826 Totally multiplicative sequence with a(p) = 5.

Original entry on oeis.org

1, 5, 5, 25, 5, 25, 5, 125, 25, 25, 5, 125, 5, 25, 25, 625, 5, 125, 5, 125, 25, 25, 5, 625, 25, 25, 125, 125, 5, 125, 5, 3125, 25, 25, 25, 625, 5, 25, 25, 625, 5, 125, 5, 125, 125, 25, 5, 3125, 25, 125

Offset: 1

Jaroslav Krizek, Sep 28 2009

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A000351, A001222, A061142, A165824, A165825.

5^PrimeOmega[Range[100]] (* G. C. Greubel, Apr 09 2016 *)

for(n=1, 100, print1(direuler(p=2, n, 1/(1-5*X))[n], ", ")) \\ Vaclav Kotesovec, Feb 28 2023

a(n) = A000351(A001222(n)) = 5^bigomega(n) = 5^A001222(n).

Dirichlet g.f.: Product_{p prime} 1 / (1 - 5 * p^(-s)). - Ilya Gutkovskiy, Oct 30 2019

A175169 Numbers k that divide the sum of digits of 2^k.

Original entry on oeis.org

1, 2, 5, 70

Offset: 1

N. J. A. Sloane, Dec 03 2010

No other terms <= 200000. - Harvey P. Dale, Dec 16 2010
No other terms <= 1320000. - Robert G. Wilson v, Dec 18 2010
There are almost certainly no further terms.

Sum of digits of k^n mod n: (k=2) A000079, A001370, A175434, A175169; (k=3) A000244, A004166, A175435, A067862; (k=5) A000351, A066001, A175456; (k=6) A000400, A066002, A175457, A067864; (k=7) A000420, A066003, A175512, A067863; (k=8) A062933; (k=13) A001022, A175527, A175528, A175525; (k=21) A175589; (k=167) A175558, A175559, A175560, A175552.

Select[Range[1000],Divisible[Total[IntegerDigits[2^#]],#]&] (* Harvey P. Dale, Dec 16 2010 *)

is(n)=sumdigits(2^n)%n==0 \\ Charles R Greathouse IV, Sep 06 2016

A175525 Numbers k that divide the sum of digits of 13^k.

Original entry on oeis.org

1, 2, 5, 140, 158, 428, 788, 887, 914, 1814, 1895, 1976, 2579, 2732, 3074, 3299, 3641, 4658, 4874, 5378, 5423, 5504, 6170, 6440, 6944, 8060, 8249, 8915, 9041, 9158, 9725, 9824, 10661, 11291, 13820, 15305, 17051, 17393, 18716, 19589, 20876, 21641, 23756, 24188, 25961, 28409, 30632, 31307, 32387, 33215, 34970, 35240, 36653, 36977, 41558, 43970, 44951, 47444, 51764, 52655, 53375, 53852, 54104, 56831, 57506, 59153, 66479, 68063, 73562, 78485, 79286, 87908, 92093, 102029, 106934, 114854, 116321, 134051, 139397, 184037, 192353, 256469, 281381, 301118, 469004

Offset: 1

T. D. Noe, Dec 03 2010

Almost certainly there are no further terms.
Comments from Donovan Johnson on the computation of this sequence, Dec 05 2010 (Start):
The number of digits of 13^k is approximately 1.114*k, so I defined an array d() that is a little bigger than 1.114 times the maximum k value to be checked. The elements of d() each are the value of a single digit of the decimal expansion of 13^k with d(1) being the least significant digit.
It's easier to see how the program works if I start with k = 2.
For k = 1, d(2) would have been set to 1 and d(1) would have been set to 3.
k = 2:
x = 13*d(1) = 13*3 = 39
y = 39\10 = 3 (integer division)
x-y*10 = 39-30 = 9, d(1) is set to 9
x = 13*d(2)+y = 13*1+3 = 16, y is the carry from previous digit
y = 16\10 = 1
x-y*10 = 16-10 = 6, d(2) is set to 6
x = 13*d(3)+y = 13*0+1 = 1, y is the carry from previous digit
y = 1\10 = 0
x-y*10 = 1-0 = 1, d(3) is set to 1
These steps would of course be inside a loop and that loop would be inside a k loop. A pointer to the most significant digit increases usually by one and sometimes by two for each successive k value checked. The number of steps of the inner loop is the size of the pointer. A scan is done from the first element to the pointer element to get the digit sum.
(End)
No other terms < 3*10^6. - Donovan Johnson, Dec 07 2010

Sum of digits of k^n mod n: (k=2) A000079, A001370, A175434, A175169; (k=3) A000244, A004166, A175435, A067862; (k=5) A000351, A066001, A175456; (k=6) A000400, A066002, A175457, A067864; (k=7) A000420, A066003, A175512, A067863; (k=8) A062933; (k=13) A001022, A175527, A175528, A175525; (k=21) A175589; (k=167) A175558, A175559, A175560, A175552.

Select[Range[1000], Mod[Total[IntegerDigits[13^#]], #] == 0 &]

a(47)-a(79) from N. J. A. Sloane, Dec 04 2010

a(80)-a(85) from Donovan Johnson, Dec 05 2010

A223321 T(n,k)=Rolling icosahedron footprints: number of nXk 0..11 arrays starting with 0 where 0..11 label vertices of an icosahedron and every array movement to a horizontal or antidiagonal neighbor moves along an icosahedral edge.

Original entry on oeis.org

1, 5, 12, 25, 125, 144, 125, 1625, 3125, 1728, 625, 21125, 105625, 78125, 20736, 3125, 274625, 3570125, 6865625, 1953125, 248832, 15625, 3570125, 122039125, 603351125, 446265625, 48828125, 2985984, 78125, 46411625, 4176940625, 54279694625

Offset: 1

R. H. Hardin Mar 19 2013

Table starts
........1...........5..............25................125....................625
.......12.........125............1625..............21125.................274625
......144........3125..........105625............3570125..............122039125
.....1728.......78125.........6865625..........603351125............54279694625
....20736.....1953125.......446265625.......101966340125.........24143758634125
...248832....48828125.....29007265625.....17232311481125......10739266230499625
..2985984..1220703125...1885472265625...2912260640310125....4776881955584279125
.35831808.30517578125.122555697265625.492172048212411125.2124782217358970404625

			Some solutions for n=3 k=4
..0..1..8..9....0..1..0..7....0..1..0..2....0..1..0..6....0..6..2..4
..0..2..8..2....0..5..0..5....0..6..0..2....0..6.10..5....0..1..2..4
..6..2..4..2....0..1..0..7....0..7..0..7....0..6.10..5....0..6.10..4
Vertex neighbors:
0 -> 1 2 5 6 7
1 -> 0 2 3 7 8
2 -> 0 1 4 6 8
3 -> 1 7 8 9 11
4 -> 2 6 8 9 10
5 -> 0 6 7 10 11
6 -> 0 2 4 5 10
7 -> 0 1 3 5 11
8 -> 1 2 3 4 9
9 -> 3 4 8 10 11
10 -> 4 5 6 9 11
11 -> 3 5 7 9 10

R. H. Hardin, Table of n, a(n) for n = 1..97

Column 1 is A001021(n-1)
Column 2 is A013710(n-1)
Column 3 is 25*65^(n-1)
Column 4 is 125*169^(n-1)
Row 1 is A000351(n-1)

Empirical for column k:
k=1: a(n) = 12*a(n-1)
k=2: a(n) = 25*a(n-1)
k=3: a(n) = 65*a(n-1)
k=4: a(n) = 169*a(n-1)
k=5: a(n) = 479*a(n-1) -15210*a(n-2)
k=6: a(n) = 1366*a(n-1) -232713*a(n-2) +9253764*a(n-3)
k=7: [order 8]
Empirical for row n:
n=1: a(n) = 5*a(n-1)
n=2: a(n) = 13*a(n-1) for n>2
n=3: a(n) = 38*a(n-1) -129*a(n-2) for n>4
n=4: [order 7] for n>10
n=5: [order 32] for n>36

cp's OEIS Frontend

A036291 a(n) = n*5^n.

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A047851 a(n) = A047848(3,n).

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A135158 a(n) = 5^n - 3^n - 2^n.

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A135159 a(n) = 5^n - 3^n + 2^n.

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A135160 a(n) = 5^n + 3^n - 2^n.

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A145232 a(n) = Fibonacci(5^n).

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A165826 Totally multiplicative sequence with a(p) = 5.

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