A055842 -id:A055842 - cp's OEIS frontend

A005054 a(0) = 1; a(n) = 4*5^(n-1) for n >= 1.

1, 4, 20, 100, 500, 2500, 12500, 62500, 312500, 1562500, 7812500, 39062500, 195312500, 976562500, 4882812500, 24414062500, 122070312500, 610351562500, 3051757812500, 15258789062500, 76293945312500, 381469726562500, 1907348632812500, 9536743164062500

Offset: 0

N. J. A. Sloane

Consider the sequence formed by the final n decimal digits of {2^k: k >= 0}. For n=1 this is 1, 2, 4, 8, 6, 2, 4, ... (A000689) with period 4. For any n this is periodic with period a(n). Cf. A000855 (n=2), A126605 (n=3, also n=4). - N. J. A. Sloane, Jul 08 2022
First differences of A000351.
Length of repeating cycle of the final n+1 digits in Fermat numbers. - Lekraj Beedassy, Robert G. Wilson v and Eric W. Weisstein, Jul 05 2004
Number of n-digit endings for a power of 2 whose exponent is greater than or equal to n. - J. Lowell
For n>=1, a(n) is equal to the number of functions f:{1,2,...,n}->{1,2,3,4,5} such that for a fixed x in {1,2,...,n} and a fixed y in {1,2,3,4,5} we have f(x) != y. - Aleksandar M. Janjic and Milan Janjic, Mar 27 2007
Equals INVERT transform of A033887: (1, 3, 13, 55, 233, ...) and INVERTi transform of A001653: (1, 5, 29, 169, 985, 5741, ...). - Gary W. Adamson, Jul 22 2010
a(n) = (n+1) terms in the sequence (1, 3, 4, 4, 4, ...) dot (n+1) terms in the sequence (1, 1, 4, 20, 100, ...). Example: a(4) = 500 = (1, 3, 4, 4, 4) dot (1, 1, 4, 20, 100) = (1 + 3 + 16, + 80 + 400), where (1, 3, 16, 80, 400, ...) = A055842, finite differences of A005054 terms. - Gary W. Adamson, Aug 03 2010
a(n) is the number of compositions of n when there are 4 types of each natural number. - Milan Janjic, Aug 13 2010
Apart from the first term, number of monic squarefree polynomials over F_5 of degree n. - Charles R Greathouse IV, Feb 07 2012
For positive integers that can be either of two colors (designated by ' or ''), a(n) is the number of compositions of 2n that are cardinal palindromes; that is, palindromes that only take into account the cardinality of the numbers and not their colors. Example: 3', 2'', 1', 1, 2', 3'' would count as a cardinal palindrome. - Gregory L. Simay, Mar 01 2020
a(n) is the length of the period of the sequence Fibonacci(k) (mod 5^(n-1)) (for n>1) and the length of the period of the sequence Lucas(k) (mod 5^n) (Kramer and Hoggatt, 1972). - Amiram Eldar, Feb 02 2022

T. Koshy, "The Ends Of A Fermat Number", pp. 183-4 Journal Recreational Mathematics, vol. 31(3) 2002-3 Baywood NY.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 458.
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets.
Judy Kramer and V. E. Hoggatt, Jr., Special Cases of Fibonacci Periodicity (Part 1, Part 2), The Fibonacci Quarterly, Vol. 10, No. 5 (1972), pp. 519-522, 530.
Eric Weisstein's World of Mathematics, Fermat Number.
Index entries for linear recurrences with constant coefficients, signature (5).

Cf. A000010, A000032, A000045, A000351, A000215, A055842.
Cf. A001653, A033887. - Gary W. Adamson, Jul 22 2010
See also A000689, A000855, A126605, A216099, A216164. - N. J. A. Sloane, Jul 08 2022

[(4*5^n+0^n)/5: n in [0..30]]; // Vincenzo Librandi, Jun 08 2013

a:= n-> ceil(4*5^(n-1)):
seq(a(n), n=0..30);  # Alois P. Heinz, Jul 08 2022

CoefficientList[Series[(1 - x) / (1 - 5 x), {x, 0, 50}], x] (* Vincenzo Librandi, Jun 08 2013 *)

Vec((1-x)/(1-5*x) + O(x^100)) \\ Altug Alkan, Dec 07 2015

a(n) = (4*5^n + 0^n) / 5. - R. J. Mathar, May 13 2008
G.f.: (1-x)/(1-5*x). - Philippe Deléham, Nov 02 2009
G.f.: 1/(1 - 4*Sum_{k>=1} x^k).
a(n) = 5*a(n-1) for n>=2. - Vincenzo Librandi, Dec 31 2010
a(n) = phi(5^n) = A000010(A000351(n)).
E.g.f.: (4*exp(5*x)+1)/5. - Paul Barry, Apr 20 2003
a(n + 1) = (((1 + sqrt(-19))/2)^n + ((1 - sqrt(-19))/2)^n)^2 - (((1 + sqrt(-19))/2)^n - ((1 - sqrt(-19))/2)^n)^2. - Raphie Frank, Dec 07 2015

Better definition from R. J. Mathar, May 13 2008

Edited by N. J. A. Sloane, Jul 08 2022

A189449 T(n,k)=Number of nXk array permutations with each element moving zero or one space horizontally or diagonally.

Original entry on oeis.org

1, 2, 1, 3, 5, 1, 5, 16, 13, 1, 8, 61, 80, 34, 1, 13, 225, 666, 400, 89, 1, 21, 841, 5080, 7300, 2000, 233, 1, 34, 3136, 40106, 118128, 80282, 10000, 610, 1, 55, 11705, 313136, 2008890, 2735828, 883049, 50000, 1597, 1, 89, 43681, 2455013, 33735505, 100047288

Offset: 1

R. H. Hardin Apr 22 2011

Table starts
.1.....2.......3...........5..............8................13
.1.....5......16..........61............225...............841
.1....13......80.........666...........5080.............40106
.1....34.....400........7300.........118128...........2008890
.1....89....2000.......80282........2735828.........100047288
.1...233...10000......883049.......63367633........4982424404
.1...610...50000.....9712873.....1467726607......248187289625
.1..1597..250000...106834338....33995627281....12362821462925
.1..4181.1250000..1175098084...787410049440...615823316590381
.1.10946.6250000.12925203122.18238068717816.30675714678416140

			Some solutions for 5X3
..1..0..2....0..5..2....0..5..2....0..2..1....0..2..1....4..5..2....1..0..2
..4..3..5....7..4..1....7..4..1....3..4..5....3..5..4....3..0..1....4..3..5
..6..7..8....6..3..8...10..3..8....7.11..8....6..8..7....6..8..7...10..7..8
.10..9.11....9.11.10....9..6.11...13..6.10...10..9.11...13.10.11...13..6.11
.12.14.13...12.14.13...12.13.14...12..9.14...12.13.14...12..9.14...12..9.14

R. H. Hardin, Table of n, a(n) for n = 1..179

Column 2 is A001519(n+1)
Column 3 is A055842
Row 1 is A000045(n+1)

A201780 Riordan array ((1-x)^2/(1-2x), x/(1-2x)).

Original entry on oeis.org

1, 0, 1, 1, 2, 1, 2, 5, 4, 1, 4, 12, 13, 6, 1, 8, 28, 38, 25, 8, 1, 16, 64, 104, 88, 41, 10, 1, 32, 144, 272, 280, 170, 61, 12, 1, 64, 320, 688, 832, 620, 292, 85, 14, 1, 128, 704, 1696, 2352, 2072, 1204, 462, 113, 16, 1

Offset: 0

Philippe Deléham, Dec 05 2011

Diagonals ascending: 1, 0, 1, 1, 2, 2, 4, 5, 1, 8, 12, 4, ... (see A201509).

			Triangle begins:
  1;
  0,  1;
  1,  2,  1;
  2,  5,  4,  1;
  4, 12, 13,  6,  1;
  8, 28, 38, 25,  8,  1;

Benjamin Braun, W. K. Hough, Matching and Independence Complexes Related to Small Grids, arXiv preprint arXiv:1606.01204 [math.CO], 2016.
Wesley K. Hough, On Independence, Matching, and Homomorphism Complexes, (2017), Theses and Dissertations--Mathematics, 42.

Diagonals: A000012, A005843, A001844, A035597,
Columns: A034008, A045623, A084851, A055585,
Row sums: A052156

CoefficientList[#, y]& /@ CoefficientList[(1-x)^2/(1-(y+2)*x) + O[x]^10, x] // Flatten (* Jean-François Alcover, Nov 03 2018 *)

T(n,k) = 2*T(n-1,k) + T(n-1,k-1) with T(0,0) = 0, T(1,0) = 0, T(2,0) = 0 and T(n,k)= 0 if k < 0 or if n < k.
Sum_{k=0..n} T(n,k)*x^k = A154955(n+1), A034008(n), A052156(n), A055841(n), A055842(n), A055846(n), A055270(n), A055847(n), A055995(n), A055996(n), A056002(n), A056116(n) for x = -1,0,1,2,3,4,5,6,7,8,9,10 respectively.
G.f.: (1-x)^2/(1-(y+2)*x).

A275142 T(n,k)=Number of nXk 0..2 arrays with no element equal to any value at offset (-2,-2) (-1,-2) or (0,-1) and new values introduced in order 0..2.

Original entry on oeis.org

1, 1, 2, 2, 6, 5, 4, 16, 36, 14, 8, 48, 80, 216, 41, 16, 144, 224, 400, 1296, 122, 32, 432, 528, 1088, 2000, 7776, 365, 64, 1296, 1216, 2320, 5248, 10000, 46656, 1094, 128, 3888, 2816, 6464, 9744, 25344, 50000, 279936, 3281, 256, 11664, 6544, 17872, 32384, 41360

Offset: 1

R. H. Hardin, Jul 17 2016

Table starts
....1........1.......2........4........8........16........32.........64
....2........6......16.......48......144.......432......1296.......3888
....5.......36......80......224......528......1216......2816.......6544
...14......216.....400.....1088.....2320......6464.....17872......49792
...41.....1296....2000.....5248.....9744.....32384....107472.....362176
..122.....7776...10000....25344....41360....165568....663904....2695808
..365....46656...50000...122368...175120....841536...4055152...19906560
.1094...279936..250000...590848...741904...4283968..24875600..147762240
.3281..1679616.1250000..2852864..3142672..21800000.152379136.1093999424
.9842.10077696.6250000.13774848.13312656.110943552.933805200.8109111360

			Some solutions for n=5 k=4
..0..1..2..1. .0..1..0..1. .0..1..0..2. .0..1..2..1. .0..1..2..0
..2..0..1..0. .0..2..1..2. .2..0..1..2. .2..0..2..0. .0..1..2..0
..2..0..1..2. .1..2..1..0. .1..2..1..2. .2..0..1..2. .2..0..2..0
..1..2..0..1. .0..1..2..1. .0..2..0..1. .2..0..1..2. .2..0..1..2
..0..2..0..1. .2..1..2..0. .0..1..2..0. .1..2..0..2. .0..1..0..1

R. H. Hardin, Table of n, a(n) for n = 1..721

Column 1 is A007051(n-1).
Column 2 is A000400(n-1).
Column 3 is A055842.
Row 1 is A000079(n-2).

Empirical for column k:
k=1: a(n) = 4*a(n-1) -3*a(n-2)
k=2: a(n) = 6*a(n-1)
k=3: a(n) = 5*a(n-1) for n>2
k=4: a(n) = 4*a(n-1) +4*a(n-2) for n>3
k=5: a(n) = 3*a(n-1) +5*a(n-2) +a(n-3) for n>4
k=6: a(n) = 3*a(n-1) +10*a(n-2) +4*a(n-3) -4*a(n-4) for n>6
k=7: a(n) = 3*a(n-1) +18*a(n-2) +11*a(n-3) -23*a(n-4) -4*a(n-5) for n>7
Empirical for row n:
n=1: a(n) = 2*a(n-1) for n>2
n=2: a(n) = 3*a(n-1) for n>3
n=3: a(n) = 3*a(n-1) -2*a(n-2) +a(n-3) for n>5
n=4: a(n) = 5*a(n-1) -9*a(n-2) +10*a(n-3) -6*a(n-4) +a(n-5) for n>9
n=5: [order 8] for n>12
n=6: [order 13] for n>18
n=7: [order 21] for n>27

A275504 T(n,k)=Number of nXk 0..2 arrays with no element equal to any value at offset (-2,-1) (-2,0) or (-1,-1) and new values introduced in order 0..2.

Original entry on oeis.org

1, 2, 2, 5, 9, 3, 14, 54, 16, 6, 41, 324, 80, 28, 12, 122, 1944, 400, 136, 56, 24, 365, 11664, 2000, 656, 232, 104, 48, 1094, 69984, 10000, 3168, 988, 516, 200, 96, 3281, 419904, 50000, 15296, 4180, 2628, 1168, 380, 192, 9842, 2519424, 250000, 73856, 17712

Offset: 1

R. H. Hardin, Jul 30 2016

Table starts
...1....2.....5.....14......41......122.......365.......1094........3281
...2....9....54....324....1944....11664.....69984.....419904.....2519424
...3...16....80....400....2000....10000.....50000.....250000.....1250000
...6...28...136....656....3168....15296.....73856.....356608.....1721856
..12...56...232....988....4180....17712.....75024.....317812.....1346268
..24..104...516...2628...13384....68080....346528....1763408.....8974288
..48..200..1168...7140...43780...268152...1643372...10069540....61703488
..96..380..2660..19368..143784..1063756...7886280...58423188...432942008
.192..724..6024..52864..470352..4220952..37846556..339516412..3045734096
.384.1380.13716.144228.1549756.16808164.182923008.1989999904.21655912500

			Some solutions for n=4 k=4
..0..0..1..1. .0..0..0..0. .0..0..0..0. .0..1..2..0. .0..1..0..0
..0..1..2..0. .1..2..1..1. .1..1..2..2. .0..1..2..0. .1..1..2..2
..1..1..2..0. .1..2..1..2. .1..2..2..1. .1..2..0..1. .1..2..2..1
..2..2..0..1. .0..0..0..2. .2..2..0..0. .1..2..0..1. .0..0..0..0

R. H. Hardin, Table of n, a(n) for n = 1..312

Column 1 is A003945(n-2).
Row 1 is A007051(n-1).
Row 3 is A055842.
Row 4 is A108051(n+1).

Empirical for column k:
k=1: a(n) = 2*a(n-1) for n>3
k=2: a(n) = a(n-1) +2*a(n-2) -a(n-4) for n>6
k=3: a(n) = 2*a(n-1) +2*a(n-2) -2*a(n-3) -4*a(n-4) +3*a(n-5) +a(n-6) -a(n-7) for n>11
k=4: [order 16] for n>20
k=5: [order 32] for n>36
k=6: [order 64] for n>68
Empirical for row n:
n=1: a(n) = 4*a(n-1) -3*a(n-2)
n=2: a(n) = 6*a(n-1) for n>2
n=3: a(n) = 5*a(n-1) for n>2
n=4: a(n) = 4*a(n-1) +4*a(n-2)
n=5: a(n) = 3*a(n-1) +5*a(n-2) +a(n-3)
n=6: a(n) = 3*a(n-1) +10*a(n-2) +4*a(n-3) -4*a(n-4) for n>5
n=7: a(n) = 3*a(n-1) +18*a(n-2) +11*a(n-3) -23*a(n-4) -4*a(n-5) for n>6

A256100 In S = A007376 (read as a sequence) the digit S(n) appears a(n) times in the sequence S(1), ..., S(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 3, 4, 5, 2, 6, 2, 7, 2, 8, 2, 9, 2, 10, 2, 11, 2, 12, 2, 3, 2, 4, 13, 5, 6, 7, 3, 8, 3, 9, 3, 10, 3, 11, 3, 12, 3, 13, 3, 4, 3, 5, 14, 6, 14, 7, 8, 9, 4, 10, 4, 11, 4, 12, 4, 13, 4, 14, 4, 5, 4, 6, 15, 7, 15, 8, 15, 9, 10, 11, 5, 12, 5, 13, 5, 14, 5, 15, 5, 6

Offset: 1

Wolfdieter Lang, Apr 08 2015

The motivation to consider this sequence came from the proposal A256379 by Anthony Sand.
This sequence can also be read as an irregular triangle (array) in which a(n, k) is the number of appearances of the k-th digit of n in the digits of 1, ... ,n-1 and the first k digits of n. See the example for the head of this array. The row length is A055842(n), n >= 1.
This can also be described as the ordinal transform of A007376. - Franklin T. Adams-Watters, Oct 10 2015

			a(10) = 2 because A007376(10) = 1 and that sequence up to n=10 is 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, and 1 appears twice.
a(24) = 10 because A007376(24) = 1 and this is the tenth 1 in A007376 up to, and including, A007376(24).
Read as a tabf array a(n, k) with row length A055842(n) this begins:
   n\k  1  2  ...
   1:   1
   2:   1
   3:   1
   4:   1
   5:   1
   6:   1
   7:   1
   8:   1
   9:   1
  10:   2  1
  11:   3  4
  12:   5  2
  13:   6  2
  14:   7  2
  15:   8  2
  16:   9  2
  17:  10  2
  18:  11  2
  19:  12  2
  20:   3  2
  ...

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A007376, A065648.

a256100 n = a256100_list !! (n-1)
a256100_list = f a007376_list $ take 10 $ repeat 1 where
   f (d:ds) counts = y : f ds (xs ++ (y + 1) : ys) where
                           (xs, y:ys) = splitAt d counts
-- Reinhard Zumkeller, Aug 13 2015

lim = 120; s = Flatten[IntegerDigits /@ Range@ lim]; f[n_] := Block[{d = IntegerDigits /@ Take[s, n] // Flatten // FromDigits}, DigitCount[d][[If[ s[[n]] == 0, 10, s[[n]] ]] ] ]; Array[f, lim] (* Michael De Vlieger, Apr 08 2015, after Robert G. Wilson v at A007376 *)

a(n) gives the number of digits A007376(n) in the sequence starting with A007376(1) and ending with A007376(n).

cp's OEIS Frontend

A005054 a(0) = 1; a(n) = 4*5^(n-1) for n >= 1.

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A189449 T(n,k)=Number of nXk array permutations with each element moving zero or one space horizontally or diagonally.

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A201780 Riordan array ((1-x)^2/(1-2x), x/(1-2x)).

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A275142 T(n,k)=Number of nXk 0..2 arrays with no element equal to any value at offset (-2,-2) (-1,-2) or (0,-1) and new values introduced in order 0..2.

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A275504 T(n,k)=Number of nXk 0..2 arrays with no element equal to any value at offset (-2,-1) (-2,0) or (-1,-1) and new values introduced in order 0..2.

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A256100 In S = A007376 (read as a sequence) the digit S(n) appears a(n) times in the sequence S(1), ..., S(n).

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