A000244 -id:A000244 - cp's OEIS frontend

A159991 Powers of 60: a(n) = 60^n.

1, 60, 3600, 216000, 12960000, 777600000, 46656000000, 2799360000000, 167961600000000, 10077696000000000, 604661760000000000, 36279705600000000000, 2176782336000000000000, 130606940160000000000000, 7836416409600000000000000, 470184984576000000000000000

Offset: 0

Reinhard Zumkeller, May 01 2009

			G.f. = 1 + 60*x + 3600*x^2 + 216000*x^3 + 12960000*x^4 + 77600000*x^5 + ... - _Michael Somos_, Jan 01 2019

Vincenzo Librandi, Table of n, a(n) for n = 0..150
Wikipedia, Sexagesimal
Index entries for linear recurrences with constant coefficients, signature (60).

Subsequence of A051037.

Cf. A159990, A159993, A159995, A088157, A091649, A125628, A091720, A091721, A091722, A060707, A070197.

List([0..20],n->60^n); # Muniru A Asiru, Nov 21 2018

[60^n: n in [0..20]]; // Vincenzo Librandi, May 02 2011

[60^n$n=0..20]; # Muniru A Asiru, Nov 21 2018

60^Range[0,15] (* Harvey P. Dale, Jun 02 2011 *)

A159991(n):=60^n$
makelist(A159991(n),n,0,30); /* Martin Ettl, Nov 05 2012 */

a(n)=60^n \\ Charles R Greathouse IV, May 02 2011

a(n)=60^n \\ Charles R Greathouse IV, Jun 19 2015

powers(60,8) \\ Charles R Greathouse IV, Jun 19 2015

for n in range(0,20): print(60**n, end=', ') # Stefano Spezia, Nov 21 2018

a(n) = A000400(n)*A011557(n) = A000351(n)*A001021(n) = A000302(n)*A001024(n) = A000244(n)*A009964(n). (Corrected by Robert B Fowler, Jan 25 2023)
From Muniru A Asiru, Nov 21 2018: (Start)
a(n) = 60^n.
a(n) = 60*a(n-1) for n > 0, a(0) = 1.
G.f.: 1/(1-60*x).
E.g.f: exp(60*x). (End)
a(n) = 1/a(-n) for all n in Z. - Michael Somos, Jan 01 2019

A334774 Triangle read by rows: T(n,k) is the number of permutations of 2 indistinguishable copies of 1..n with exactly k local maxima.

Original entry on oeis.org

1, 3, 3, 9, 57, 24, 27, 705, 1449, 339, 81, 7617, 48615, 49695, 7392, 243, 78357, 1290234, 3650706, 2234643, 230217, 729, 791589, 30630618, 197457468, 314306943, 128203119, 9689934, 2187, 7944321, 686779323, 9080961729, 30829608729, 31435152267, 9159564513, 529634931

Offset: 1

Andrew Howroyd, May 11 2020

Also the number of permutations of 2 indistinguishable copies of 1..n with exactly k-1 peaks. A peak is an interior maximum.

			Triangle begins:
    1;
    3,      3;
    9,     57,       24;
   27,    705,     1449,       339;
   81,   7617,    48615,     49695,      7392;
  243,  78357,  1290234,   3650706,   2234643,    230217;
  729, 791589, 30630618, 197457468, 314306943, 128203119, 9689934;
  ...
The T(2,1) = 3 permutations of 1122 with 1 local maxima are 1122, 1221, 2211.
The T(2,2) = 3 permutations of 1122 with 2 local maxima are 1212, 2112, 2121.
The T(2,1) = 3 permutations of 1122 with 0 peaks are 2211, 2112, 1122.
The T(2,2) = 3 permutations of 1122 with 1 peak are 2121, 1221, 1212.

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)

Columns k=1..6 are A000244(n-1), 3*A152494, 3*A152495, 3*A152496, 3*A152497, 3*A152498.
Row sums are A000680.
Main diagonal is A334775.
The version for permutations of 1..n is A008303(n,k-1).
Cf. A154283, A334773, A334776, A334777, A334778.

PeaksBySig(sig, D)={
  my(F(lev,p,q) = my(key=[lev,p,q], z); if(!mapisdefined(FC, key, &z),
    my(m=sig[lev]); z = if(lev==1, if(p==0, binomial(m-1, q), 0), sum(i=0, p, sum(j=0, min(m-i, q), self()(lev-1, p-i, q-j+i) * binomial(m+2*(q-j)+1, 2*q+i-j+1) * binomial(q-j+i, i) * binomial(q+1, j) )));
    mapput(FC, key, z)); z);
  local(FC=Map());
  vector(#D, i, F(#sig, D[i], 0));
}
Row(n)={ PeaksBySig(vector(n,i,2), [0..n-1]) }
{ for(n=1, 8, print(Row(n))) }

T(n,k) = F(2,n,k-1,0) where F(m,n,p,q) = Sum_{i=0..p} Sum_{j=0..min(m-i, q)} F(m, n-1, p-i, q-j+i) * binomial(m+2*(q-j)+1, 2*q+i-j+1) * binomial(q-j+i, i) * binomial(q+1, j) for n > 1 with F(m,1,0,q) = binomial(m-1, q), F(m,1,p,q) = 0 for p > 0.
A334776(n) = Sum_{k=1..n} (k-1)*T(n,k).
A334777(n) = Sum_{k=1..n} k*T(n,k).

A009998 Triangle in which j-th entry in i-th row is (j+1)^(i-j).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 8, 9, 4, 1, 1, 16, 27, 16, 5, 1, 1, 32, 81, 64, 25, 6, 1, 1, 64, 243, 256, 125, 36, 7, 1, 1, 128, 729, 1024, 625, 216, 49, 8, 1, 1, 256, 2187, 4096, 3125, 1296, 343, 64, 9, 1, 1, 512, 6561, 16384, 15625, 7776, 2401, 512, 81, 10, 1

Offset: 0

N. J. A. Sloane

Read as a square array this is the Hilbert transform of triangle A123125 (see A145905 for the definition of this term). For example, the fourth row of A123125 is (0,1,4,1) and the expansion (x + 4*x^2 + x^3)/(1-x)^4 = x + 8*x^2 + 27*x^3 + 64*x^4 + ... generates the entries in the fourth row of this array read as a square. - Peter Bala, Oct 28 2008

			Triangle begins:
  1;
  1,  1;
  1,  2,  1;
  1,  4,  3,  1;
  1,  8,  9,  4,  1;
  1, 16, 27, 16,  5,  1;
  1, 32, 81, 64, 25,  6,  1;
  ...
From _Gus Wiseman_, May 01 2021: (Start)
The rows of the triangle are obtained by reading antidiagonals upward in the following table of A(k,n) = n^k, with offset k = 0, n = 1:
         n=1:     n=2:     n=3:     n=4:     n=5:     n=6:
   k=0:   1        1        1        1        1        1
   k=1:   1        2        3        4        5        6
   k=2:   1        4        9       16       25       36
   k=3:   1        8       27       64      125      216
   k=4:   1       16       81      256      625     1296
   k=5:   1       32      243     1024     3125     7776
   k=6:   1       64      729     4096    15625    46656
   k=7:   1      128     2187    16384    78125   279936
   k=8:   1      256     6561    65536   390625  1679616
   k=9:   1      512    19683   262144  1953125 10077696
  k=10:   1     1024    59049  1048576  9765625 60466176
(End)

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 24.

T. D. Noe, Rows n=0..50 of triangle, flattened
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].
Peter Luschny, Figurate number - a very short introduction

Row sums give A026898.
Cf. A088956, A123125, A179927.
Column n = 2 of the array is A000079.
Column n = 3 of the array is A000244.
Row k = 2 of the array is A000290.
Row k = 3 of the array is A000578.
Diagonal n = k of the array is A000312.
Diagonal n = k + 1 of the array is A000169.
Diagonal n = k + 2 of the array is A000272.
The transpose of the array is A009999.
The numbers of divisors of the entries are A343656 (row sums: A343657).
A007318 counts k-sets of elements of {1..n}.
A059481 counts k-multisets of elements of {1..n}.
Cf. A002109, A062319, A066959, A143773, A176029, A204688, A326358, A327527.

a009998 n k = (k + 1) ^ (n - k)
a009998_row n = a009998_tabl !! n
a009998_tabl = map reverse a009999_tabl
-- Reinhard Zumkeller, Feb 02 2014

E := (n,x) -> `if`(n=0,1,x*(1-x)*diff(E(n-1,x),x)+E(n-1,x)*(1+(n-1)*x));
G := (n,x) -> E(n,x)/(1-x)^(n+1);
A009998 := (n,k) -> coeff(series(G(n-k,x),x,18),x,k);
seq(print(seq(A009998(n,k),k=0..n)),n=0..6);
# Peter Luschny, Aug 02 2010

Flatten[Table[(j+1)^(i-j),{i,0,20},{j,0,i}]] (* Harvey P. Dale, Dec 25 2012 *)

T(i,j)=(j+1)^(i-j) \\ Charles R Greathouse IV, Feb 06 2017

T(n,n) = 1; T(n,k) = (k+1)*T(n-1,k) for k=0..n-1. - Reinhard Zumkeller, Feb 02 2014

T(n,m) = (m+1)*Sum_{k=0..n-m}((n+1)^(k-1)*(n-m)^(n-m-k)*(-1)^(n-m-k)*binomial(n-m-1,k-1)). - Vladimir Kruchinin, Sep 12 2015

a(62) corrected to 512 by T. D. Noe, Dec 20 2007

A109395 Denominator of phi(n)/n = Product_{p|n} (1 - 1/p); phi(n)=A000010(n), the Euler totient function.

Original entry on oeis.org

1, 2, 3, 2, 5, 3, 7, 2, 3, 5, 11, 3, 13, 7, 15, 2, 17, 3, 19, 5, 7, 11, 23, 3, 5, 13, 3, 7, 29, 15, 31, 2, 33, 17, 35, 3, 37, 19, 13, 5, 41, 7, 43, 11, 15, 23, 47, 3, 7, 5, 51, 13, 53, 3, 11, 7, 19, 29, 59, 15, 61, 31, 7, 2, 65, 33, 67, 17, 69, 35, 71, 3, 73, 37, 15, 19, 77, 13, 79, 5, 3

Offset: 1

Franz Vrabec, Aug 26 2005

a(n)=2 iff n=2^k (k>0); otherwise a(n) is odd. If p is prime, a(p)=p; the converse is false, e.g.: a(15)=15. It is remarkable that this sequence often coincides with A006530, the largest prime P dividing n. Theorem: a(n)=P if and only if for every prime p < P in n there is some prime q in n with p|(q-1). - Franz Vrabec, Aug 30 2005

			a(10) = 10/gcd(10,phi(10)) = 10/gcd(10,4) = 10/2 = 5.

Antti Karttunen, Table of n, a(n) for n = 1..16384 (terms 1..1000 from T. D. Noe)
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A076512 for the numerator.
Cf. A000010, A009195, A054741, A059956, A082695, A318304, A318305, A323170.
Phi(m)/m = k: A000079 \ {1} (k=1/2), A033845 (k=1/3), A000244 \ {1} (k=2/3), A033846 (k=2/5), A000351 \ {1} (k=4/5), A033847 (k=3/7), A033850 (k=4/7), A000420 \ {1} (k=6/7), A033848 (k=5/11), A001020 \ {1} (k=10/11), A288162 (k=6/13), A001022 \ {1} (12/13), A143207 (k=4/15), A033849 (k=8/15), A033851 (k=24/35).

Table[Denominator[EulerPhi[n]/n], {n, 81}] (* Alonso del Arte, Sep 03 2011 *)

a(n)=n/gcd(n,eulerphi(n)) \\ Charles R Greathouse IV, Feb 20 2013

A007947(n) = factorback(factorint(n)[, 1]); \\ From A007947
A173557(n) = my(f=factor(n)[, 1]); prod(k=1, #f, f[k]-1); \\ From A173557
A109395(n) = denominator(A173557(n)/A007947(n)); \\ Antti Karttunen, Feb 09 2019

a(n) = n/gcd(n, phi(n)) = n/A009195(n).
From Antti Karttunen, Feb 09 2019: (Start)
a(n) = denominator of A173557(n)/A007947(n).
a(2^n) = 2 for all n >= 1.
(End)
From Amiram Eldar, Jul 31 2020: (Start)
Asymptotic mean of phi(n)/n: lim_{m->oo} (1/m) * Sum_{n=1..m} A076512(n)/a(n) = 6/Pi^2 (A059956).
Asymptotic mean of n/phi(n): lim_{m->oo} (1/m) * Sum_{n=1..m} a(n)/A076512(n) = zeta(2)*zeta(3)/zeta(6) (A082695). (End)

A303428 Number of ways to write n as x(3x-2) + y(3y-2) + 3^u + 3^v, where x,y,u,v are integers with x <= y and 0 <= u <= v.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 2, 2, 1, 3, 3, 5, 2, 3, 3, 2, 2, 5, 4, 5, 2, 3, 5, 2, 3, 5, 4, 7, 2, 4, 5, 3, 4, 6, 4, 7, 3, 6, 6, 4, 4, 5, 5, 9, 5, 6, 6, 2, 5, 5, 7, 8, 4, 5, 4, 4, 4, 6, 6, 8, 3, 6, 6, 3, 4, 6, 7, 8, 5, 8, 6, 5, 4, 6, 7, 8, 6, 6, 6, 2

Offset: 1

Zhi-Wei Sun, Apr 23 2018

nonn

Conjecture: a(n) > 0 for all n > 1. Moreover, any integer n > 1 can be written as x*(3*x+2) + y*(3*y+2) + 3^z + 3^w, where x is an integer and y,z,w are nonnegative integers.
a(n) > 0 for all n = 2..3*10^8. Those x*(3*x-2) with x integral are called generalized octagonal numbers (A001082). 76683391 is the least integer n > 1 not representable as the sum of two generalized octagonal numbers and two powers of 2.
See also A303389, A303401 and A303432 for similar conjectures.

			a(2) = 1 with 2 = 0*(3*0-2) + 0*(3*0-2) + 3^0 + 3^0.
a(3) = 1 with 3 = 0*(3*0-2) + 1*(3*1-2) + 3^0 + 3^0.
a(4) = 2 with 4 = 1*(3*1-2) + 1*(3*1-2) + 3^0 + 3^0 = 0*(3*0-2) + 0*(3*0-2) + 3^0 + 3^1.
a(5) = 1 with 5 = 0*(3*0-2) + 1*(3*1-2) + 3^0 + 3^1.
a(9) = 1 with 9 = (-1)*(3*(-1)-2) + 0*(3*0-2) + 3^0 + 3^1.
a(4360) = 4 with 4360 = (-35)*(3*(-35)-2) + (-13)*(3*(-13)-2) + 3^0 + 3^4 = (-37)*(3*(-37)-2) + (-7)*(3*(-7)-2) + 3^2 + 3^2 = (-27)*(3*(-27)-2) + (-23)*(3*(-23)-2) + 3^5 + 3^5 = (-25)*(3*(-25)-2) + (-1)*(3*(-1)-2) + 3^5 + 3^7.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, A result similar to Lagrange's theorem, J. Number Theory 162(2016), 190-211.
Zhi-Wei Sun, Refining Lagrange's four-square theorem, J. Number Theory 175(2017), 167-190.
Zhi-Wei Sun, New conjectures on representations of integers (I), Nanjing Univ. J. Math. Biquarterly 34(2017), no. 2, 97-120.
Zhi-Wei Sun, Restricted sums of four squares, arXiv:1701.05868 [math.NT], 2017-2018.

Cf. A000244, A000567, A001082, A045944, A280472, A303233, A303338, A303363, A303389, A303393, A303399, A303401, A303432.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
f[n_]:=f[n]=FactorInteger[n];
g[n_]:=g[n]=Sum[Boole[Mod[Part[Part[f[n],i],1],4]==3&&Mod[Part[Part[f[n],i],2],2]==1],{i,1,Length[f[n]]}]==0;
QQ[n_]:=QQ[n]=(n==0)||(n>0&&g[n]);
tab={};Do[r=0;Do[If[QQ[3(n-3^j-3^k)+2],Do[If[SQ[3(n-3^j-3^k-x(3x-2))+1],r=r+1],{x,-Floor[(Sqrt[3(n-3^j-3^k)/2+1]-1)/3],(Sqrt[3(n-3^j-3^k)/2+1]+1)/3}]],
{j,0,Log[3,n/2]},{k,j,Log[3,n-3^j]}];tab=Append[tab,r],{n,1,80}];Print[tab]

A003992 Square array read by upwards antidiagonals: T(n,k) = n^k for n >= 0, k >= 0.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 4, 1, 0, 1, 4, 9, 8, 1, 0, 1, 5, 16, 27, 16, 1, 0, 1, 6, 25, 64, 81, 32, 1, 0, 1, 7, 36, 125, 256, 243, 64, 1, 0, 1, 8, 49, 216, 625, 1024, 729, 128, 1, 0, 1, 9, 64, 343, 1296, 3125, 4096, 2187, 256, 1, 0, 1, 10, 81, 512, 2401, 7776, 15625, 16384, 6561, 512, 1, 0

Offset: 0

Marc LeBrun

If the array is transposed, T(n,k) is the number of oriented rows of n colors using up to k different colors. The formula would be T(n,k) = [n==0] + [n>0]*k^n. The generating function for column k would be 1/(1-k*x). For T(3,2)=8, the rows are AAA, AAB, ABA, ABB, BAA, BAB, BBA, and BBB. - Robert A. Russell, Nov 08 2018

T(n,k) is the number of multichains of length n from {} to [k] in the Boolean lattice B_k. - Geoffrey Critzer, Apr 03 2020

			Rows begin:
[1, 0,  0,   0,    0,     0,      0,      0, ...],
[1, 1,  1,   1,    1,     1,      1,      1, ...],
[1, 2,  4,   8,   16,    32,     64,    128, ...],
[1, 3,  9,  27,   81,   243,    729,   2187, ...],
[1, 4, 16,  64,  256,  1024,   4096,  16384, ...],
[1, 5, 25, 125,  625,  3125,  15625,  78125, ...],
[1, 6, 36, 216, 1296,  7776,  46656, 279936, ...],
[1, 7, 49, 343, 2401, 16807, 117649, 823543, ...], ...

T. D. Noe, Rows n = 0..50 of triangle, flattened

Rows 0-49 are A000007, A000012, A000079, A000244, A000302, A000351, A000400, A000420, A001018, A001019, A011557, A001020, A001021, A001022, A001023, A001024, A001025, A001026, A001027, A001029, A009964-A009992, A087752.
Columns 0-26 are A000012, A001477, A000290, A000578, A000583, A000584, A001014, A001015, A001016, A001017, A008454, A008455, A008456, A010801-A010813, A089081.
Main diagonal is A000312. Other diagonals include A000169, A007778, A000272, A008788. Antidiagonal sums are in A026898.
Cf. A099555.
Transpose is A004248. See A051128, A095884, A009999 for other versions.
Cf. A277504 (unoriented), A293500 (chiral).

[[(n-k)^k: k in [0..n]]: n in [0..10]]; // G. C. Greubel, Nov 08 2018

Table[If[k == 0, 1, (n - k)^k], {n, 0, 11}, {k, 0, n}]//Flatten

T(n,k) = (n-k)^k \\ Charles R Greathouse IV, Feb 07 2017

E.g.f.: Sum T(n,k)*x^n*y^k/k! = 1/(1-x*exp(y)). - Paul D. Hanna, Oct 22 2004

E.g.f.: Sum T(n,k)*x^n/n!*y^k/k! = e^(x*e^y). - Franklin T. Adams-Watters, Jun 23 2006

More terms from David W. Wilson

Edited by Paul D. Hanna, Oct 22 2004

A011764 a(n) = 3^(2^n) (or: write in base 3, read in base 9).

Original entry on oeis.org

3, 9, 81, 6561, 43046721, 1853020188851841, 3433683820292512484657849089281, 11790184577738583171520872861412518665678211592275841109096961

Offset: 0

Stephan Y Solomon (ilans(AT)way.com)

a(n) is the second-highest value k such that A173419(k) = n+2. - Charles R Greathouse IV, Oct 03 2012
Let b(0) = 6; b(n+1) = smallest number such that b(n+1) + Product_{i=0..n} b(i) divides b(n+1)*Product_{i=0..n} b(i). Then b(n+1) = a(n) for n >= 0. - Derek Orr, Dec 13 2014
Changing "+" to "-": Let b(0) = 6; b(n+1) = smallest number such that b(n+1) - Product_{i=0..n} b(i) divides b(n+1)*Product_{i=0..n} b(i). Then b(n+2) = a(n) for n >= 0. - Derek Orr, Jan 04 2015
With offset = 1, a(n) is the number of collections C of subsets of {1,2,...,n} such that if S is in C then the complement of S is not in C. - Geoffrey Critzer, Feb 06 2017

Vincenzo Librandi, Table of n, a(n) for n = 0..11

Cf. A001146, A078885, A176594.

Subsequence of A000244 (powers of 3).

[3^(2^n): n in [0..8]]; // Vincenzo Librandi, Sep 15 2011

3^(2^Range[0,10]) (* Harvey P. Dale, Oct 14 2012 *)

a(n)=3^2^n \\ Charles R Greathouse IV, Oct 03 2012

def A011764(n): return 3**(1<Chai Wah Wu, Oct 09 2024

a(0) = 3 and a(n+1) = a(n)^2. - Benoit Jubin, Jun 27 2009
Sum_{n>=0} 1/a(n) = A078885. - Amiram Eldar, Nov 09 2020
Product_{n>=0} (1 + 1/a(n)) = 3/2. - Amiram Eldar, Jan 29 2021
a(n) = A000244(A000079(n)), or A011764 = A000244 o A000079. - M. F. Hasler, Jul 20 2023

A015523 a(n) = 3a(n-1) + 5a(n-2), with a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 3, 14, 57, 241, 1008, 4229, 17727, 74326, 311613, 1306469, 5477472, 22964761, 96281643, 403668734, 1692414417, 7095586921, 29748832848, 124724433149, 522917463687, 2192374556806, 9191710988853, 38537005750589

Offset: 0

Olivier Gérard

From Johannes W. Meijer, Aug 01 2010: (Start)
a(n) represents the number of n-move routes of a fairy chess piece starting in a given corner square (m = 1, 3, 7 and 9) on a 3 X 3 chessboard. This fairy chess piece behaves like a king on the eight side and corner squares but on the central square the king goes crazy and turns into a red king, see A179596.
For n >= 1, the sequence above corresponds to 24 red king vectors, i.e., A[5] vectors, with decimal values 27, 30, 51, 54, 57, 60, 90, 114, 120, 147, 150, 153, 156, 177, 180, 210, 216, 240, 282, 306, 312, 402, 408 and 432. These vectors lead for the side squares to A152187 and for the central square to A179606.
This sequence belongs to a family of sequences with g.f. 1/(1-3*x-k*x^2). Red king sequences that are members of this family are A007482 (k=2), A015521 (k=4), A015523 (k=5; this sequence), A083858 (k=6), A015524 (k=7) and A015525 (k=8). We observe that there is no red king sequence for k=3. Other members of this family are A049072 (k=-4), A057083 (k=-3), A000225 (k=-2), A001906 (k=-1), A000244 (k=0), A006190 (k=1), A030195 (k=3), A099012 (k=9), A015528 (k=10) and A015529 (k=11).
Inverse binomial transform of A052918 (with extra leading 0).
(End)
First differences in A197189. - Bruno Berselli, Oct 11 2011
Pisano period lengths:  1, 3, 4, 6, 4, 12, 3, 12, 12, 12, 120, 12, 12, 3, 4, 24, 288, 12, 72, 12, ... - R. J. Mathar, Aug 10 2012
This is the Lucas U(P=3, Q=-5) sequence, and hence for n >= 0, a(n+2)/a(n+1) equals the continued fraction 3 + 5/(3 + 5/(3 + 5/(3 + ... + 5/3))) with n 5's. - Greg Dresden, Oct 06 2019

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,5).

Cf. A072263, A072264, A152187, A179606, A197189.

[ n eq 1 select 0 else n eq 2 select 1 else 3*Self(n-1)+5*Self(n-2): n in [1..30] ]; // Vincenzo Librandi, Aug 23 2011

Join[{a = 0, b = 1}, Table[c = 3 * b + 5 * a; a = b; b = c, {n, 100}]] (* Vladimir Joseph Stephan Orlovsky, Jan 16 2011 *)
a[0] := 0; a[1] := 1; a[n_] := a[n] = 3a[n - 1] + 5a[n - 2]; Table[a[n], {n, 0, 49}] (* Alonso del Arte, Jan 16 2011 *)

x='x+O('x^30); concat([0], Vec(x/(1-3*x-5*x^2))) \\ G. C. Greubel, Jan 01 2018

[lucas_number1(n,3,-5) for n in range(0, 24)] # Zerinvary Lajos, Apr 22 2009

a(n) = 3*a(n-1) + 5*a(n-2).
From Paul Barry, Jul 20 2004: (Start)
a(n) = ((3/2 + sqrt(29)/2)^n - (3/2 - sqrt(29)/2)^n)/sqrt(29).
a(n) = Sum_{k=0..floor((n-1)/2)} binomial(n-k-1,k)*5^k*3^(n-2*k-1). (End)
G.f.: x/(1 - 3*x - 5*x^2). - R. J. Mathar, Nov 16 2007
From Johannes W. Meijer, Aug 01 2010: (Start)
Limit_{k->oo} a(n+k)/a(k) = (A072263(n) + a(n)*sqrt(29))/2.
Limit_{n->oo} A072263(n)/a(n) = sqrt(29). (End)
G.f.: G(0)*x/(2-3*x), where G(k) = 1 + 1/(1 - x*(29*k-9)/(x*(29*k+20) - 6/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 17 2013
E.g.f.: 2*exp(3*x/2)*sinh(sqrt(29)*x/2)/sqrt(29). - Stefano Spezia, Oct 06 2019

A055643 Babylonian numbers: integers in base 60 with each sexagesimal digit represented by 2 decimal digits, leading zeros omitted.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110

Offset: 0

Henry Bottomley, Jun 06 2000

From Wolfdieter Lang, Jan 16 2018: (Start)
The symbols used for 0..9 in this base 60 notation are 00, 01, ..., 09, but leading zeros are omitted.
For the Sumerian-Babylonian sexagesimal-decimal number system which uses two positions for each base-60 position filled with only one-digit numbers alternating between ranges of 0 to 9 and 0 to 5 see the link below.
(End)
For n < 1440, US and NATO military time designation of n minutes since midnight. - J. Lowell, Dec 29 2020

Mohammad K. Azarian, Meftah al-hesab: A Summary, MJMS, Vol. 12, No. 2, Spring 2000, pp. 75-95. Mathematical Reviews, MR 1 764 526. Zentralblatt MATH, Zbl 1036.01002.
Mohammad K. Azarian, A Summary of Mathematical Works of Ghiyath ud-din Jamshid Kashani, Journal of Recreational Mathematics, Vol. 29(1), pp. 32-42, 1998.
Georges Ifrah, Histoire Universelle des Chiffres, Paris, 1981.
Georges Ifrah, From one to zero, A universal history of numbers, Viking Penguin Inc., 1985.
Georges Ifrah, Universalgeschichte der Zahlen, Campus Verlag, Frankfurt, New York, 2. Auflage, 1987, pp. 210-221.

Michael De Vlieger, Table of n, a(n) for n = 0..10000
Wolfdieter Lang, Sumerian-Babylonian sexagesimal-decimal number system.

Cf. A049872, A131650,
Note also that A250073 = a(A000079(n)), A250089 = a(A051037(n)), A254334 = a(A000244(n)), A254335 = a(A000351(n)), A254336 = a(A011557(n)).
See also A281863 (value of the 0,1,2,...,n-th digit of a(n), counted from the right), A282622 (length of a(n), #digits, for n >= 1).

Array[FromDigits@ Apply[Join, PadLeft[#, 2] & /@ IntegerDigits@ IntegerDigits[#, 60]] &, 71, 0] (* Michael De Vlieger, Jan 11 2018 *)

A055643(n)=fromdigits(digits(n,60),100) \\ M. F. Hasler, Jan 09 2018

def a(n): return n if n < 60 else 100*a(n//60) + n%60
print([a(n) for n in range(71)]) # Michael S. Branicky, Oct 22 2022

a(60*n+r) = 100*a(n) + r, 0 <= r <= 59. - Jianing Song, Oct 22 2022

a(69) and a(70) from WG Zeist, Sep 08 2012

A076264 Number of ternary (0,1,2) sequences without a consecutive '012'.

Original entry on oeis.org

1, 3, 9, 26, 75, 216, 622, 1791, 5157, 14849, 42756, 123111, 354484, 1020696, 2938977, 8462447, 24366645, 70160958, 202020427, 581694636, 1674922950, 4822748423, 13886550633, 39984728949, 115131438424, 331507764639

Offset: 0

John L. Drost, Nov 05 2002

A transform of A000244 under the mapping g(x)->(1/(1+x^3))g(x/(1+x^3)). - Paul Barry, Oct 20 2004
b(n) := (-1)^n*a(n) appears in the formula for the nonpositive powers of rho(9) := 2*cos(Pi/9), when written in the power basis of the algebraic number field Q(rho(9)) of degree 3. See A187360 for the minimal polynomial C(9, x) of rho(9), and a link to the Q(2*cos(pi/n)) paper. 1/rho(9) = -3*1 + 0*rho(9) + 1*rho(9)^2 (see A230079, row n=5). 1/rho(9)^n = b(n)*1 + b(n-2)*rho(9) + b(n-1)*rho(9)^2, n >= 0, with b(-1) = 0 = b(-2). - Wolfdieter Lang, Nov 04 2013
The limit b(n+1)/b(n) = -a(n+1)/a(n) for n -> infinity is -tau(9) := -(1 + rho(9)) = 1/(2*cos(Pi*5/9)), approximately -2.445622407. tau(9) is known to be the length ratio (longest diagonal)/side in the regular 9-gon. This limit follows from the b(n)-recurrence and the solutions of X^3 + 3*X^2 - 1 = 0, which are given by the inverse of the known solutions of the minimal polynomial C(9, x) of rho(9) (see A187360). The other two X solutions are 1/rho(9) = -3 + rho(9)^2, approximately 0.5320888860 and 1/(2*cos(Pi*7/9)) = 1 + rho(9) - rho(9)^2, approximately -0.6527036445, and they are therefore irrelevant for this sequence. - Wolfdieter Lang, Nov 08 2013
a(n) is also the number of ternary (0,1,2) sequences of length n without a consecutive '110' because the patterns A=012 and B=110 have the same autocorrelation, i.e., AA=100=BB, in the sense of Guibas and Odlysko (1981). (A cyclic version of this sequence can be found in sequence A274018.) - Petros Hadjicostas, Sep 12 2017

			1/rho(9)^3 = -26*1 - 3*rho(9) + 9*rho(9)^2, (approximately 0.15064426) with rho(9) given in the Nov 04 2013 comment above. - _Wolfdieter Lang_, Nov 04 2013
G.f. = 1 + 3*x + 9*x^2 + 26*x^3 + 75*x^4 + 216*x^5 + 622*x^6 + 1791*x^7 + ...

A. Tucker, Applied Combinatorics, 4th ed. p. 277

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Taras Goy and Mark Shattuck, Toeplitz-Hessenberg determinant formulas for the sequence F_n-1, Online J. Anal. Comb. (2025) Vol. 19, Paper 1, 1-26. See p. 12.
Leo J. Guibas and Andrew M. Odlyzko, String overlaps, pattern matching, and nontransitive games, J. Combin. Theory Ser. A 30 (1981), 183-208. [Comment: The authors use generating functions in terms of z^{-1}. To get the g.f. of the sequence, as shown below in the FORMULA section, let x=z^{-1} and perform simple algebra. There are some minor typos in Theorem 2.1, p. 191, that can be easily corrected by looking at the proof. - _Petros Hadjicostas_, Sep 12 2017]
Yun-Tak Oh, Hosho Katsura, Hyun-Yong Lee, and Jung Hoon Han, Proposal of a spin-one chain model with competing dimer and trimer interactions, arXiv:1709.01344 [cond-mat.str-el], 2017.
Index entries for linear recurrences with constant coefficients, signature (3,0,-1).

The g.f. corresponds to row 3 of triangle A225682.

List([0..25],n->Sum([0..Int(n/3)],k->Binomial(n-2*k,k)*(-1)^k*3^(n-3*k))); # Muniru A Asiru, Feb 20 2018

LinearRecurrence[{3,0,-1},{1,3,9},30] (* Harvey P. Dale, Feb 28 2016 *)

{a(n) = if( n<0, 0, polcoeff( 1 / (1 - 3*x + x^3) + x * O(x^n), n))};

a(n) is asymptotic to g*c^n where c = cos(Pi/18)/cos(7*Pi/18) and g is the largest real root of 81*x^3 - 81*x^2 - 9*x + 1 = 0. - Benoit Cloitre, Nov 06 2002
G.f.: 1/(1 - 3x + x^3).
a(n) = 3*a(n-1) - a(n-3), n > 0.
a(n) = Sum_{k=0..floor(n/3)} binomial(n-2k, k)(-1)^k*3^(n-3k). - Paul Barry, Oct 20 2004
a(n) = middle term in M^(n+1) * [1 0 0], where M = the 3 X 3 matrix [2 1 1 / 1 1 0 / 1 0 0]. Right term = A052536(n), left term = A052536(n+1). - Gary W. Adamson, Sep 05 2005

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A159991 Powers of 60: a(n) = 60^n.

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A334774 Triangle read by rows: T(n,k) is the number of permutations of 2 indistinguishable copies of 1..n with exactly k local maxima.

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A009998 Triangle in which j-th entry in i-th row is (j+1)^(i-j).

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A109395 Denominator of phi(n)/n = Product_{p|n} (1 - 1/p); phi(n)=A000010(n), the Euler totient function.

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A303428 Number of ways to write n as x*(3*x-2) + y*(3*y-2) + 3^u + 3^v, where x,y,u,v are integers with x <= y and 0 <= u <= v.

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A003992 Square array read by upwards antidiagonals: T(n,k) = n^k for n >= 0, k >= 0.

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A303428 Number of ways to write n as x(3x-2) + y(3y-2) + 3^u + 3^v, where x,y,u,v are integers with x <= y and 0 <= u <= v.

A015523 a(n) = 3a(n-1) + 5a(n-2), with a(0)=0, a(1)=1.