A009965 -id:A009965 - cp's OEIS frontend

A218724 a(n) = (21^n - 1)/20.

0, 1, 22, 463, 9724, 204205, 4288306, 90054427, 1891142968, 39714002329, 833994048910, 17513875027111, 367791375569332, 7723618886955973, 162195996626075434, 3406115929147584115, 71528434512099266416, 1502097124754084594737, 31544039619835776489478

Offset: 0

M. F. Hasler, Nov 04 2012

Partial sums of powers of 21 (A009965); q-integers for q=21: diagonal k=1 in triangle A022185.
Partial sums are in A014905. Also, the sequence is related to A014938 by A014938(n) = n*a(n) - Sum_{i=0..n-1} a(i) for n > 0. - Bruno Berselli, Nov 06 2012
For n >= 1, 4*a(n) is the total number of holes in a certain box fractal (start with 21 boxes, 4 holes) after n iterations. See illustration in links. - Kival Ngaokrajang, Jan 27 2015

Vincenzo Librandi, Table of n, a(n) for n = 0..700
Kival Ngaokrajang, Illustration of initial terms
Index entries related to partial sums
Index entries for linear recurrences with constant coefficients, signature (22,-21).

Cf. similar sequences of the form (k^n-1)/(k-1): A000225, A003462, A002450, A003463, A003464, A023000, A023001, A002452, A002275, A016123, A016125, A091030, A135519, A135518, A131865, A091045, A218721, A218722, A064108, A218725-A218734, A132469, A218736-A218753, A133853, A094028, A218723.

Cf. A009965, A014905, A014938, A022185.

[n le 2 select n-1 else 22*Self(n-1) - 21*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 07 2012

LinearRecurrence[{22, -21}, {0, 1}, 40] (* Vincenzo Librandi, Nov 07 2012 *)

A218724(n):=(21^n-1)/20$ makelist(A218724(n),n,0,30); /* Martin Ettl, Nov 05 2012 */

PARI
```
A218724(n)=21^n\20
```

a(n) = floor(21^n/20).
G.f.: x/((1-x)*(1-21*x)). - Bruno Berselli, Nov 06 2012
a(n) = 22*a(n-1) - 21*a(n-2). - Vincenzo Librandi, Nov 07 2012
a(n) = 21*a(n-1) + 1. - Kival Ngaokrajang, Jan 27 2015
a(n) = a(n-1) + 21^(n-1), n >= 1, a(0) = 0.  - Wolfdieter Lang, Feb 02 2015
E.g.f.: exp(11*x)*sinh(10*x)/10. - Elmo R. Oliveira, Aug 29 2024

A329332 Table of powers of squarefree numbers, powers of A019565(n) in increasing order in row n. Square array A(n,k) n >= 0, k >= 0 read by descending antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 8, 9, 6, 1, 1, 16, 27, 36, 5, 1, 1, 32, 81, 216, 25, 10, 1, 1, 64, 243, 1296, 125, 100, 15, 1, 1, 128, 729, 7776, 625, 1000, 225, 30, 1, 1, 256, 2187, 46656, 3125, 10000, 3375, 900, 7, 1, 1, 512, 6561, 279936, 15625, 100000, 50625, 27000, 49, 14

Offset: 0

Peter Munn, Nov 10 2019

The A019565 row order gives the table neat relationships with A003961, A003987, A059897, A225546, A319075 and A329050. See the formula section.

Transposition of this table, that is reflection about its main diagonal, has subtle symmetries. For example, consider the unique factorization of a number into powers of distinct primes. This can be restated as factorization into numbers from rows 2^n (n >= 0) with no more than one from each row. Reflecting about the main diagonal, this factorization becomes factorization (of a related number) into numbers from columns 2^k (k >= 0) with no more than one from each column. This is also unique and is factorization into powers of squarefree numbers with distinct exponents that are powers of two. See the example section.

			Square array A(n,k) begins:
n\k |  0   1     2      3        4          5           6             7
----+------------------------------------------------------------------
   0|  1   1     1      1        1          1           1             1
   1|  1   2     4      8       16         32          64           128
   2|  1   3     9     27       81        243         729          2187
   3|  1   6    36    216     1296       7776       46656        279936
   4|  1   5    25    125      625       3125       15625         78125
   5|  1  10   100   1000    10000     100000     1000000      10000000
   6|  1  15   225   3375    50625     759375    11390625     170859375
   7|  1  30   900  27000   810000   24300000   729000000   21870000000
   8|  1   7    49    343     2401      16807      117649        823543
   9|  1  14   196   2744    38416     537824     7529536     105413504
  10|  1  21   441   9261   194481    4084101    85766121    1801088541
  11|  1  42  1764  74088  3111696  130691232  5489031744  230539333248
  12|  1  35  1225  42875  1500625   52521875  1838265625   64339296875
Reflection of factorization about the main diagonal: (Start)
The canonical (prime power) factorization of 864 is 2^5 * 3^3 = 32 * 27. Reflecting the factors about the main diagonal of the table gives us 10 * 36 = 10^1 * 6^2 = 360. This is the unique factorization of 360 into powers of squarefree numbers with distinct exponents that are powers of two.
Reflection about the main diagonal is given by the self-inverse function A225546(.). Clearly, all positive integers are in the domain of A225546, whether or not they appear in the table. It is valid to start from 360, observe that A225546(360) = 864, then use 864 to derive 360's factorization into appropriate powers of squarefree numbers as above.
(End)

The range of values is A072774.
Rows (abbreviated list): A000079(1), A000244(2), A000400(3), A000351(4), A011557(5), A001024(6), A009974(7), A000420(8), A001023(9), A009965(10), A001020(16), A001022(32), A001026(64).
A019565 is column 1, A334110 is column 2, and columns that are sorted in increasing order (some without the 1) are: A005117(1), A062503(2), A062838(3), A113849(4), A113850(5), A113851(6), A113852(7).
Other subtables: A182944, A319075, A329050.
Re-ordered subtable of A297845, A306697, A329329.
A000290, A003961, A003987, A059897 and A225546 are used to express relationships between terms of this sequence.
Cf. A285322.

A(n,k) = A019565(n)^k.
A(k,n) = A225546(A(n,k)).
A(n,2k) = A000290(A(n,k)) = A(n,k)^2.
A(2n,k) = A003961(A(n,k)).
A(n,2k+1) = A(n,2k) * A(n,1).
A(2n+1,k) = A(2n,k) * A(1,k).
A(A003987(n,m), k) = A059897(A(n,k), A(m,k)).
A(n, A003987(m,k)) = A059897(A(n,m), A(n,k)).
A(2^n,k) = A319075(k,n+1).
A(2^n, 2^k) = A329050(n,k).
A(n,k) = A297845(A(n,1), A(1,k)) = A306697(A(n,1), A(1,k)), = A329329(A(n,1), A(1,k)).
Sum_{n>=0} 1/A(n,k) = zeta(k)/zeta(2*k), for k >= 2. - Amiram Eldar, Dec 03 2022

A072978 Numbers of the form m*2^Omega(m), where m>1 is odd and Omega(m)=A001222(m), the number of prime factors of m.

Original entry on oeis.org

1, 6, 10, 14, 22, 26, 34, 36, 38, 46, 58, 60, 62, 74, 82, 84, 86, 94, 100, 106, 118, 122, 132, 134, 140, 142, 146, 156, 158, 166, 178, 194, 196, 202, 204, 206, 214, 216, 218, 220, 226, 228, 254, 260, 262, 274, 276, 278, 298, 302, 308, 314, 326, 334, 340, 346

Offset: 1

Reinhard Zumkeller, Aug 20 2002

(number of odd prime factors) = (number of even prime factors).
A000400, A011557, A001023, A001024, A009965, A009966 and A009975 are subsequences. - Reinhard Zumkeller, Jan 06 2008
Subsequence of A028260. - Reinhard Zumkeller, Sep 20 2008

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

Cf. A001222, A007814, A087436.

Cf. A000400, A011557, A001023, A001024, A009965, A009966, A009975, A028260.

Join[{1}, Select[Range[2, 500, 2], First[#] == Total[Rest[#]] & [FactorInteger[#][[All, 2]]] &]] (* Paolo Xausa, Feb 19 2025 *)

isok(k) = {my(v = valuation(k, 2)); bigomega(k >> v) == v;} \\ Amiram Eldar, May 15 2025

from math import prod, isqrt
from sympy import primerange, integer_nthroot, primepi
def A072978(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def g(x,a,b,c,m): yield from (((d,) for d in enumerate(primerange(b,isqrt(x//c)+1),a)) if m==2 else (((a2,b2),)+d for a2,b2 in enumerate(primerange(b,integer_nthroot(x//c,m)[0]+1),a) for d in g(x,a2,b2,c*b2,m-1)))
    def h(x,n): return sum(primepi(x//prod(c[1] for c in a))-a[-1][0] for a in g(x,1,3,1,n))
    def f(x): return int(n+x-primepi(x>>1)-sum(h(x>>m,m) for m in range(2,x.bit_length()+1))) if x>1 else 1
    return bisection(f,n,n) # Chai Wah Wu, Apr 10 2025

A007814(a(n)) = A087436(a(n)). - Reinhard Zumkeller, Jan 06 2008

A222281 T(n,k) = number of n X k 0..5 arrays with no entry increasing mod 6 by 5 rightwards or downwards, starting with upper left zero.

Original entry on oeis.org

1, 5, 5, 25, 105, 25, 125, 2205, 2205, 125, 625, 46305, 194485, 46305, 625, 3125, 972405, 17153945, 17153945, 972405, 3125, 15625, 20420505, 1513010465, 6354787485, 1513010465, 20420505, 15625, 78125, 428830605, 133450391205

Offset: 1

R. H. Hardin, Feb 14 2013

1/6 the number of 6-colorings of the grid graph P_n X P_k. - Andrew Howroyd, Jun 26 2017

			Table starts
........1................5......................25..........................125
........5..............105....................2205........................46305
.......25.............2205..................194485.....................17153945
......125............46305................17153945...................6354787485
......625...........972405..............1513010465................2354171487645
.....3125.........20420505............133450391205..............872117822449905
....15625........428830605..........11770577485085...........323081602357856985
....78125.......9005442705........1038187247574145........119687637492011211885
...390625.....189114296805.......91570083319317865......44339047670574481807485
..1953125....3971400232905.....8076654937439905005...16425682631297501047982145
..9765625...83399404891005...712376276332499775685.6084998755694142903356375385
.48828125.1751387502711105.62832938018547611186345
...
Some solutions for n=3, k=4:
..0..0..0..0....0..0..0..0....0..0..0..0....0..3..0..0....0..0..0..0
..4..2..0..1....1..2..0..4....0..0..0..1....0..0..3..1....0..2..3..0
..0..4..1..4....1..4..1..2....3..4..4..1....3..0..4..4....4..5..1..3

Andrew Howroyd, Table of n, a(n) for n = 1..325 (terms 1..111 from R. H. Hardin)

Columns 1-7 are A000351(n-1), 5*A009965(n-1), A222276, A222277, A222278, A222279, A222280.
Main diagonal is A068256.
Cf. A078099 (3 colorings), A222444 (4 colorings), A222144 (5 colorings), A198982 (unlabeled 6 colorings), A222340 (7 colorings), A222462 (8 colorings).

T(n, k) = 5 * (24*A198982(n,k) - 12*A198715(n,k) - 8*A207997(n,k) - 3) for n*k > 1. - Andrew Howroyd, Jun 27 2017

A105317 Powers of Fibonacci numbers.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 8, 9, 13, 16, 21, 25, 27, 32, 34, 55, 64, 81, 89, 125, 128, 144, 169, 233, 243, 256, 377, 441, 512, 610, 625, 729, 987, 1024, 1156, 1597, 2048, 2187, 2197, 2584, 3025, 3125, 4096, 4181, 6561, 6765, 7921, 8192, 9261, 10946, 15625, 16384, 17711

Offset: 1

Reinhard Zumkeller, Apr 25 2005

nonn

The subset of nontrivial Fibonacci powers [numbers A000045(k)^n which are not in A000045] starts 4, 9, 16, 25, 27, 32, 64, 81, 125, 128, 169, 243, 256, 441, 512, 625, 729, 1024, 1156... - R. J. Mathar, Jan 26 2015. These are the initial terms of A254719. - Reinhard Zumkeller, Feb 06 2015

			2197 = 13^3 = A000045(7)^3, therefore 2197 is a term.

R. Zumkeller, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Eric Weisstein's World of Mathematics, Fibonacci Number

Subsequences: A056570-A056574, A007598, A000045, A000079, A000244, A000351, A001018, A001022 and A009965.

Cf. A103323, A254719.

import Data.Set (singleton, deleteFindMin, insert)
a105317 n = a105317_list !! (n-1)
a105317_list = 0 : 1 : h 1 (drop 4 a000045_list) (singleton (2, 2)) where
  h y xs'@(x:xs) s
    | x < ff    = h y xs (insert (x, x) s)
    | ff == y   = h y xs' s'
    | otherwise = ff : h ff xs' (insert (f * ff, f) s')
    where ((ff, f), s') = deleteFindMin s
-- Reinhard Zumkeller, Feb 06 2015

N:= 10^6: # to get all terms <= N
select(`<=`,{0,1,seq(seq(combinat:-fibonacci(i)^j, i = 3 ..floor(log[phi](sqrt(5)*N^(1/j)+1))),j=1..ilog2(N))},N);
# if using Maple 11 or earlier, uncomment the next line
# sort(convert(%,list)); # Robert Israel, Jan 26 2015

lim = 10^5; t = Table[f = Fibonacci[n]; f^Range[Floor[Log[lim]/Log[f]]], {n, 3, Ceiling[Log[GoldenRatio, lim] + 1]}]; Union[{0, 1}, Flatten[t]] (* T. D. Noe, Sep 27 2011 *)

list(lim)=my(v=List([0]),k=1,f,t); while(k<=lim, listput(v,k); k*=2); k=3; while(k<=lim, listput(v,k); k*=3); k=5; while(k<=lim, listput(v,k); k*=5); k=6; while((f=fibonacci(k++))<=lim, t=1; while((t*=f)<=lim, listput(v,t))); Set(v) \\ Charles R Greathouse IV, Oct 03 2016

A009986 Powers of 42.

Original entry on oeis.org

1, 42, 1764, 74088, 3111696, 130691232, 5489031744, 230539333248, 9682651996416, 406671383849472, 17080198121677824, 717368321110468608, 30129469486639681536, 1265437718438866624512, 53148384174432398229504, 2232232135326160725639168, 93753749683698750476845056

Offset: 0

N. J. A. Sloane

Same as Pisot sequences E(1, 42), L(1, 42), P(1, 42), T(1, 42). Essentially same as Pisot sequences E(42, 1764), L(42, 1764), P(42, 1764), T(42, 1764). See A008776 for definitions of Pisot sequences.

The compositions of n in which each natural number is colored by one of p different colors are called p-colored compositions of n. For n >= 1, a(n) equals the number of 42-colored compositions of n such that no adjacent parts have the same color. - Milan Janjic, Nov 17 2011

T. D. Noe, Table of n, a(n) for n = 0..100
Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (42).

Cf. A000079, A000400, A000420, A008776, A009965.

[42^n: n in [0..20]]; // Vincenzo Librandi, Nov 21 2010

42^Range[0, 14] (* Michael De Vlieger, Jan 13 2018 *)

a(n) = 42^n; \\ Michel Marcus, Jan 14 2018

G.f.: 1/(1-42*x). - Philippe Deléham, Nov 24 2008
a(n) = 42^n; a(n) = 42*a(n-1), a(0)=1. - Vincenzo Librandi, Nov 21 2010
From Elmo R. Oliveira, Jul 10 2025: (Start)
E.g.f.: exp(42*x).
a(n) = A000079(n)*A009965(n) = A000400(n)*A000420(n). (End)

A067422 Seventh column of triangle A067417.

Original entry on oeis.org

1, 9, 189, 3969, 83349, 1750329, 36756909, 771895089, 16209796869, 340405734249, 7148520419229, 150118928803809, 3152497504879989, 66202447602479769, 1390251399652075149, 29195279392693578129

Offset: 0

Wolfdieter Lang, Jan 25 2002

Vincenzo Librandi, Table of n, a(n) for n = 0..700
Index entries for linear recurrences with constant coefficients, signature (21).

Cf. A067421 (sixth column), A067423 (eighth column), A009965 (powers of 21).

[Ceiling(9*(3*7)^(n-1)): n in [0..20]]; // Vincenzo Librandi, Oct 02 2011

a(n) = A067417(n+6, 6).
a(n) = 9*(3*7)^(n-1), n >= 1, a(0)=1.
G.f.: (1-12*x)/(1-21*x).

A013726 a(n) = 21^(2*n + 1).

Original entry on oeis.org

21, 9261, 4084101, 1801088541, 794280046581, 350277500542221, 154472377739119461, 68122318582951682301, 30041942495081691894741, 13248496640331026125580781, 5842587018385982521381124421, 2576580875108218291929075869661, 1136272165922724266740722458520501

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (441).

Bisection of A009965 (21^n).

Cf. A005408, A013708-A013725, A013727-A013729.

[21^(2*n+1): n in [0..15]]; // Vincenzo Librandi, Jun 26 2011

seq(21^(2*n+1),n=0..10); # Nathaniel Johnston, Jun 25 2011

Table[21^(2 n + 1), {n, 0, 15}] (* Wesley Ivan Hurt, Feb 10 2014 *)

a(n)=21^(2*n+1) \\ Charles R Greathouse IV, Jul 11 2016

From Philippe Deléham, Nov 28 2008: (Start)
a(n) = 441*a(n-1); a(0)=21.
G.f.: 21/(1-441*x). (End)
a(n) = A009965(A005408(n)). - Wesley Ivan Hurt, Feb 10 2014
From Elmo R. Oliveira, Jul 10 2025: (Start)
E.g.f.: 21*exp(441*x).
a(n) = A013708(n)*A013712(n). (End)

A100401 Digital root of 3^n.

Original entry on oeis.org

1, 3, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9

Offset: 0

Cino Hilliard, Dec 30 2004

This sequence also gives the digital root of 12^n, 21^n, 30^n, 39^n, 48^n, 57^n, ... (any k^n where k is congruent to 3 mod 9). - Timothy L. Tiffin, Dec 02 2023

			For n=14, the digits of 3^14 = 4782969 sum to 45, whose digits sum to 9. So, a(14) = 9.

Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A000244, A001021, A007953, A009965, A009974, A009983, A009992, A010888, A100403, A225374.

Table[PowerMod[3, n, 18], {n, 0, 100}] (* Timothy L. Tiffin, Dec 03 2023 *)
PadRight[{1,3}, 100, 9] (* Timothy L. Tiffin, Dec 03 2023 *)

a(n) = if( n<2, [1,3][n+1], 9); \\ Joerg Arndt, Dec 03 2023

[power_mod(3,n,18) for n in range(105)] # Zerinvary Lajos, Nov 25 2009

a(n) = 3^n mod 18. - Zerinvary Lajos, Nov 25 2009
From Timothy L. Tiffin, Nov 30 2023: (Start)
a(n) = 9 for n >= 2.
G.f.: (1+2x+6x^2)/(1-x).
a(n) = A100403(n) for n <> 1. (End)
a(n) = A010888(A000244(n)). - Michel Marcus, Dec 01 2023
a(n) = A010888(A001021(n)) = A010888(A009965(n)) = A010888(A009974(n)) = A010888(A009983(n)) = A010888(A009992(n)) = A010888(A225374(n)). - Timothy L. Tiffin, Dec 02 2023
E.g.f.: 9*exp(x) - 6*x - 8. - Elmo R. Oliveira, Aug 08 2024
a(n) = A007953(3*a(n-1)) = A010888(3*a(n-1)). - Stefano Spezia, Mar 20 2025

A319092 Triangle read by rows: T(0,0) = 1; T(n,k) = T(n-1, k) + 2T(n-1, k-1) + 3T(n-1, k-2) + 4T(n-1, k-3) + 5T(n-1, k-4) + 6T(n-1, k-5) for k = 0..5n; T(n,k)=0 for n or k < 0.

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 6, 1, 4, 10, 20, 35, 56, 70, 76, 73, 60, 36, 1, 6, 21, 56, 126, 252, 441, 684, 954, 1204, 1365, 1344, 1169, 882, 540, 216, 1, 8, 36, 120, 330, 792, 1688, 3232, 5619, 8944, 13088, 17568, 21642, 24456, 25236, 23528, 19489, 14232, 8856, 4320, 1296, 1, 10, 55, 220, 715, 2002, 4970

Offset: 0

Shara Lalo, Oct 01 2018

Row n gives the coefficients in the expansion of (1 + 2*x + 3*x^2 + 4*x^3 + 5*x^4 + 6*x^5)^n, where n is a nonnegative integer.
The row sum is s(n)=21^n (see A009965).
In the center-justified triangle, the sum of numbers along "first layer" skew diagonals pointing top-right are the coefficients in the expansion of 1/(1 - x - 2*x^2 -3*x^3 - 4*x^4 - 5*x^5 - 6*x^6) and the sum of numbers along "first layer" skew diagonals pointing top-left are the coefficients in the expansion of 1/(1 - 6*x - 5*x^2 - 4*x^3 - 3*x^4 - 2*x^5 - x^6), see links.

			Triangle begins:
1;
1, 2,  3,  4,   5,   6;
1, 4, 10, 20,  35,  56,  70,  76,  73,  60,    36;
1, 6, 21, 56, 126, 252, 441, 684, 954, 1204, 1365, 1344, 1169, 882, 540, 216;
...

Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3.

Cf. A009965, A319093, A319094, A319095.

t[n_, k_] := t[n, k] = Sum[(2^(k + q - 2*r)*3^(j + r - 2*q)*4^(i + q - 2*j)*5^(j - 2*i)*6^i*n!)/((n - k + r)!*(k + q - 2*r)!*(j + r - 2*q)!*(i + q -2*j)!*(j - 2*i)!*i!), {i, 0, k}, {j, 2*i, k}, {q, 3*i, k}, {r, 4*i, k}]; Flatten[Table[t[n, k], {n, 0, 5}, {k, 0, 5 n}]]
t[0, 0] = 1; t[n_, k_] :=  t[n, k] =   If[n < 0 || k < 0, 0, t[n - 1, k] + 2 t[n - 1, k - 1] + 3 t[n - 1, k - 2] + 4 t[n - 1, k - 3] + 5 t[n - 1, k - 4] + 6 t[n - 1, k - 5]]; Table[t[n, k], {n, 0, 5}, {k, 0, 5 n}  ]  // Flatten

row(n) = Vecrev((1+2*x+3*x^2+4*x^3+5*x^4+6*x^5)^n);
tabl(nn) = for (n=0, nn, print(row(n))); \\ Michel Marcus, Oct 15 2018

T(n,k) = Sum_{i=0..k} Sum_{j=2*i..k} Sum_{q=3*i..k} Sum_{r=4*i..k}(f) for k=0..5*n; f=((2^(k + q - 2*r)*3^(j + r - 2*q)*4^(i + q - 2*j)*5^(j - 2*i)*6^i*n!)/((n - k + r)!*(k + q - 2*r)!*(j + r - 2*q)!*(i + q - 2*j)!*(j - 2*i)!*i!) ); f=0 for (n - k + r)<0 or (k + q - 2*r)<0; (j + r - 2*q)<0 or (i + q - 2*j) <0 or (j - 2*i)<0. A novel formula proven by Shara Lalo and Zagros Lalo. Also see formula in Links section.

G.f.: 1/(1 - x*t- 2*x^2*t - 3*x^3*t - 4*x^4*t - 5*x^5*t - 6*x^6*t).

cp's OEIS Frontend

A218724 a(n) = (21^n - 1)/20.

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A329332 Table of powers of squarefree numbers, powers of A019565(n) in increasing order in row n. Square array A(n,k) n >= 0, k >= 0 read by descending antidiagonals.

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A072978 Numbers of the form m*2^Omega(m), where m>1 is odd and Omega(m)=A001222(m), the number of prime factors of m.

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A222281 T(n,k) = number of n X k 0..5 arrays with no entry increasing mod 6 by 5 rightwards or downwards, starting with upper left zero.

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A105317 Powers of Fibonacci numbers.

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A009986 Powers of 42.

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A067422 Seventh column of triangle A067417.

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A013726 a(n) = 21^(2*n + 1).

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A319092 Triangle read by rows: T(0,0) = 1; T(n,k) = T(n-1, k) + 2T(n-1, k-1) + 3T(n-1, k-2) + 4T(n-1, k-3) + 5T(n-1, k-4) + 6T(n-1, k-5) for k = 0..5n; T(n,k)=0 for n or k < 0.