A001021 -id:A001021 - cp's OEIS frontend

A067419 Fourth column of triangle A067417.

1, 6, 72, 864, 10368, 124416, 1492992, 17915904, 214990848, 2579890176, 30958682112, 371504185344, 4458050224128, 53496602689536, 641959232274432, 7703510787293184, 92442129447518208, 1109305553370218496, 13311666640442621952, 159739999685311463424

Offset: 0

Wolfdieter Lang, Jan 25 2002

Vincenzo Librandi, Table of n, a(n) for n = 0..900
Index entries for linear recurrences with constant coefficients, signature (12).

Cf. A067403 (third column), A067420 (fifth column), A001021 (powers of 12).

[Ceiling(6*(3*4)^(n-1)): n in [0..20]]; // Vincenzo Librandi, Oct 02 2011

Join[{1}, NestList[12*# &, 6, 20]] (* Paolo Xausa, Sep 03 2024 *)

a(n) = A067417(n+3, 3).
a(n) = 6*(3*4)^(n-1), n >= 1, a(0)=1.
G.f.: (1-6*x)/(1-12*x).
a(n) = Sum_{k=0..n} A134309(n,k)*6^k = Sum_{k=0..n} A055372(n,k)*5^k. - Philippe Deléham, Feb 04 2012

A194887 Numbers that are the sum of two powers of 12.

Original entry on oeis.org

2, 13, 24, 145, 156, 288, 1729, 1740, 1872, 3456, 20737, 20748, 20880, 22464, 41472, 248833, 248844, 248976, 250560, 269568, 497664, 2985985, 2985996, 2986128, 2987712, 3006720, 3234816, 5971968, 35831809, 35831820, 35831952, 35833536, 35852544, 36080640

Offset: 1

Jeremy Gardiner, Oct 09 2011

Parity of this sequence is A073424.

			12^0 + 12^2 = 145

Zak Seidov, Table of n, a(n) for n = 1..1000

Cf. A001021, A033048, A052216, A073424.

t = 12^Range[0, 9]; Select[Union[Flatten[Table[i + j, {i, t}, {j, t}]]], # <= t[[-1]] + 1 &] (* T. D. Noe, Oct 09 2011 *)
Total/@Tuples[12^Range[0,10],2]//Union (* Harvey P. Dale, Jul 20 2019 *)

Typo in example corrected by Zak Seidov, Oct 23 2011

A100851 Triangle read by rows: T(n,k) = 2^n * 3^k, 0 <= k <= n, n >= 0.

Original entry on oeis.org

1, 2, 6, 4, 12, 36, 8, 24, 72, 216, 16, 48, 144, 432, 1296, 32, 96, 288, 864, 2592, 7776, 64, 192, 576, 1728, 5184, 15552, 46656, 128, 384, 1152, 3456, 10368, 31104, 93312, 279936, 256, 768, 2304, 6912, 20736, 62208, 186624, 559872, 1679616, 512, 1536, 4608, 13824, 41472, 124416, 373248, 1119744, 3359232, 10077696

Offset: 0

Reinhard Zumkeller, Nov 20 2004

			From _Stefano Spezia_, Apr 28 2024: (Start)
Triangle begins:
   1;
   2,  6;
   4, 12,  36;
   8, 24,  72, 216;
  16, 48, 144, 432, 1296;
  32, 96, 288, 864, 2592, 7776;
  ...
(End)

G. C. Greubel, Rows n = 0..100 of the triangle, flattened
Eric Weisstein's World of Mathematics, Smooth Number.

Cf. A000079, A000400, A001021, A003586, A005010, A007283, A016129.
Cf. A016149, A051958, A053524, A100852, A248337.
Cf. A065333(T(n, k))=1.

A100851:= func< n,k | 2^n*3^k >;
[A100851(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 11 2024

A100851[n_, k_]= 2^n*3^k;
Table[A100851[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Nov 11 2024 *)

def A100851(n,k): return 2^n*3^k
flatten([[A100851(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Nov 11 2024

T(n,0) = A000079(n).
T(n,1) = A007283(n) for n>0.
T(n,2) = A005010(n) for n>1.
T(n,n) = A000400(n) = A100852(n,n).
Sum_{k=0..n} T(n, k) = A016129(n).
T(2*n, n) = A001021(n). - Reinhard Zumkeller, Mar 04 2006
G.f.: 1/((1 - 2*x)*(1 - 6*x*y)). - Stefano Spezia, Apr 28 2024
From G. C. Greubel, Nov 11 2024: (Start)
Sum_{k=0..n} (-1)^k*T(n, k) = A053524(n+1).
Sum_{k=0..floor(n/2)} T(n-k, k) = (1/2)*((1-(-1)^n)*A248337((n+1)/2) + (1 + (-1)^n)*A016149(n/2)).
Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = (1/2)*(-1)^floor(n/2)*( (1+(-1)^n) *A051958((n+2)/2) + 2*(1-(-1)^n)*A051958((n+1)/2)). (End)
Sum_{n>=0, k=0..n} 1/T(n,k) = 12/5. - Amiram Eldar, May 12 2025

A117961 Hexagonal numbers with prime indices.

Original entry on oeis.org

6, 15, 45, 91, 231, 325, 561, 703, 1035, 1653, 1891, 2701, 3321, 3655, 4371, 5565, 6903, 7381, 8911, 10011, 10585, 12403, 13695, 15753, 18721, 20301, 21115, 22791, 23653, 25425, 32131, 34191, 37401, 38503, 44253, 45451, 49141, 52975, 55611

Offset: 1

Jonathan Vos Post, Apr 05 2006

See also: A034953 Triangular numbers (A000217) with prime indices. A001248 Squares of primes. A116995 Pentagonal numbers with prime indices. A000384 Hexagonal numbers: n(2n-1). There are no prime hexagonal numbers. The n-th Hexagonal number A000384(n) = n*(2*n-1) is semiprime iff both n and 2*n-1 are prime iff A000384(n) is an element of A001358 iff n is an element of A005382.

Cf. A000005, A000040, A000384, A001021, A034953, A001248, A116995.

With[{hex=Table[n(2n-1),{n,250}]},Flatten[Table[Take[hex,{Prime[n]}],{n, 40}]]] (* Harvey P. Dale, Dec 04 2011 *)

a(n) = A000040(n)*(2*A000040(n)-1). a(n) = A000384(prime(n)). a(n) = number of divisors of 12^(prime(n)-1) = A000005(A001021(A000040(n)-1)).

A013717 a(n) = 12^(2*n + 1).

Original entry on oeis.org

12, 1728, 248832, 35831808, 5159780352, 743008370688, 106993205379072, 15407021574586368, 2218611106740436992, 319479999370622926848, 46005119909369701466112, 6624737266949237011120128

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 (144).

Bisection of A001021.

Cf. A013708-A013716, A013718-A013729.

[12^(2*n+1): n in [0..15]]; // Vincenzo Librandi, Jun 26 2011

seq(12^(2*n+1),n=0..11); # Nathaniel Johnston, Jun 25 2011

a(n)=12^(2*n+1) \\ Charles R Greathouse IV, Jul 11 2016

From Philippe Deléham, Nov 25 2008: (Start)
a(n) = 144*a(n-1), a(0)=12.
G.f.: 12/(1-144*x). (End)

A038492 Sums of 2 distinct powers of 12.

Original entry on oeis.org

13, 145, 156, 1729, 1740, 1872, 20737, 20748, 20880, 22464, 248833, 248844, 248976, 250560, 269568, 2985985, 2985996, 2986128, 2987712, 3006720, 3234816, 35831809, 35831820, 35831952, 35833536, 35852544, 36080640, 38817792, 429981697, 429981708, 429981840, 429983424, 430002432, 430230528

Offset: 1

Olivier Gérard

Amiram Eldar, Table of n, a(n) for n = 1..10000

Base-12 interpretation of A038444.

Cf. A001021, A038493.

Take[Union[Plus@@@Subsets[12^Range[0,20],{2}]],50] (* Harvey P. Dale, Dec 16 2010 *)

from math import isqrt
def A038492(n): return 12**(m:=isqrt(n<<3)+1>>1)+12**(n-1-(m*(m-1)>>1)) # Chai Wah Wu, Apr 04 2025

More terms from Vincenzo Librandi, Aug 06 2009

Offset corrected by Amiram Eldar, Jul 14 2022

A067413 Sixth column of triangle A067410.

Original entry on oeis.org

1, 7, 84, 1008, 12096, 145152, 1741824, 20901888, 250822656, 3009871872, 36118462464, 433421549568, 5201058594816, 62412703137792, 748952437653504, 8987429251842048, 107849151022104576, 1294189812265254912

Offset: 0

Wolfdieter Lang, Jan 25 2002

The fifth column is [1,6,60,600,6000,60000,...].

Index entries for linear recurrences with constant coefficients, signature (12).

Cf. A067412 (fourth column), A067414 (seventh column), A001021 (powers of 12).

a(n)= A067410(n+5, 5). a(n)= 7*12^(n-1), n>=1, a(0)=1.

G.f.: (1-5*x)/(1-12*x).

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

A013750 a(n) = 12^(3*n + 1).

Original entry on oeis.org

12, 20736, 35831808, 61917364224, 106993205379072, 184884258895036416, 319479999370622926848, 552061438912436417593344, 953962166440690129601298432, 1648446623609512543951043690496

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 (1728).

Subsequence of A001021.

[12^(3*n+1): n in [0..15]]; // Vincenzo Librandi, Jun 27 2011

a(n)=12^(3*n+1) \\ Charles R Greathouse IV, Jun 27 2011

From Philippe Deléham, Nov 30 2008: (Start)
a(n) = 1728*a(n-1); a(0)=12.
G.f.: 12/(1-1728*x).
a(n) = A013751(n)/12. (End)

A013751 a(n) = 12^(3*n + 2).

Original entry on oeis.org

144, 248832, 429981696, 743008370688, 1283918464548864, 2218611106740436992, 3833759992447475122176, 6624737266949237011120128, 11447545997288281555215581184

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 (1728).

Subsequence of A001021.

[12^(3*n+2): n in [0..15]]; // Vincenzo Librandi, Jun 27 2011

a(n)=12^(3*n+2) \\ Charles R Greathouse IV, Jun 27 2011

From Philippe Deléham, Nov 30 2008: (Start)
a(n) = 1728*a(n-1); a(0)=144.
G.f.: 144/(1-1728*x).
a(n) = 12*A013750(n). (End)

cp's OEIS Frontend

A067419 Fourth column of triangle A067417.

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A194887 Numbers that are the sum of two powers of 12.

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A100851 Triangle read by rows: T(n,k) = 2^n * 3^k, 0 <= k <= n, n >= 0.

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A117961 Hexagonal numbers with prime indices.

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A013717 a(n) = 12^(2*n + 1).

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A038492 Sums of 2 distinct powers of 12.

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A067413 Sixth column of triangle A067410.

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A100401 Digital root of 3^n.

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