A001021 -id:A001021 - cp's OEIS frontend

A038493 Sums of 3 distinct powers of 12.

157, 1741, 1873, 1884, 20749, 20881, 20892, 22465, 22476, 22608, 248845, 248977, 248988, 250561, 250572, 250704, 269569, 269580, 269712, 271296, 2985997, 2986129, 2986140, 2987713, 2987724, 2987856, 3006721, 3006732, 3006864, 3008448, 3234817, 3234828, 3234960

Offset: 1

Olivier Gérard

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

Base-12 interpretation of A038445.

Cf. A001021, A038492.

Union[Total/@Subsets[12^Range[0,6],{3}]] (* Harvey P. Dale, Sep 06 2012 *)

from math import isqrt, comb
from sympy import integer_nthroot
def A038493(n): return 12**((r:=n-1-comb((m:=integer_nthroot(6*n,3)[0])+(t:=(n>comb(m+2,3)))+1,3))-comb((k:=isqrt(b:=r+1<<1))+(b>k*(k+1)),2))+12**((a:=isqrt(s:=n-comb(m-(t^1)+2,3)<<1))+((s<<2)>(a<<2)*(a+1)+1))+12**(m+t+1) # Chai Wah Wu, Apr 04 2025

Offset corrected by Amiram Eldar, Jul 14 2022

A117962 Partial sums of hexagonal numbers with prime indices.

Original entry on oeis.org

6, 21, 66, 157, 388, 713, 1274, 1977, 3012, 4665, 6556, 9257, 12578, 16233, 20604, 26169, 33072, 40453, 49364, 59375, 69960, 82363, 96058, 111811, 130532, 150833, 171948, 194739, 218392, 243817, 275948, 310139, 347540, 386043, 430296, 475747

Offset: 1

Jonathan Vos Post, Apr 05 2006

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.

			a(4) = hexagonal(2) + hexagonal(3) + hexagonal(5) + hexagonal(7) = 6 + 15 + 45 + 91 = 157 is prime.
a(12) = 6 + 15 + 45 + 91 + 231 + 325 + 561 + 703 + 1035 + 1653 + 1891 + 2701 = 9257 is prime.
a(26) = 150833 is prime.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

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). A117961 Hexagonal numbers with prime indices. A117965 Prime partial sums of hexagonal numbers with prime indices.

Cf. A000005, A000040, A000384, A001021, A034953, A001248, A116995, A117961, A117965.

Accumulate[Table[n(2n-1),{n,Prime[Range[50]]}]] (* Harvey P. Dale, Jan 30 2014 *)

a(n) = SUM[i=1..n] A117961(i). a(n) = SUM[i=1..n] A000040(i)*(2*A000040(i)-1). a(n) = SUM[i=1..n] A000384(prime(n)). a(n) = Partial sum of number of divisors of 12^(prime(n)-1) = SUM[i=1..n] A000005(A001021(A000040(n)-1)).

A180691 Smallest power of 12 that begins with n.

Original entry on oeis.org

1, 20736, 35831808, 429981696, 5159780352, 61917364224, 743008370688, 8916100448256, 953962166440690129601298432, 106993205379072, 11447545997288281555215581184, 12, 137370551967459378662586974208, 144, 15407021574586368, 1648446623609512543951043690496

Offset: 1

Daniel Mondot, Sep 17 2010

Cf. A001021, A180689, A180692, A180693.

tlv=12^Range[0,40]; f[n_]:=Module[{idn=IntegerDigits[n],len}, len=Length[idn]; First[Select[tlv, Take[IntegerDigits[#],len]==idn&]]]; Table[f[i],{i,20}]//Quiet  (* Harvey P. Dale, Feb 14 2011 *)

More terms from Harvey P. Dale, Feb 14 2011

A196791 a(n) = A047848(9, n).

Original entry on oeis.org

1, 2, 14, 158, 1886, 22622, 271454, 3257438, 39089246, 469070942, 5628851294, 67546215518, 810554586206, 9726655034462, 116719860413534, 1400638324962398, 16807659899548766, 201691918794585182, 2420303025535022174, 29043636306420266078, 348523635677043192926

Offset: 0

Vincenzo Librandi, Oct 11 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..900
Index entries for linear recurrences with constant coefficients, signature (13,-12).

Cf. A047848, A047849, A047850, A047851, A047852, A047853, A047854, A047855, A047856.

Cf. A001021 (first differences).

Magma
```
[(12^n+10)/11: n in [0..20]];
```

LinearRecurrence[{13,-12},{1,2},30] (* Harvey P. Dale, Sep 07 2015 *)
(12^Range[0,40] +10)/11 (* G. C. Greubel, Jan 17 2025 *)

def A196791(n): return (pow(12, n) + 10)//11
print([A196791(n) for n in range(41)]) # G. C. Greubel, Jan 17 2025

a(n) = (12^n + 10)/11.
a(n) = 12*a(n-1) - 10, with a(0) = 1.
G.f.: (1-11*x)/((1-x)*(1-12*x)). - Bruno Berselli, Oct 11 2011
From Elmo R. Oliveira, Aug 30 2024: (Start)
E.g.f.: exp(x)*(exp(11*x) + 10)/11.
a(n) = 13*a(n-1) - 12*a(n-2) for n > 1. (End)

A013618 Triangle of coefficients in expansion of (1+11x)^n.

Original entry on oeis.org

1, 1, 11, 1, 22, 121, 1, 33, 363, 1331, 1, 44, 726, 5324, 14641, 1, 55, 1210, 13310, 73205, 161051, 1, 66, 1815, 26620, 219615, 966306, 1771561, 1, 77, 2541, 46585, 512435, 3382071, 12400927, 19487171, 1, 88, 3388, 74536, 1024870, 9018856, 49603708, 155897368, 214358881

Offset: 0

N. J. A. Sloane

T(n,k) equals the number of n-length words on {0,1,...,11} having n-k zeros. - Milan Janjic, Jul 24 2015

Seiichi Manyama, Rows n = 0..139, flattened

Cf. A001020 (right edge).

T:= n-> (p-> seq(coeff(p, x, k), k=0..n))((1+11*x)^n):
seq(T(n), n=0..10);  # Alois P. Heinz, Jul 24 2015

G.f.: 1 / (1 - x(1+11y)).

T(n,k) = 11^k*C(n,k) = Sum_{i=n-k..n} C(i,n-k)*C(n,i)*10^(n-i). Row sums are 12^n = A001021. - Mircea Merca, Apr 28 2012

A013797 a(n) = 12^(4*n + 3).

Original entry on oeis.org

1728, 35831808, 743008370688, 15407021574586368, 319479999370622926848, 6624737266949237011120128, 137370551967459378662586974208, 2848515765597237675947403497177088, 59066822915424320448445358917464096768

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 (20736).

Subsequence of A001021.

[12^(4*n+3): n in [0..10]]; // Vincenzo Librandi, Jun 28 2011

12^(4*Range[0,20]+3) (* or *) NestList[20736#&,1728,20] (* Harvey P. Dale, Apr 29 2012 *)

a(n) = 20736*a(n-1); a(0) = 1728. - Harvey P. Dale, Apr 29 2012

A021724 Expansion of 1/((1-x)(1-3x)(1-10x)(1-12x)).

Original entry on oeis.org

1, 26, 465, 7150, 101621, 1378026, 18123145, 233349350, 2958918141, 37094306626, 461004657425, 5690785933950, 69876732453061, 854393804284826, 10411455807073305, 126524771262956950, 1534170271000826381

Offset: 0

N. J. A. Sloane

From Bruno Berselli, May 08 2013: (Start)
Naturally, the sequence is related to:
A018207, 1/((1-3x)(1-10x)(1-12x)):  A018207(n) = a(n)-a(n-1), n>0;
A016267, 1/((1-x)(1-10x)(1-12x)):  A016267(n) = a(n)-3*a(n-1), n>0;
A016217, 1/((1-x)(1-3x)(1-12x)):  A016217(n) = a(n)-10*a(n-1), n>0;
A016215, 1/((1-x)(1-3x)(1-10x)):  A016215(n) = a(n)-12*a(n-1), n>0;
A016196, 1/((1-10x)(1-12x)):  A016196(n) = a(n)-4*a(n-1)+3*a(n-2), n>1;
A016147, 1/((1-3x)(1-12x)):  A016147(n) = a(n)-11*a(n-1)+10*a(n-2), n>1;
A016145, 1/((1-3x)(1-10x)):  A016145(n) = a(n)-13*a(n-1)+12*a(n-2), n>1;
A016125, 1/((1-x)(1-12x)):  A016125(n) = a(n)-13*a(n-1)+30*a(n-2), n>1;
A002275, x/((1-x)(1-10x)):  A002275(n) = a(n-1)-15*a(n-2)+36*a(n-3), n>2;
A003462, x/((1-x)(1-3x)):  A003462(n) = a(n-1)-22*a(n-2)+120*a(n-3), n>2;
A000012, 1/(1-x):  A000012(n) = a(n)-25*a(n-1)+186*a(n-2)-360*a(n-3), n>2;
A000244, 1/(1-3x):  A000244(n) = a(n)-23*a(n-1)+142*a(n-2)-120*a(n-3), n>2;
A011557, 1/(1-10x):  A011557(n) = a(n)-16*a(n-1)+51*a(n-2)-36*a(n-3), n>2;
A001021, 1/(1-12x):  A001021(n) = a(n)-14*a(n-1)+43*a(n-2)-30*a(n-3), n>2. (End)

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (26,-211,546,-360).

Cf. A000012, A000244, A001021, A002275, A003462, A011557, A016125, A016145, A016147, A016196, A016215, A016217, A016267, A018207.

m:=20; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-x)*(1-3*x)*(1-10*x)*(1-12*x)))); // Bruno Berselli, May 07 2013

CoefficientList[Series[1/((1 - x) (1 - 3 x) (1 - 10 x) (1 - 12 x)), {x, 0, 20}], x] (* Bruno Berselli, May 07 2013 *)
LinearRecurrence[{26,-211,546,-360},{1,26,465,7150},120] (* Harvey P. Dale, Jul 06 2019 *)

Vec(1/((1-x)*(1-3*x)*(1-10*x)*(1-12*x))+O(x^20)) \\ Bruno Berselli, May 07 2013

G.f.: 1/((1-x)*(1-3*x)*(1-10*x)*(1-12*x)).

a(n) = -1/198 +3^(n+1)/14 -2^(n+2)*5^(n+3)/63 +2^(2n+5)*3^(n+1)/11. [Bruno Berselli, May 07 2013]

A038335 Triangle whose (i,j)-th entry is binomial(i,j)12^(i-j)9^j.

Original entry on oeis.org

1, 12, 9, 144, 216, 81, 1728, 3888, 2916, 729, 20736, 62208, 69984, 34992, 6561, 248832, 933120, 1399680, 1049760, 393660, 59049, 2985984, 13436928, 25194240, 25194240, 14171760, 4251528, 531441, 35831808, 188116992, 423263232

Offset: 0

N. J. A. Sloane

			       1
      12        9
     144      216       81
    1728     3888     2916      729
   20736    62208    69984    34992     6561
  248832   933120  1399680  1049760   393660    59049
 2985984 13436928 25194240 25194240 14171760  4251528   531441

B. N. Cyvin et al., Isomer enumeration of unbranched catacondensed polygonal systems with pentagons and heptagons, Match, No. 34 (Oct 1996), pp. 109-121.

Cf. A009965 (row sums), A001021 (column 0), A001019 (diagonal)

A038335 := proc(i,j)
    binomial(i,j)*12^(i-j)*9^j ;
end proc: # R. J. Mathar, Nov 22 2022

Flatten[Table[Binomial[i,j]12^(i-j) 9^j,{i,0,10},{j,0,i}]] (* Harvey P. Dale, Oct 17 2013 *)

A156330 Numerator of Euler(n, 1/12).

Original entry on oeis.org

1, -5, -11, 415, 1705, -120125, -737891, 73544935, 602197585, -77251102325, -790660144571, 123981052932655, 1522721707926265, -282190333761783725, -4043468004740204051, 864617687280807347575, 14158848206836206915745, -3431276846283480837508325

Offset: 0

N. J. A. Sloane, Nov 07 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..200

For denominators see A001021.

Numerator[EulerE[Range[0,20],1/12]] (* Vincenzo Librandi, May 05 2012 *)

G.f.: conjecture T(0)/(1+5*x), where T(k) = 1 - 36*x^2*(k+1)^2/(36*x^2*(k+1)^2 + (1+5*x)^2/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Nov 12 2013

a(n) = (-6)^n*skp(n, 5/6), where skp(n,x) are the Swiss-Knife polynomials A153641. - Peter Luschny, Apr 19 2014

A156332 Numerator of Euler(n, 5/12).

Original entry on oeis.org

1, -1, -35, 107, 6265, -32041, -2749355, 19696067, 2247032305, -20698163281, -2950725914675, 33220406931227, 5682862415856745, -75612617322835321, -15090424387627057595, 231673592430307689587, 52841539466887256047585, -919407851749260210944161

Offset: 0

N. J. A. Sloane, Nov 07 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..200

For denominators see A001021.

Numerator[EulerE[Range[0,20],5/12]] (* Vincenzo Librandi, May 05 2012 *)

a(n) = a(n) = (-6)^n*skp(n, 1/6), where skp(n,x) are the Swiss-Knife polynomials A153641. - Peter Luschny, Apr 19 2014

cp's OEIS Frontend

A038493 Sums of 3 distinct powers of 12.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Python

Extensions

A117962 Partial sums of hexagonal numbers with prime indices.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A180691 Smallest power of 12 that begins with n.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Extensions

A196791 a(n) = A047848(9, n).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

Python

Formula

A013618 Triangle of coefficients in expansion of (1+11x)^n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Formula

A013797 a(n) = 12^(4*n + 3).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A021724 Expansion of 1/((1-x)(1-3x)(1-10x)(1-12x)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A038335 Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*9^j.

Views

Author

Keywords

Examples

References

Crossrefs

A038335 Triangle whose (i,j)-th entry is binomial(i,j)12^(i-j)9^j.