A067403 -id:A067403 - cp's OEIS frontend

A067411 Third column of triangle A067410 and second column of A067417.

1, 4, 24, 144, 864, 5184, 31104, 186624, 1119744, 6718464, 40310784, 241864704, 1451188224, 8707129344, 52242776064, 313456656384, 1880739938304, 11284439629824, 67706637778944, 406239826673664

Offset: 0

Wolfdieter Lang, Jan 25 2002

Let f(k) be the sum of the smallest three positive divisors of k, g(k) be the sum of the largest two positive divisors of k, this sequence from a(2) onwards contains the numbers k for which g(k) is a positive integer power of f(k). - Yifan Xie, Jan 27 2024

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Paul Barry and A. Hennessy, The Euler-Seidel Matrix, Hankel Matrices and Moment Sequences, J. Int. Seq. 13 (2010) # 10.8.2, Example 14.
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
Craig Knecht, Number of tilings for a stackable six sphinx tile shape.
Index entries for linear recurrences with constant coefficients, signature (6).

A002001, A067412 (second and fourth column of A067410), A000244, A067403 (first and third column of A067417), A000400 (powers of 6).

Row sums of A038195.

CoefficientList[Series[(1-2x)/(1-6x),{x,0,30}],x] (* Harvey P. Dale, Feb 26 2015 *)

a(n) = if(n<=0, 0, 4*6^(n-1) ); \\ Joerg Arndt, Feb 23 2014

a(n) = A067410(n+2, 2) = A067417(n+1, 1).
a(n) = 4 * 6^(n-1), for n >= 1, a(0)=1.
G.f.: (1-2*x)/(1-6*x).
E.g.f.: (2*exp(6*x)+1) / 3 = exp(3*x)*(cosh(3*x) + sinh(3*x)/3). - Paul Barry, Nov 20 2003
a(n) = Sum_{k=0..n} C(n,k) * A001045(n+k+1). - Paul Barry, Apr 19 2010

Incorrect formula deleted by Harvey P. Dale, Feb 26 2015

Formula restored by Sean A. Irvine, Jan 10 2021

A067402 Triangle with columns built from certain power sequences.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 12, 5, 1, 1, 48, 45, 7, 1, 1, 192, 405, 112, 9, 1, 1, 768, 3645, 1792, 225, 11, 1, 1, 3072, 32805, 28672, 5625, 396, 13, 1, 1, 12288, 295245, 458752, 140625, 14256, 637, 15, 1, 1, 49152, 2657205, 7340032, 3515625, 513216, 31213, 960, 17, 1

Offset: 0

Wolfdieter Lang, Jan 25 2002

			Triangle starts:
      m=0   1   2   3   4
  n=0:  1;
  n=1:  1,  1;
  n=2:  1,  3,  1;
  n=3:  1, 12,  5,  1;
  n=4:  1, 48, 45,  7,  1;
  ...

Indranil Ghosh, Rows 0..125, flattened

Cf. A009998 (triangle built from powers of (m+1)).

Columns m=0..8: A000012, A002001, A067403-A067409.

A[n_,m_]:=If[n==m,1,(2m+1)(m+1)^(2(n-m-1))]; Flatten[Table[A[n,m],{n,0,9},{m,0,n}]] (* Stefano Spezia, Sep 30 2022 *)

a(n, m) = 1 if n=m; a(n, m) = (2*m+1)*(m+1)^(2*(n-m-1)) if n>m>=0.

G.f. for column m: (x^m)*(1-x*m^2)/(1-x*(m+1)^2).

More terms from Larry Reeves (larryr(AT)acm.org), Jun 11 2002

A067417 Triangle with columns built from certain power sequences.

Original entry on oeis.org

1, 3, 1, 9, 4, 1, 27, 24, 5, 1, 81, 144, 45, 6, 1, 243, 864, 405, 72, 7, 1, 729, 5184, 3645, 864, 105, 8, 1, 2187, 31104, 32805, 10368, 1575, 144, 9, 1, 6561, 186624, 295245, 124416, 23625, 2592, 189, 10, 1, 19683, 1119744, 2657205, 1492992, 354375, 46656, 3969, 240, 11, 1

Offset: 0

Wolfdieter Lang, Jan 25 2002

			Triangle starts:
   1;
   3,  1;
   9,  4, 1;
  27, 24, 5, 1;
  ...

Indranil Ghosh, Rows 0..125, flattened

Cf. A009998 (triangle built from powers of (m+1)), A067402, A067410.

Columns m=0..8 are A000244, A067411, A067403, A067419, A067420, A067421, A067422, A067423, A067424.

A[n_,m_]:=If[n==m,1,(m+3)(3(m+1))^(n-m-1)]; Flatten[Table[A[n,m],{n,0,9},{m,0,n}]] (* Stefano Spezia, Sep 30 2022 *)

a(n, m) = 1 if n = m; a(n, m) = (m+3)*(3*(m+1))^(n-m-1) if n > m >= 0.

G.f. for column m: (x^m)*(1-2*m*x)/(1-3*(m+1)*x).

A067419 Fourth column of triangle A067417.

Original entry on oeis.org

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

A067404 Fourth column of triangle A067402.

Original entry on oeis.org

1, 7, 112, 1792, 28672, 458752, 7340032, 117440512, 1879048192, 30064771072, 481036337152, 7696581394432, 123145302310912, 1970324836974592, 31525197391593472, 504403158265495552, 8070450532247928832, 129127208515966861312, 2066035336255469780992, 33056565380087516495872

Offset: 0

Wolfdieter Lang, Jan 25 2002

Index entries for linear recurrences with constant coefficients, signature (16).

Cf. A067403 (third column), A067405 (fifth column), A001025 (powers of 16).

Cf. A067402.

CoefficientList[Series[(1-9x)/(1-16x),{x,0,20}],x] (* Harvey P. Dale, Jun 02 2017 *)

a(n) = A067402(n+3, 3).
a(n) = 7*16^(n-1), n>=1, a(0) = 1.
G.f.: (1-9*x)/(1-16*x).
E.g.f.: (9 + 7*exp(16*x))/16. - Stefano Spezia, Sep 30 2022

More terms from Harvey P. Dale, Jun 02 2017

A270369 Expansion of g.f. (1-7x)/(1-9x).

Original entry on oeis.org

1, 2, 18, 162, 1458, 13122, 118098, 1062882, 9565938, 86093442, 774840978, 6973568802, 62762119218, 564859072962, 5083731656658, 45753584909922, 411782264189298, 3706040377703682, 33354363399333138, 300189270593998242, 2701703435345984178, 24315330918113857602, 218837978263024718418

Offset: 0

Colin Barker, Mar 18 2016

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (9).

Cf. A001019 (powers of 9), A054879 (partial sums), A132025.

Cf. similar sequences with g.f. (1-k*x)/(1-9*x) and k=0..8: A001019 (k=0; k=8 gives two initial 1's ), A055275 (k=1), A270472 (k=2), A092810 (k=3), A067403 (k=4), A270473 (k=5), A102518 (k=6), this sequence (k=7).

CoefficientList[Series[(1-7x)/(1-9x),{x,0,20}],x] (* or *) Join[ {1}, NestList[9#&,2,20]] (* Harvey P. Dale, Oct 15 2017 *)

PARI
```
Vec((1-7*x)/(1-9*x) + O(x^30))
```

G.f.: (1-7*x)/(1-9*x).
a(n) = 9*a(n-1) for n>1.
a(n) = 2*9^(n-1) for n>0.
From Amiram Eldar, May 08 2023: (Start)
Sum_{n>=0} 1/a(n) = 25/16.
Sum_{n>=0} (-1)^n/a(n) = 11/20.
Product_{n>=1} (1 - 1/a(n)) = A132025. (End)
E.g.f.: (2*exp(9*x) + 7)/9. - Elmo R. Oliveira, Mar 25 2025

A270473 Expansion of g.f. (1-5x)/(1-9x).

Original entry on oeis.org

1, 4, 36, 324, 2916, 26244, 236196, 2125764, 19131876, 172186884, 1549681956, 13947137604, 125524238436, 1129718145924, 10167463313316, 91507169819844, 823564528378596, 7412080755407364, 66708726798666276, 600378541187996484, 5403406870691968356, 48630661836227715204

Offset: 0

Colin Barker, Mar 17 2016

Also squares that can be expressed as the sum of two powers of three (3^x + 3^y), except a(0). - Karl-Heinz Hofmann, Sep 03 2022

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (9).

Cf. A001019 (powers of 9), A083884 (partial sums).
Cf. A067403: (1-4*x)/(1-9*x); A102518: (1-6*x)/(1-9*x).
Cf. A356879, A356880.

Join[{1},NestList[9#&,4,20]] (* Harvey P. Dale, Oct 23 2022 *)

PARI
```
Vec((1-5*x)/(1-9*x) + O(x^30))
```

G.f.: (1-5*x)/(1-9*x).
a(n) = 9*a(n-1) for n>1.
a(n) = 4*9^(n-1) for n>0.
E.g.f.: (4*exp(9*x) + 5)/9. - Stefano Spezia, Jul 09 2024

A331741 Squares s such that A331733(s) = sigma(A225546(n)) is congruent to 2 modulo 4.

Original entry on oeis.org

16, 144, 400, 784, 1936, 2704, 3600, 4624, 5776, 7056, 8464, 10000, 13456, 15376, 17424, 19600, 21904, 24336, 26896, 29584, 35344, 38416, 41616, 44944, 48400, 51984, 55696, 59536, 67600, 71824, 76176, 80656, 85264, 90000, 94864, 99856, 104976, 110224, 115600, 121104, 126736, 132496, 138384, 144400, 150544, 163216, 169744, 176400

Offset: 1

Antti Karttunen, Feb 03 2020

nonn

Squares s for which A331733(s) is two times an odd number, i.e., squares s such that A007814(A331733(s)) == 1.
For each term k present, A006519(k) = 2^(2^e), with A000040(1+e) == 1 (mod 4). See A191218, A228058.
Equal to the sequence A225546(A191218(n)), for n >= 1, when its elements are sorted into ascending order.

Cf. A006519, A007814, A010052, A067403, A191218, A225546, A228058, A331733, A331734.

Select[Range[100]^2, Mod[DivisorSigma[1, If[# == 1, 1, Apply[Times, Flatten@ Map[Function[{p, e}, Map[Prime[Log2@# + 1]^(2^(PrimePi@ p - 1)) &, DeleteCases[NumberExpand[e, 2], 0]]] @@ # &, FactorInteger[#]]]]], 4] == 2 &] (* Michael De Vlieger, Feb 08 2020 *)

k=0; for(n=1,500,if(!(n%2)&&(1==A007814(A331733(n^2))),k++; write("b331741.txt", k, " ", n^2); print(n^2, " -> ", factor(n^2),", ")));

{n: A010052(n)*A007814(A331733(n)) == 1}.

A384853 Squared length of interior diagonal of n-th (U, V)-crossbox, where U = (1, 0, 1) and V = (0, 1, 0), as in Comments.

Original entry on oeis.org

1, 5, 9, 21, 57, 165, 489, 1461, 4377, 13125, 39369, 118101, 354297, 1062885, 3188649, 9565941, 28697817, 86093445, 258280329, 774840981, 2324522937, 6973568805, 20920706409, 62762119221, 188286357657, 564859072965, 1694577218889, 5083731656661

Offset: 1

Clark Kimberling, Jul 02 2025

nonn

Suppose that U and V are 3-dimensional vectors over the field of real numbers. Define f(1) = U, f(2) = V, f(3) = UxV, where x = cross product, and for n>=2, define f(n) = h(n - 1), g(n) = f(n - 1) + g(n - 1) - h(n - 1), h(n) = f(n) x g(n).
The parallelopiped having edge vectors f(n), g(n), h(n) is the n-th (U,V)-crossbox, with volume |f(n).(g(n) x h(n))|, where . = dot product, and interior diagonal length ||g(n)||. These two sequences, after removal of their first 2 terms, are given for selected U and V by the following table, except for the 3 initial terms:
    U             V         volume   squared diagonal length, ||g(n)||^2
(1, 0, 0)     (0, 1, 0)     A000079           A052548
(1, 0, 0)     (0, 1, 1)     A008776         3*A052919
(1, 0, 0)     (1, 0, 1)     A000351           A178676
(1, 0, 0)     (1, 1, 1)     A167747         5*A204061
(1, 0, 0)     (0, 2, 0)     A005054         4*A199215
(1, 0, 0)     (1, 2, 0)     A013731         8*A199552
(1, 0, 0)     (2, 1, 0)     A011557        10*A000533
(1, 0, 0)     (1, 1, 2)     A067403        18*A135423
(1, 0, 0)     (2, 1, 1)     A334603        11*A199750
(1, 0, 1)     (0, 1, 0)     A008776           this sequence
(1, 1, 0)     (0, 1, 1)     A081341         6*A199318
(1, 1, 0)     (1, 1, 1)     A270369         9*A199559
(1, 2, 3)     (3, 2, 1)   2*A009992   48 + 96*A009992

			Taking U = (1, 0, 1) and V = (0, 1, 0), successive edge vectors are given by
(f(n)) = ( (1, 0, 1), (-1,0,1), (-1,2,-1), (3,0,-3), (3,-6,3), ...)
(g(n)) = ( (0,1,0), (2,1,0), (2,-1,2), (-2,1,4), (-2,7,-2), (10,1,-8), ...)
(h(n)) = ( (-1.0,1), (-1,2,-1), (3,0,-3), (3,-6,3), (-9,0,9),...)
The successive volumes are (2, 6, 18, 54, 162, 486, 1458, 4374, 13122,...).
The lengths of diagonals of the first five crossboxes are 1, sqrt(5), 3, sqrt(21), sqrt(57), so the first five squared lengths are 1, 5, 9, 21, 57.

Index entries for linear recurrences with constant coefficients, signature (4,-3).

Cf. A000079, A052548, A008776.

f[1] = {1, 0, 1}; g[1] = {0, 1, 0}; h[1] = Cross[f[1], g[1]];
f[n_] := f[n] = h[n - 1];
g[n_] := g[n] = f[n - 1] + g[n - 1] - h[n - 1];
h[n_] := h[n] = Cross[f[n], g[n]];
v[n_] := f[n] . Cross[g[n], h[n]] (* signed volume of nth parallelopiped P(n) *)
d[n_] := Norm[g[n]] (* length of interior diagonal of P(n) *)
Column[Table[{f[n], g[n], h[n]}, {n, 1, 16}]]  (* edge vectors of P(n) *)
Table[v[n], {n, 1, 16}]  (* A008776 *)
u = Table[d[n]^2, {n, 1, 30}] (* A384853 *)
Join[{1},Table[1+2*(3^(n-1)+1),{n,40}]] (* or *) LinearRecurrence[{4,-3},{1,5,9},50] (* Harvey P. Dale, Jul 20 2025 *)

a(0) = 1, a(n) = 1 + 2 * (3^(n-1)+1) for n>=1.
a(n) = 4*a(n-1) - 3*a(n-2) for n>=4.
In general, suppose that U = (a,b,c) and V = (s,t,u), and let D = -(a^2 + b^2 + c^2 + s^2 + t^2 + u^2 + 2 (a s + b t + c u)). Then, linear recurrences are given for n>=3 by f(n) = D*f (n - 2), g(n) = g(n - 1) + D*g(n - 2) - D*g(n - 3), h(n) = D*h(n - 2). If w(n) denotes the volume of the n-th (U,V)-crossbox, then w(n) = D*w(n-1) for n>=2.

cp's OEIS Frontend

A067411 Third column of triangle A067410 and second column of A067417.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A067402 Triangle with columns built from certain power sequences.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A067417 Triangle with columns built from certain power sequences.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A067419 Fourth column of triangle A067417.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A067404 Fourth column of triangle A067402.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A270369 Expansion of g.f. (1-7*x)/(1-9*x).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A270473 Expansion of g.f. (1-5*x)/(1-9*x).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A331741 Squares s such that A331733(s) = sigma(A225546(n)) is congruent to 2 modulo 4.

Views

Author

Keywords

Comments

A270369 Expansion of g.f. (1-7x)/(1-9x).

A270473 Expansion of g.f. (1-5x)/(1-9x).