A067412 -id:A067412 - cp's OEIS frontend

A016957 a(n) = 6*n + 4.

4, 10, 16, 22, 28, 34, 40, 46, 52, 58, 64, 70, 76, 82, 88, 94, 100, 106, 112, 118, 124, 130, 136, 142, 148, 154, 160, 166, 172, 178, 184, 190, 196, 202, 208, 214, 220, 226, 232, 238, 244, 250, 256, 262, 268, 274, 280, 286, 292, 298, 304, 310, 316, 322, 328

Offset: 0

N. J. A. Sloane

Number of 2 X n binary matrices avoiding simultaneously the right-angled numbered polyomino patterns (ranpp) (00;1), (01,1) and (11;0). An occurrence of a ranpp (xy;z) in a matrix A=(a(i,j)) is a triple (a(i1,j1), a(i1,j2), a(i2,j1)) where i1 < i2, j1 < j2 and these elements are in the same relative order as those in the triple (x,y,z). In general, the number of m X n 0-1 matrices in question is given by (n+2)*2^(m-1) + 2*m*(n-1) - 2 for m > 1 and n > 1. - Sergey Kitaev, Nov 12 2004
If Y is a 4-subset of an n-set X then, for n >= 4, a(n-4) is the number of 3-subsets of X having at least two elements in common with Y. - Milan Janjic, Dec 08 2007
4th transversal numbers (or 4-transversal numbers): Numbers of the 4th column of positive numbers in the square array of nonnegative and polygonal numbers A139600. Also, numbers of the 4th column in the square array A057145. - Omar E. Pol, May 02 2008
a(n) is the maximum number such that there exists an edge coloring of the complete graph with a(n) vertices using n colors and every subgraph whose edges are of the same color (subgraph induced by edge color) is planar. - Srikanth K S, Dec 18 2010
Also numbers having two antecedents in the Collatz problem: 12*n+8 and 2*n+1 (respectively A017617(n) and A005408(n)). - Michel Lagneau, Dec 28 2012
a(n) = 6n+4 has three undirected edges e1 = (3n+2, 6n+4), e2 = (6n+4, 12n+8) and e3 = (2n+1, 6n+4) in the Collatz graph of A006370. - Heinz Ebert, Mar 16 2021
Conjecture: this sequence contains some but not all, even numbers with odd abundance A088827. They appear in this sequence at indices A186424(n) - 1. - John Tyler Rascoe, Jul 09 2022

A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 189. - From N. J. A. Sloane, Dec 01 2012

Heinz Ebert, A Graph Theoretical Approach to the Collatz Problem, arXiv:1905.07575 [math.GM], 2019-2020.
Tanya Khovanova, Recursive Sequences.
Sergey Kitaev, On multi-avoidance of right angled numbered polyomino patterns, Integers: Electronic Journal of Combinatorial Number Theory, Vol. 4 (2004), Article A21, 20pp.
Index to sequences related to polygonal numbers.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A000217, A005408, A006370, A008588, A008615, A016921, A016789, A016933, A016945, A016958, A016969, A017617, A017329, A057145, A067412, A088827, A139600, A139606, A157176, A186424.

a016957 = (+ 4) . (* 6)  -- Reinhard Zumkeller, Jul 05 2013

seq(6*n+4, n = 0 .. 50) # Matt C. Anderson, Jun 09 2017

Range[4, 1000, 6] (* Vladimir Joseph Stephan Orlovsky, May 27 2011 *)

makelist(6*n+4, n, 0, 30); /* Martin Ettl, Nov 12 2012 */

a(n)=6*n+4 \\ Charles R Greathouse IV, Jul 10 2016

A008615(a(n)) = n+1. - Reinhard Zumkeller, Feb 27 2008
a(n) = A016789(n)*2. - Omar E. Pol, May 02 2008
A157176(a(n)) = A067412(n+1). - Reinhard Zumkeller, Feb 24 2009
a(n) = sqrt(A016958(n)). - Zerinvary Lajos, Jun 30 2009
a(n) = 2*(6*n+1) - a(n-1) (with a(0)=4). - Vincenzo Librandi, Nov 20 2010
a(n) = floor((sqrt(36*n^2 - 36*n + 1) + 6*n + 1)/2). - Srikanth K S, Dec 18 2010
From Colin Barker, Jan 30 2012: (Start)
G.f.: 2*(2+x)/(1-2*x+x^2).
a(n) = 2*a(n-1) - a(n-2). (End)
A089911(2*a(n)) = 9. - Reinhard Zumkeller, Jul 05 2013
a(n) = 3 * A005408(n) + 1. - Fred Daniel Kline, Oct 24 2015
a(n) = A057145(n+2,4). - R. J. Mathar, Jul 28 2016
a(4*n+2) = 4 * a(n). - Zhandos Mambetaliyev, Sep 22 2018
Sum_{n>=0} (-1)^n/a(n) = sqrt(3)*Pi/18 - log(2)/6. - Amiram Eldar, Dec 10 2021
E.g.f.: 2*exp(x)*(2 + 3*x). - Stefano Spezia, May 29 2024

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

Original entry on oeis.org

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

A067410 Triangle with columns built from certain power sequences.

Original entry on oeis.org

1, 2, 1, 4, 3, 1, 8, 12, 4, 1, 16, 48, 24, 5, 1, 32, 192, 144, 40, 6, 1, 64, 768, 864, 320, 60, 7, 1, 128, 3072, 5184, 2560, 600, 84, 8, 1, 256, 12288, 31104, 20480, 6000, 1008, 112, 9, 1, 512, 49152, 186624, 163840, 60000, 12096, 1568, 144, 10, 1

Offset: 0

Wolfdieter Lang, Jan 25 2002

			Triangle starts:
  1;
  2,  1;
  4,  3, 1;
  8, 12, 4, 1;
  ...

Indranil Ghosh, Rows 0..125, flattened

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

Columns m=0..8 are A000079, A002001, A067411, A067412, A090019, A067413, A067414, A067415, A067416.

A[n_,m_]:=If[n==m,1,(m+2)(2(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+2)*(2*(m+1))^(n-m-1) if n > m >= 0.

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

A157176 a(n+1) = a(n - n mod 2) + a(n - n mod 3), a(0) = 1.

Original entry on oeis.org

1, 2, 2, 3, 5, 8, 8, 16, 16, 24, 40, 64, 64, 128, 128, 192, 320, 512, 512, 1024, 1024, 1536, 2560, 4096, 4096, 8192, 8192, 12288, 20480, 32768, 32768, 65536, 65536, 98304, 163840, 262144, 262144, 524288, 524288, 786432, 1310720, 2097152, 2097152, 4194304, 4194304

Offset: 0

Reinhard Zumkeller, Feb 24 2009

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,8).

Cf. A001018, A008588, A013730, A016921, A016933, A016945, A016957, A016969, A067412, A103333.

LinearRecurrence[{0,0,0,0,0,8},{1, 2, 2, 3, 5, 8},45] (* Stefano Spezia, May 29 2024 *)

a(n+6) = 8*a(n).
a(6*k) = 8^k; a(A008588(n))=A001018(n);
a(6*k+1) = a(6*k+2) = 2*8^k; a(A016921(n))=a(A016933(n))=A013730(n);
a(6*k+3) = 3*8^k; a(A016945(n))=A103333(n+1);
a(6*k+4) = 5*8^k; a(A016957(n))=A067412(n+1);
a(6*k+5) = 8^(k+1); a(A016969(n))=A001018(n+1).
G.f.: (1 + 2*x + 2*x^2 + 3*x^3 + 5*x^4 + 8*x^5)/((1 - 2*x^2)*(1 + 2*x^2 + 4*x^4)). - Stefano Spezia, May 29 2024

a(43)-a(44) from Stefano Spezia, May 29 2024

A067425 Triangle with columns built from certain power sequences.

Original entry on oeis.org

1, 4, 1, 16, 5, 1, 64, 40, 6, 1, 256, 320, 72, 7, 1, 1024, 2560, 864, 112, 8, 1, 4096, 20480, 10368, 1792, 160, 9, 1, 16384, 163840, 124416, 28672, 3200, 216, 10, 1, 65536, 1310720, 1492992, 458752, 64000

Offset: 0

Wolfdieter Lang, Jan 25 2002

The fifth column (m=4) gives [1, 8, 160, 3200, 64000, 1280000, 25600000, ...].

			Triangle starts:
   1;
   4,  1;
  16,  5,  1;
  64, 40,  6,  1;
  ...

Indranil Ghosh, Rows 0..125, flattened

Cf. A009998, A067410, A067417.
Columns 0..3 are A000302 (powers of 4), A067412, A067419, A067404.
Columns 5..8 are A067426, A067427, A067428, A067429.

A067425[n_, m_] := If[n == m, 1, (m + 4)*(4*(m + 1))^(n - m - 1)];
Table[A067425[n, m], {n, 0, 10}, {m, 0, n}] (* Paolo Xausa, Oct 16 2024 *)

T(n,m) = 1 if n = m; T(n,m) = (m+4)*(4*(m+1))^(n-m-1) if n > m >= 0, else 0.

G.f. for column m: (x^m)*(1-3*m*x)/(1-4*(m+1)*x).

A164737 a(n) = 8*a(n-2) for n > 2; a(1) = 5, a(2) = 12.

Original entry on oeis.org

5, 12, 40, 96, 320, 768, 2560, 6144, 20480, 49152, 163840, 393216, 1310720, 3145728, 10485760, 25165824, 83886080, 201326592, 671088640, 1610612736, 5368709120, 12884901888, 42949672960, 103079215104, 343597383680, 824633720832

Offset: 1

Klaus Brockhaus, Aug 24 2009

nonn

Interleaving of 5*A001018 and 12*A001018.

Binomial transform is A096980 without initial terms 1. Second binomial transform is A164593. Third binomial transform is A101386.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,8).

Cf. A001018 (powers of 8), A067412, A096980, A101386, A164593.

[ n le 2 select 7*n-2 else 8*Self(n-2): n in [1..26] ];

seq(coeff(series( x*(5+12*x)/(1-8*x^2) , x, n+1), x, n), n=1..30); # G. C. Greubel, Apr 16 2020

LinearRecurrence[{0,8}, {5,12}, 30] (* G. C. Greubel, Apr 16 2020 *)

[(13 -7*(-1)^n)*2^((6*n -11 +3*(-1)^n)/4) for n in (1..30)] # G. C. Greubel, Apr 16 2020

a(n) = (13 - 7*(-1)^n)*2^(1/4*(6*n - 11 + 3*(-1)^n)).

G.f.: x*(5 + 12*x)/(1 - 8*x^2).

cp's OEIS Frontend

A016957 a(n) = 6*n + 4.

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

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A067410 Triangle with columns built from certain power sequences.

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A157176 a(n+1) = a(n - n mod 2) + a(n - n mod 3), a(0) = 1.

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A067425 Triangle with columns built from certain power sequences.

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A164737 a(n) = 8*a(n-2) for n > 2; a(1) = 5, a(2) = 12.

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A067413 Sixth column of triangle A067410.

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