A002023 -id:A002023 - cp's OEIS frontend

A283070 Sierpinski tetrahedron or tetrix numbers: a(n) = 2*4^n + 2.

4, 10, 34, 130, 514, 2050, 8194, 32770, 131074, 524290, 2097154, 8388610, 33554434, 134217730, 536870914, 2147483650, 8589934594, 34359738370, 137438953474, 549755813890, 2199023255554, 8796093022210, 35184372088834, 140737488355330, 562949953421314

Offset: 0

Peter M. Chema, Feb 28 2017

Number of vertices required to make a Sierpinski tetrahedron or tetrix of side length 2^n. The sum of the vertices (balls) plus line segments (rods) of one tetrix equals the vertices of its larger, adjacent iteration. See formula.
Equivalently, the number of vertices in the (n+1)-Sierpinski tetrahedron graph. - Eric W. Weisstein, Aug 17 2017
Also the independence number of the (n+2)-Sierpinski tetrahedron graph. - Eric W. Weisstein, Aug 29 2021
Final digit alternates 4 and 0.

Colin Barker, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Independence Number
Eric Weisstein's World of Mathematics, Sierpinski Tetrahedron Graph
Eric Weisstein's World of Mathematics, Tetrix
Eric Weisstein's World of Mathematics, Vertex Count
Index entries for linear recurrences with constant coefficients, signature (5,-4).

Subsequence of A016957.
Cf. A052539, A279511, A279512.
First bisection of A052548, A087288; second bisection of A049332, A133140, A135440.
Cf. A002023 (edge count).

Table[2 4^n + 2, {n, 0, 30}] (* Bruno Berselli, Feb 28 2017 *)
2 (4^Range[0, 20] + 1) (* Eric W. Weisstein, Aug 17 2017 *)
LinearRecurrence[{5, -4}, {4, 10}, 20] (* Eric W. Weisstein, Aug 17 2017 *)
CoefficientList[Series[-((2 (-2 + 5 x))/(1 - 5 x + 4 x^2)), {x, 0, 20}], x] (* Eric W. Weisstein, Aug 17 2017 *)

a(n)=2*4^n+2 \\ Charles R Greathouse IV, Feb 28 2017

Vec(2*(2 - 5*x) / ((1 - x)*(1 - 4*x)) + O(x^30)) \\ Colin Barker, Mar 02 2017

def a(n): return 2*4**n + 2
print([a(n) for n in range(25)]) # Michael S. Branicky, Aug 29 2021

G.f.: 2*(2 - 5*x)/((1 - x)*(1 - 4*x)).
a(n) = 5*a(n-1) - 4*a(n-2) for n > 1.
a(n+1) = a(n) + A002023(n).
a(n) = 2*A052539(n) = A188161(n) - 1 = A087289(n) + 1 = A056469(2*n+2) = A261723(4*n+1).
E.g.f.: 2*(exp(4*x) + exp(x)). - G. C. Greubel, Aug 17 2017

Entry revised by Editors of OEIS, Mar 01 2017

A292543 Number of 5-cycles in the n-Sierpinski tetrahedron graph.

Original entry on oeis.org

0, 96, 384, 1536, 6144, 24576, 98304, 393216, 1572864, 6291456, 25165824, 100663296, 402653184, 1610612736, 6442450944, 25769803776, 103079215104, 412316860416, 1649267441664, 6597069766656, 26388279066624, 105553116266496, 422212465065984, 1688849860263936

Offset: 1

Eric W. Weisstein, Sep 18 2017

Eric Weisstein's World of Mathematics, Graph Cycle
Eric Weisstein's World of Mathematics, Sierpinski Tetrahedron Graph
Index entries for linear recurrences with constant coefficients, signature (4).

Cf. A002023 (6*4^n).

Cf. A292540 (3-cycles), A292542 (4-cycles), A292545 (6-cycles).

Table[If[n == 1, 0, 6 4^n], {n, 20}]
Join[{0}, LinearRecurrence[{4}, {96}, 20]]
CoefficientList[Series[96 x/(1 - 4 x), {x, 0, 20}], x]

a(n) = 6*4^n = A002023(n) for n > 1.
a(n) = 4*a(n-1) for n > 2.
G.f.: 96*x^/(1 - 4*x).

A336641 Numbers k such that A007913(k) divides sigma(k) and A008833(k)-1 either divides A326127(k) (= sigma(k)-core(k)-k), or both are zero.

Original entry on oeis.org

6, 24, 28, 96, 120, 150, 294, 384, 496, 1014, 1536, 3276, 3750, 3780, 6144, 8128, 14406, 20328, 24576, 32760, 93750, 98304, 171366, 306180, 393216, 705894, 1241460, 1572864, 2343750, 6291456, 16380000, 24800580, 25165824, 28960854, 30387840, 33550336, 34588806, 58593750, 100663296, 165143160, 332226048, 402653184

Offset: 1

Antti Karttunen, Jul 28 2020

nonn

Numbers k such that A326128(k) = A326129(k) form a subsequence of this sequence. So far it is not known whether it contains any other terms apart from those of A000396. See comments in A326129.
Sequence is infinite because all numbers of the form 6*4^n (A002023) are present.
Question: Are there any odd terms?

Index entries for sequences where odd perfect numbers must occur, if they exist at all

Cf. A000203, A007913, A228058, A326126, A326127, A326128, A326129, A336642.
Cf. A000396, A002023 (subsequences).
Cf. also A336550 for a similar construction.

isA336641(n) = { my(c=core(n), s=sigma(n), u=((n/c)-1)); (!(s%c) && (gcd(u,(s-c-n))==u)); };

A084431 Expansion of g.f. (1 + 6x + 5x^2)/((1-2x)(1+2*x)).

Original entry on oeis.org

1, 6, 9, 24, 36, 96, 144, 384, 576, 1536, 2304, 6144, 9216, 24576, 36864, 98304, 147456, 393216, 589824, 1572864, 2359296, 6291456, 9437184, 25165824, 37748736, 100663296, 150994944, 402653184, 603979776, 1610612736, 2415919104

Offset: 0

Paul Barry, Jun 26 2003

Binomial transform is A085287.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,4).

Bisections are A002023 and A002063.

Cf. A085287.

[(-10*0^n-3*(-2)^n+21*2^n)/8: n in [0..30]]; // Vincenzo Librandi, Nov 16 2011

CoefficientList[Series[(1+6x+5x^2)/((1-2x)(1+2x)),{x,0,30}],x] (* or *) Join[{1},Flatten[NestList[4#&,{6,9},15]]] (* Harvey P. Dale, Nov 05 2011 *)

a(n) = (-10*0^n - 3*(-2)^n + 21*2^n)/8.
a(n) = 4*a(n-2), n > 1. - Harvey P. Dale, Nov 05 2011
E.g.f.: (9*cosh(2*x) + 12*sinh(2*x) - 5)/4. - Stefano Spezia, Sep 20 2023

A156067 a(0)=1. a(n)= -2^(n-1)-3*(-1)^n, n>1.

Original entry on oeis.org

1, 2, -5, -1, -11, -13, -35, -61, -131, -253, -515, -1021, -2051, -4093, -8195, -16381, -32771, -65533, -131075, -262141, -524291, -1048573, -2097155, -4194301, -8388611, -16777213, -33554435, -67108861, -134217731, -268435453, -536870915, -1073741821, -2147483651

Offset: 0

Paul Curtz, Feb 03 2009

The main diagonal of the array of A153130 and its successive differences.

A154589 is the second upper diagonal of the array.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,2).

Join[{1},LinearRecurrence[{1,2},{2,-5},40]] (* Harvey P. Dale, Dec 11 2011 *)

a(n)= +a(n-1) +2*a(n-2), n>2.
G.f.: x*(-2+7*x) / ( (1+x)*(2*x-1) ).
a(n) == A153130(n) (mod 9).
a(n+1)-2*a(n) = (-1)^n*9, n>0.
a(n) = A154589(n)-3*(-1)^n.
a(n)+a(n+3) = -A005010(n-1) = -9*A131577(n).
a(2*n)+a(2*n+1) = -3*2^(2n-1) = -A002023(n-2).

A321358 a(n) = (2*4^n + 7)/3.

Original entry on oeis.org

3, 5, 13, 45, 173, 685, 2733, 10925, 43693, 174765, 699053, 2796205, 11184813, 44739245, 178956973, 715827885, 2863311533, 11453246125, 45812984493, 183251937965, 733007751853, 2932031007405, 11728124029613, 46912496118445, 187649984473773, 750599937895085, 3002399751580333

Offset: 0

Paul Curtz, Nov 07 2018

Difference table:
3,  5, 13,  45,  173,  685,  2733, ...   (this sequence)
2,  8, 32, 128,  512, 2048,  8192, ...    A004171
6, 24, 96, 384, 1536, 6144, 24576, ...    A002023

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-4).

Cf. A010701, A010727, A020988, A083594, A002023, A004171, A155980, A171382, A193579.

a[n_]:= (2*4^n + 7)/3; Array[a, 20, 0] (* or *)
CoefficientList[Series[1/3 (7 E^x + 2 E^(4 x)), {x, 0, 20}], x]*Table[n!, {n, 0, 20}] (* Stefano Spezia, Nov 10 2018 *)

a(n) = (2*4^n + 7)/3; \\ Michel Marcus, Nov 08 2018

Vec((3 - 10*x) / ((1 - x)*(1 - 4*x)) + O(x^30)) \\ Colin Barker, Nov 10 2018

O.g.f.: (3 - 10*x) / ((1 - x)*(1 - 4*x)). - Colin Barker, Nov 10 2018
E.g.f.: (1/3)*(7*exp(x) + 2*exp(4*x)). - Stefano Spezia, Nov 10 2018
a(n) = 5*a(n-1) - 4*a(n-2), a(0) = 3, a(1) = 5.
a(n) = 4*a(n-1) - 7, a(0) = 3.
a(n) = (2/3)*(4^n-1)/3 + 3.
a(n) = A171382(2*n) = A155980(2*n+2).
a(n) = A193579(n)/3.
a(n) = A007583(n) + 2 = A001045(2*n+1) + 2.

More terms from Michel Marcus, Nov 08 2018

A337217 One half of the even numbers of A094739.

Original entry on oeis.org

1, 3, 5, 7, 11, 15, 21, 23, 29, 35, 39, 71, 95

Offset: 1

Wolfdieter Lang, Aug 20 2020

This finite sequence a(n), for n = 1, 2, ..., 13, appears as eq. (2.3) given by Kaplansky on p. 87.
It enters Theorem 2.1 of Kaplansky, p. 87, with proof on p. 90 (here reformulated): The positive integers uniquely represented by x^2 + y^2 + 2*z^2, with  0 <= x <= y and 0 <= z, consist of the 13 numbers a(n) and 4^k*6 = A002023(k), for integers k >= 0. See a comment in A002023 for this uniquely representable positive integers of this ternary form.
It also enters Theorem 2.3 of Kaplansky, p. 88, with proof on p.91 (here reformulated): The positive integers uniquely represented by x^2 + 2*y^2 + 4*z^2, with nonnegative integers x, y, z consist of the 13 odd numbers a(n) and the four even numbers 2, 10, 26, and 74. This is the finite sequence
1, 2, 3, 5, 7, 10, 11, 15, 21, 23, 26, 29, 35, 39, 71, 74, 95.

Irving Kaplansky, Integers Uniquely Represented by Certain Ternary Forms, in "The Mathematics of Paul Erdős I", Ronald. L. Graham and Jaroslav Nešetřil (Eds.), Springer, 1997, pp. 86 - 94.

Cf. A002023, A094739, A337218.

A355581 Exponentially-odd 3-smooth numbers: number of the form 2^i * 3^j where i and j are either 0 or odd.

Original entry on oeis.org

1, 2, 3, 6, 8, 24, 27, 32, 54, 96, 128, 216, 243, 384, 486, 512, 864, 1536, 1944, 2048, 2187, 3456, 4374, 6144, 7776, 8192, 13824, 17496, 19683, 24576, 31104, 32768, 39366, 55296, 69984, 98304, 124416, 131072, 157464, 177147, 221184, 279936, 354294, 393216, 497664

Offset: 1

Amiram Eldar, Jul 08 2022

			6 is a term since 6 = 2^1 * 3^1 and the exponents of 2 and 3 are both odd: 1.
24 is a term since 24 = 2^3 * 3^1 and the exponents of 2 and 3 are both odd: 3 and 1, respectively.

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

Intersection of A003586 and A268335.
Subsequences: A002023, A013711, A092810.
Cf. A355580.

q[n_] := Module[{e = IntegerExponent[n, {2, 3}]}, (e[[1]] == 0 || OddQ[e[[1]]]) && (e[[2]] == 0 || OddQ[e[[2]]]) && Times@@({2, 3}^e) == n]; Select[Range[500000], q]

is(n) = {my(e2 = valuation(n, 2), e3 = valuation(n, 3)); (e2 == 0 || e2%2) && (e3 == 0 || e3%2) && n == 2^e2 * 3^e3};

from itertools import count, takewhile
def aupto(lim):
    pows2 = list(takewhile(lambda x: xMichael S. Branicky, Jul 08 2022

Sum_{n>=1} 1/a(n) = 55/24.

A141516 The main diagonal of the array of A141425 and its higher order differences.

Original entry on oeis.org

1, 2, 1, -7, -23, -1, 7, -103, -251, -133, -149, -1387, -3143, -3001, -4913, -19663, -42611, -55693, -101549, -291667, -612863, -960001, -1831433, -4460023, -9185771, -15980053, -31162949, -69500347, -141392183, -261261001

Offset: 0

Paul Curtz, Aug 11 2008

The sequence A141425 and higher order differences in subsequent rows starts (see A141533):
1, 2, 4,  5,  7, 8, 1, 2, 4,  5,  7, 8, 1, 2, 4,...
1, 2, 1,  2,  1,-7, 1, 2, 1,  2,  1,-7, 1, 2, 1, 2,...
1,-1, 1, -1, -8, 8, 1,-1, 1, -1, -8, 8, 1, -1,..
-2, 2,-2, -7, 16,-7,-2, 2,-2, -7, 16,-7, -2,..
4,-4,-5, 23,-23, 5, 4,-4,-5, 23,-23, 5, 4,..
-8,-1,28,-46, 28,-1,-8,-1,28,-46, 28,-1,..
Reading downwards the main diagonal of this array defines the sequence.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (1,-1,3,6)

A108411 := proc(n) 3^floor(n/2) ; end proc:
A141516 := proc(n) if n = 0 then 1; else (-3*(-1)^n-2^n+3*(-1)^(floor((n-1)/2))*A108411(n))/2 ; end if; end proc: # R. J. Mathar, Mar 08 2011

LinearRecurrence[{1,-1,3,6},{1,2,1,-7,-23},30] (* Harvey P. Dale, Nov 23 2022 *)

a(n) = ( -3*(-1)^n -2^n +3*(-1)^(floor((n-1)/2))*A108411(n) )/2, n>0. - R. J. Mathar, Mar 08 2011
a(2n)+a(2n+1)= -A002023(n-1) = -3*A081294(n), n>0.
a(4n)+a(4n+1)+a(4n+2)+a(4n+3) = -120*16^(n-1), n>0.
a(4n+2)+a(4n+3)+a(4n+4)+a(4n+5) = -30*A001025(n).
G.f. x*(-2+x+6*x^2+21*x^3) / ( (2*x-1)*(1+x)*(3*x^2+1) ). - R. J. Mathar, Mar 08 2011

cp's OEIS Frontend

A283070 Sierpinski tetrahedron or tetrix numbers: a(n) = 2*4^n + 2.

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A292543 Number of 5-cycles in the n-Sierpinski tetrahedron graph.

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A336641 Numbers k such that A007913(k) divides sigma(k) and A008833(k)-1 either divides A326127(k) (= sigma(k)-core(k)-k), or both are zero.

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A084431 Expansion of g.f. (1 + 6*x + 5*x^2)/((1-2*x)*(1+2*x)).

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A156067 a(0)=1. a(n)= -2^(n-1)-3*(-1)^n, n>1.

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A321358 a(n) = (2*4^n + 7)/3.

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A337217 One half of the even numbers of A094739.

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A355581 Exponentially-odd 3-smooth numbers: number of the form 2^i * 3^j where i and j are either 0 or odd.

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A084431 Expansion of g.f. (1 + 6x + 5x^2)/((1-2x)(1+2*x)).