Peter M. Chema - cp's OEIS frontend

A285551 Volume of each square prism building the next 3-dimensional box in A100538 where side lengths form the Padovan spiral number sequence (A134816), starting with 1 X 1 X 1, 1 X 1 X 2, 2 X 2 X 2, 2 X 2 X 3, 4 X 4 X 5, ...

1, 2, 8, 12, 36, 80, 175, 441, 972, 2304, 5376, 12348, 29008, 67081, 156065, 363350, 843144, 1962396, 4560200, 10600000, 24648975, 57288465, 133194600, 309636096, 719790336, 1673379352, 3890033728, 9043304417, 21023197601, 48872682810, 113615800200, 264124052396

Offset: 1

Peter M. Chema, Apr 25 2017

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Robert Dickau, Padovan's Spiral Numbers, Wolfram Demonstrations Project Published: August 19, 2010.
Index entries for linear recurrences with constant coefficients, signature (1,2,3,-2, 4,-4,-1,-1,0,-1).

Cf. A000931, A100538, A134816.

A[n_]:=Sum[Binomial[k, n - 2k], {k, 0, Floor[n/2]}]; a000931[n_]:=If[n==0, 1, If[n<3, 0, A[n  - 3]]]; a[n_]:=a000931[n + 5]^2*a000931[n + 6]; Table[a[n], {n, 0, 50}] (* Indranil Ghosh, Apr 26 2017 *)
LinearRecurrence[{1, 2, 3, -2, 4, -4, -1, -1, 0, -1}, {1, 2, 8, 12, 36, 80, 175, 441, 972, 2304}, 40] (* Vincenzo Librandi, Jul 19 2017 *)

A(n) = sum(k=0, n\2, binomial(k, n - 2*k));
a000931(n) = if(n==0, 1, if(n<3, 0, A(n - 3)));
a(n) = a000931(n + 5)^2*a000931(n + 6); \\ Indranil Ghosh, Apr 26 2017

from sympy import binomial
def A(n): return sum([binomial(k, n - 2*k) for k in range(int(n/2) + 1)])
def a000931(n): return 1 if n==0 else 0 if n<3 else A(n - 3)
def a(n): return a000931(n + 5)**2*a000931(n + 6) # Indranil Ghosh, Apr 26 2017

a(n) = A000931(n+5)^2*A000931(n+6).

a(n) = A100538(n+1) - A100538(n).

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

Original entry on oeis.org

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

A281381 a(n) = n(n + 1)(4*n + 5)/2.

Original entry on oeis.org

0, 9, 39, 102, 210, 375, 609, 924, 1332, 1845, 2475, 3234, 4134, 5187, 6405, 7800, 9384, 11169, 13167, 15390, 17850, 20559, 23529, 26772, 30300, 34125, 38259, 42714, 47502, 52635, 58125, 63984, 70224, 76857, 83895, 91350, 99234, 107559, 116337, 125580, 135300, 145509, 156219, 167442, 179190, 191475

Offset: 0

Peter M. Chema, Jan 21 2017

Shares its digital root, zero together with period 9: repeat [3, 3, 3, 6, 6, 6, 9, 9, 9] with A027480.

Final digits cycle a length period 20: repeat [0, 9, 9, 2, 0, 5, 9, 4, 2, 5, 5, 4, 4, 7, 5, 0, 4, 9, 7, 0].

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Partial sums of A195319.

Cf. A006002, A007742, A016061, A027480, A281258.

[n*(n+1)*(4*n+5)/2 : n in [0..50]]; // Wesley Ivan Hurt, Aug 30 2022

Table[n (n + 1) (4 n + 5)/2, {n, 0, 45}] (* or *)
CoefficientList[Series[3 x (3 + x)/(1 - x)^4, {x, 0, 45}], x] (* Michael De Vlieger, Jan 21 2017 *)

concat(0, Vec(3*x*(3 + x) / (1 - x)^4 + O(x^50))) \\ Colin Barker, Jan 21 2017

a(n) = n*(n + 1)*(4*n + 5)/2 \\ Charles R Greathouse IV, Feb 01 2017

a(n) = 2*n^3 + 9*n^2/2 + 5*n/2.
a(n) = 3*A016061(n).
a(n) = A006002(n+1)*(n) - A006002(n)*(n-1).
a(n) = A007742(n)*(n - 1)/2.
From Colin Barker, Jan 21 2017: (Start)
G.f.: 3*x*(3 + x) / (1 - x)^4.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>3. (End)
From Stefano Spezia, Aug 30 2022: (Start)
E.g.f.: exp(x)*x*(18 + 21*x + 4*x^2)/2.
Sum_{n>0} 1/a(n) = 2*(20*log(8) + 10*Pi - 71)/25 = 0.1603805895595720759728288896228498341201... . (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = 4*sqrt(2)*Pi/5 + 4*(3+sqrt(2))*log(2)/5 - 8*sqrt(2)*log(2-sqrt(2))/5 - 178/25. - Amiram Eldar, Sep 22 2022

A281258 Digital root of n(n+1)(n+2)/2.

Original entry on oeis.org

0, 3, 3, 3, 6, 6, 6, 9, 9, 9, 3, 3, 3, 6, 6, 6, 9, 9, 9, 3, 3, 3, 6, 6, 6, 9, 9, 9, 3, 3, 3, 6, 6, 6, 9, 9, 9, 3, 3, 3, 6, 6, 6, 9, 9, 9, 3, 3, 3, 6, 6, 6, 9, 9, 9, 3, 3, 3, 6, 6, 6, 9, 9, 9, 3, 3, 3, 6, 6, 6, 9, 9, 9, 3, 3, 3, 6, 6, 6, 9, 9, 9, 3, 3, 3, 6, 6, 6, 9, 9, 9, 3, 3, 3, 6, 6, 6, 9, 9, 9, 3, 3, 3, 6, 6, 6, 9, 9, 9, 3, 3, 3, 6, 6, 6, 9, 9, 9

Offset: 0

Peter M. Chema, Jan 18 2017

Also zero together with period 9: repeat [3, 3, 3, 6, 6, 6, 9, 9, 9].

Also the decimal expansion of 334001/1001001.

Index entries for linear recurrences with constant coefficients, signature (1,0,-1,1,0,-1,1).

Cf. A010888, A027480.

PadRight[{0},120,{9,3,3,3,6,6,6,9,9}] (* Harvey P. Dale, Mar 22 2018 *)

a(n) = A010888(A027480(n)).
From Stefano Spezia, Aug 30 2022: (Start)
G.f.: 3*x*(1 + 2*x^3 + 3*x^6)/((1 - x)*(1 + x^3 + x^6)).
a(n) = a(n-1) - a(n-3) + a(n-4) - a(n-6) + a(n-7) for n > 7. (End)

A280426 Digital roots of tetranacci numbers A000078.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 4, 8, 6, 2, 2, 9, 1, 5, 8, 5, 1, 1, 6, 4, 3, 5, 9, 3, 2, 1, 6, 3, 3, 4, 7, 8, 4, 5, 6, 5, 2, 9, 4, 2, 8, 5, 1, 7, 3, 7, 9, 8, 9, 6, 5, 1, 3, 6, 6, 7, 4, 5, 4, 2, 6, 8, 2, 9, 7, 8, 8, 5, 1, 4, 9, 1, 6, 2, 9, 9, 8, 1, 9, 9, 9, 1, 1, 2, 4, 8, 6, 2

Offset: 0

Peter M. Chema, Jan 02 2017

From a(3) onwards, periodic with period length 78, twice the periodicity of the digital roots of tribonacci numbers A222407 (also the 12th triangular number).

Cf. A000078, A010888, A222407.

a(n) = A010888(A000078(n)).

A280308 Tribonacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3) with a(0)=3, a(1)=4, a(2)=5.

Original entry on oeis.org

3, 4, 5, 12, 21, 38, 71, 130, 239, 440, 809, 1488, 2737, 5034, 9259, 17030, 31323, 57612, 105965, 194900, 358477, 659342, 1212719, 2230538, 4102599, 7545856, 13878993, 25527448, 46952297, 86358738, 158838483, 292149518, 537346739, 988334740, 1817830997, 3343512476, 6149678213, 11311021686, 20804212375

Offset: 0

Peter M. Chema, Dec 31 2016

Like other tribonacci sequences, the digital root is period length 39, and is as follows: (3, 4, 5, 3, 3, 2, 8, 4, 5, 8, 8, 3, 1, 3, 7, 2, 3, 3, 8, 5, 7, 2, 5, 5, 3, 4, 3, 1, 8, 3, 3, 5, 2, 1, 8, 2, 2, 3, 7).
Completes the set of tribonacci numbers with 3,4,5 as initial terms, the others being (3,5,4), (4,5,3), (4,3,5), (5,3,4), and (5,4,3). The sum of each of the digital root periods in the above set is 162, except (4,3,5), which sums to 180; the sum of the digital root period of A081172 is also 180.
Each digital root period for tribonacci sequences has triple patterns in cycles of 13, such as period (1,4,7) or digital root of 4^n.

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

Cf. A000073, A001590, A081172, A000213.

RecurrenceTable[{a[n] == a[n - 1] + a[n - 2] + a[n - 3], a[0] == 3, a[1] == 4, a[2] == 5}, a, {n, 38}] (* Michael De Vlieger, Dec 31 2016 *)
LinearRecurrence[{1, 1, 1}, {3, 4, 5}, 40] (* Vincenzo Librandi, Jan 01 2017 *)

G.f.: (3+x-2*x^2)/(1-x-x^2-x^3). - Vincenzo Librandi, Jan 01 2017

A275704 Digital root of n + (n+1)^2.

Original entry on oeis.org

1, 5, 2, 1, 2, 5, 1, 8, 8, 1, 5, 2, 1, 2, 5, 1, 8, 8, 1, 5, 2, 1, 2, 5, 1, 8, 8, 1, 5, 2, 1, 2, 5, 1, 8, 8, 1, 5, 2, 1, 2, 5, 1, 8, 8, 1, 5, 2, 1, 2, 5, 1, 8, 8, 1, 5, 2, 1, 2, 5, 1, 8, 8, 1, 5, 2, 1, 2, 5, 1, 8, 8, 1, 5, 2, 1, 2, 5, 1, 8, 8, 1, 5, 2, 1, 2, 5, 1, 8, 8, 1, 5, 2, 1, 2, 5, 1, 8, 8, 1, 5, 2, 1, 2, 5, 1, 8, 8, 1, 5, 2, 1, 2, 5, 1, 8, 8, 1, 5, 2, 1

Offset: 0

Peter M. Chema, Aug 05 2016

Period 9: repeat [1,5,2,1,2,5,1,8,8].
When expressed starting from the 9th term, i.e., 8,1,5,2,1,2,5,1,8,8,1,5,2,1,2,5,1,8, etc., the sequence is palindromic.
Decimal expansion is 50708396/333333333.

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,1).

Cf. A010888, A028387.

Table[FixedPoint[Total@ IntegerDigits@ # &, n + (n + 1)^2], {n, 0, 120}] (* or *)
PadRight[{}, 120, {1, 5, 2, 1, 2, 5, 1, 8, 8}] (* Michael De Vlieger, Aug 07 2016 *)

a(n) = A010888(A028387(n)).

G.f.: ( -1 - 5*x - 2*x^2 - x^3 - 2*x^4 - 5*x^5 - x^6 - 8*x^7 - 8*x^8 ) / ( (x-1)*(1+x+x^2)*(x^6+x^3+1) ). - R. J. Mathar, Jan 07 2019

A275615 Decimal expansion of 22/111.

Original entry on oeis.org

1, 9, 8, 1, 9, 8, 1, 9, 8, 1, 9, 8, 1, 9, 8, 1, 9, 8, 1, 9, 8, 1, 9, 8, 1, 9, 8, 1, 9, 8, 1, 9, 8, 1, 9, 8, 1, 9, 8, 1, 9, 8, 1, 9, 8, 1, 9, 8, 1, 9, 8, 1, 9, 8, 1, 9, 8, 1, 9, 8, 1, 9, 8, 1, 9, 8, 1, 9, 8, 1, 9, 8, 1, 9, 8, 1, 9, 8, 1, 9, 8, 1, 9, 8, 1, 9, 8, 1, 9, 8, 1, 9, 8, 1, 9, 8, 1, 9, 8, 1, 9, 8

Offset: 0

Peter M. Chema, Aug 03 2016

Period 3: repeat [1,9,8]. Also the digital root of Nexus numbers A022522.

Other variants of this period are decimal expansions of 91/111 (.819 repeating) and 109/111 (.981 repeating).

			0.198198198198198198198198198198198198198198198198198198198...

Cf. A010888, A022522.

First@ RealDigits@ N[22/111, 120] (* or *)
PadRight[{}, 120, {1, 9, 8}] (* or *)
FixedPoint[Total@ IntegerDigits@ # &, #] & /@ Table[(n + 1)^6 - n^6, {n, 0, 120}] (* Michael De Vlieger, Aug 04 2016 *)

a(n) = my(x=[1, 9, 8]); x[n%3+1] \\ Felix Fröhlich, Aug 04 2016

a(n) = A010888(A022522(n)).

a(n) = 6 - 5*cos(2*n*Pi/3) + sin(2*n*Pi/3)/sqrt(3). - Wesley Ivan Hurt, Oct 04 2018

A269222 Period 4: repeat [1,9,8,9].

Original entry on oeis.org

1, 9, 8, 9, 1, 9, 8, 9, 1, 9, 8, 9, 1, 9, 8, 9, 1, 9, 8, 9, 1, 9, 8, 9, 1, 9, 8, 9, 1, 9, 8, 9, 1, 9, 8, 9, 1, 9, 8, 9, 1, 9, 8, 9, 1, 9, 8, 9, 1, 9, 8, 9, 1, 9, 8, 9, 1, 9, 8, 9, 1, 9, 8, 9, 1, 9, 8, 9, 1, 9, 8, 9, 1, 9, 8, 9, 1, 9, 8, 9, 1, 9, 8, 9, 1, 9, 8, 9

Offset: 1

Peter M. Chema, Jul 11 2016

Digital root of Fib(18*n).

Decimal expansion of 221/1111.

			For n=2, a(2) = digital root of Fibonacci(18*2) or 14930352; therefore, a(2) = 9, since the digital root of 14930352 = 9.

Index entries for linear recurrences with constant coefficients, signature (0,0,0,1).

Cf. A008600, A010888.

Table[NestWhile[Total@ IntegerDigits@ # &, Fibonacci[18 n], IntegerLength@ # > 1 &], {n, 120}] (* Michael De Vlieger, Jul 11 2016 *)
PadRight[{},100,{1,9,8,9}] (* Harvey P. Dale, Sep 24 2021 *)

a(n)=[9,1,9,8][n%4+1] \\ Charles R Greathouse IV, Jul 21 2016

a(n) = A010888(Fibonacci(18*n)).
From Wesley Ivan Hurt, Sep 03 2022: (Start)
a(n) = a(n-4) for n >= 5.
a(n) = (9/4)*(3+(-1)^n)-7*sin(n*Pi/2)/2. (End)

A269226 Period 6: repeat [3, 9, 6, 6, 9, 3].

Original entry on oeis.org

3, 9, 6, 6, 9, 3, 3, 9, 6, 6, 9, 3, 3, 9, 6, 6, 9, 3, 3, 9, 6, 6, 9, 3, 3, 9, 6, 6, 9, 3, 3, 9, 6, 6, 9, 3, 3, 9, 6, 6, 9, 3, 3, 9, 6, 6, 9, 3, 3, 9, 6, 6, 9, 3, 3, 9, 6, 6, 9, 3, 3, 9, 6, 6, 9, 3, 3, 9, 6, 6, 9, 3, 3, 9, 6, 6, 9, 3, 3, 9, 6, 6, 9, 3, 3, 9, 6, 6, 9, 3, 3, 9, 6, 6, 9, 3

Offset: 1

Peter M. Chema, Jul 11 2016

The palindromic sequence arising when the digital root of n alternates diagonally in opposite directions on a square grid.  This is the sequence of 3-6-9 appearing every third column on a square grid when A010888 (digital root of n) alternates in both directions diagonally. Other columns are the digital root of 2^n: {1, 2, 4, 8, 7, 5}, or in its opposite direction 5^n: {5,7,8,4,2,1}.  All diagonals parallel to the digital roots of n are also {1,2,3,4,5,6,7,8,9} or {9,8,7,6,5,4,3,2,1}.
See the link below for a visual illustration.
This sequence also arises when A180592 (digital root of 2n) is substituted for A010888.
Decimal expansion of 40070/10101. - David A. Corneth, Jul 12 2016

Peter M. Chema, Illustration of initial terms
Index entries for linear recurrences with constant coefficients, signature (1,-1,1,-1,1).

Cf. A010888, A180592.

a(n)=[3, 3, 9, 6, 6, 9][n%6+1] \\ Charles R Greathouse IV, Jul 12 2016

a(n+1) = digital root of 5^n - 2^n.
a(n) = a(n-1) - a(n-2) + a(n-3) - a(n-4) + a(n-5) = a(n-6). - Charles R Greathouse IV, Jul 12 2016
a(n) = (12 - 3*cos(n*Pi/3) - 3*cos(2*n*Pi/3) - sqrt(3)*sin(n*Pi/3) - 3*sqrt(3)*sin(2*n*Pi/3))/2. - Wesley Ivan Hurt, Oct 05 2018

cp's OEIS Frontend

User: Peter M. Chema

A285551 Volume of each square prism building the next 3-dimensional box in A100538 where side lengths form the Padovan spiral number sequence (A134816), starting with 1 X 1 X 1, 1 X 1 X 2, 2 X 2 X 2, 2 X 2 X 3, 4 X 4 X 5, ...

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

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A281381 a(n) = n*(n + 1)*(4*n + 5)/2.

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A281258 Digital root of n*(n+1)*(n+2)/2.

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A280426 Digital roots of tetranacci numbers A000078.

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A280308 Tribonacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3) with a(0)=3, a(1)=4, a(2)=5.

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A275704 Digital root of n + (n+1)^2.

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A275615 Decimal expansion of 22/111.

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A281381 a(n) = n(n + 1)(4*n + 5)/2.

A281258 Digital root of n(n+1)(n+2)/2.