powers - cp's OEIS frontend

A381681 a(n) is one of two integer components (with A000254) used in computing the inverse second moment of X+n, where X~Poisson(1).

0, 1, 2, 7, 30, 159, 998, 7251, 59862, 553591, 5669406, 63698427, 779065694, 10304068863, 146547757014, 2230287456259, 36165665815878, 622513383121671, 11336090988469742, 217741030441959051, 4399571340398826126, 93286012779568250767, 2071087588405552461414, 48048511292938827392403

Offset: 0

Michael R. Powers, Mar 05 2025

nonn

Analog of A093344 (with alternating terms in inner summation).

			If X~Poisson(1), then E[(X+n)^(-2)] = (-1)^n * {(n-1)! * [-Ei(1)+gamma] - A000254(n-1) + e*a(n-1)}/e for n = 1,2,... where gamma is Euler's constant.

Michael R. Powers, Conjunctions of Three "Euler Constants" in Poisson-Related Expressions, arXiv:2503.02054 [math.NT], 2025.

A093344 gives one of two integer components (with A000254) used in computing the alternating inverse second moment of X+n for X~Poisson(1).

a(n) = n! * sum(i=1, n, (1/i)*sum(j=0, i-1, (-1)^j/j!)); \\ Michel Marcus, Mar 07 2025

a(n) = n! * Sum_{i=1..n} (1/i)*Sum_{j=0..i-1} (-1)^j/j!.

A343129 Numbers that are palindromes in base 3/2.

Original entry on oeis.org

0, 1, 2, 5, 8, 17, 35, 170, 278, 422, 494

Offset: 0

Wyatt Powers, Apr 05 2021

Next term >= 10^12. - Joerg Arndt, Apr 06 2021

Next term > 2^64. - David Radcliffe, Dec 10 2021

			    0:  0
    1:  1
    2:  2
    5:  22
    8:  212
   17:  21012
   35:  212212
  170:  2120220212
  278:  21221112212
  422:  2101100011012
  494:  2120010100212

Matvey Borodin, Hannah Han, Kaylee Ji, Tanya Khovanova, Alexander Peng, David Sun, Isabel Tu, Jason Yang, William Yang, Kevin Zhang, and Kevin Zhao, Variants of Base 3 over 2, arXiv:1901.09818 [math.NT], 2019.
B. Chen, R. Chen, J. Guo, S. Lee et al., On Base 3/2 and its sequences, arXiv:1808.04304 [math.NT], 2018.

Cf. A024629.

(* All terms <= 500: *)
b = {};
a[n_] := If[n < 1, 0, a[Quotient[n, 3] 2] 10 + Mod[n, 3]];
Do[If[PalindromeQ[a[n]], AppendTo[b, n], Nothing], {n, 0, 500}];
b

A342802 Replace 2^k with (-3)^k in binary expansion of n.

Original entry on oeis.org

0, 1, -3, -2, 9, 10, 6, 7, -27, -26, -30, -29, -18, -17, -21, -20, 81, 82, 78, 79, 90, 91, 87, 88, 54, 55, 51, 52, 63, 64, 60, 61, -243, -242, -246, -245, -234, -233, -237, -236, -270, -269, -273, -272, -261, -260, -264, -263, -162, -161, -165, -164, -153, -152, -156, -155, -189, -188, -192, -191, -180, -179, -183, -182

Offset: 0

Wyatt Powers, Mar 21 2021

All terms correspond to a sum of distinct powers of -3.

			For n = 0, a(0) = 0.
for n = 1, a(1) = -3^0 = 1.
for n = 2, a(2) = -3^1 = -3.
for n = 3, a(3) = -3^1 + -3^0 = -2.
for n = 4, a(4) = -3^2 = 9.
for n = 5, a(5) = -3^2 + -3^0 = 10.

Wyatt Powers, Table of n, a(n) for n = 0..9999

Cf. A005836 (sums of distinct powers of 3), A053985, A065369.

(* Returns first 100 numbers in the sequence; assigned to the list, a *)
a = Table[IntegerDigits[x, 2], {x, 0, 100}];
For[i = 1, i <= Length[a], i++,
  For[j = 1, j <= Length[a[[i]]], j++,
   a[[i]][[j]] = ((a[[i]][[j]])*(-3)^(Length[a[[i]]] - j))
   ]
  ];
For[i = 1, i <= Length[a], i++, a[[i]] = Total[a[[i]]]];
a

a(n) = my(b=Vecrev(binary(n))); sum(k=1, #b, b[k]*(-3)^(k-1)); \\ Michel Marcus, Mar 22 2021

a(n) = fromdigits(binary(n),-3) \\ Kevin Ryde, Mar 22 2021

def a(n):
  return sum((-3)**k for k, b in enumerate(bin(n)[2:][::-1]) if b=='1')
print([a(n) for n in range(64)]) # Michael S. Branicky, Mar 23 2021

A124437 Experience Points thresholds for levels in the pen and paper role-playing game "Das Schwarze Auge" (DSA, a.k.a. "The Dark Eye").

Original entry on oeis.org

0, 100, 300, 600, 1000, 1500, 2100, 2800, 3600, 4500, 5500, 6600, 7800, 9100, 10500, 12000, 13600, 15300, 17100, 19000, 21000, 23100, 25300, 27600, 30000, 32500, 35100, 37800, 40600, 43500, 46500, 49600, 52800, 56100, 59500, 63000, 66600

Offset: 1

Christoph D. Schmidt (snu(AT)power.ms), Dec 16 2006, Dec 21 2006

			a(7)=2100 because a(6) + (7-1)*100 = (a(5) + 500) + 600 =(((((0+100) + 200) + 300) + 400) + 500) + 600.

"Das Schwarze Auge - Basisregelwerk" (basic rule book), Fantasy Productions Verlags- und Medienvertriebsgesellschaft mbH, Erkrath, Germany, 2005, ISBN 3890644406

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

int folge(n){ if (n==1) return 0; return (folge(n-1)+(n-1)*100); }

Table[50 n (n - 1), {n, 37}] (* or *)
CoefficientList[Series[100 x^2/(1 - x)^3, {x, 0, 37}], x] (* Michael De Vlieger, Jul 11 2016 *)
LinearRecurrence[{3,-3,1},{0,100,300},40] (* Harvey P. Dale, Jun 26 2017 *)

a(n)=50*n*(n-1) \\ Charles R Greathouse IV, Jun 17 2017

a(n) = a(n-1) + (n-1)*100; a(1) = 0.
From Chai Wah Wu, Jul 11 2016: (Start)
a(n) = 50*n*(n-1).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3.
G.f.: 100*x^2/(1 - x)^3. (End)

A086901 a(1) = a(2) = 1; for n>2, a(n) = 4a(n-1) + 3a(n-2).

Original entry on oeis.org

1, 1, 7, 31, 145, 673, 3127, 14527, 67489, 313537, 1456615, 6767071, 31438129, 146053729, 678529303, 3152278399, 14644701505, 68035641217, 316076669383, 1468413601183, 6821884412881, 31692778455073, 147236767058935

Offset: 1

Rick Powers (rick.powers(AT)mnsu.edu), Sep 18 2003

			a(3) = 4*1 + 3*1 = 7;
a(4) = 4*7 + 3*1 = 31.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Sela Fried and Toufik Mansour, Staircase graph words, arXiv:2312.08273 [math.CO], 2023.
Lucyna Trojnar-Spelina and Iwona Włoch, On Generalized Pell and Pell-Lucas Numbers, Iranian Journal of Science and Technology, Transactions A: Science (2019), 1-7.

Cf. A015530, A102900, A122542.

a086901 n = a086901_list !! (n-1)
a086901_list = 1 : 1 : zipWith (+)
               (map (* 3) a086901_list) (map (* 4) $ tail a086901_list)
-- Reinhard Zumkeller, Feb 13 2015

[n le 2 select 1 else 4*Self(n-1) +3*Self(n-2): n in [1..41]]; // G. C. Greubel, Oct 28 2024

a[n_]:=(MatrixPower[{{3,2},{3,1}},n].{{2},{1}})[[2,1]]; Table[a[n],{n,0,40}] (* Vladimir Joseph Stephan Orlovsky, Feb 20 2010 *)
Transpose[NestList[Flatten[{Rest[#],ListCorrelate[{3,4},#]}]&, {1,1},40]][[1]]  (* Harvey P. Dale, Mar 23 2011 *)
LinearRecurrence[{4,3}, {1,1}, 41] (* G. C. Greubel, Oct 28 2024 *)

A086901(n)=if(n<3,1,4*A086901(n-1)+3*A086901(n-2)) \\ Michael B. Porter, Apr 04 2010

A086901=BinaryRecurrenceSequence(4,3,1,1)
[A086901(n) for n in range(41)] # G. C. Greubel, Oct 28 2024

a(n) = ((c + 5)*b^n - (b + 5)*c^n)/14, where b = 2 + sqrt(7), c = 2 - sqrt(7).
From Ralf Stephan, Feb 01 2004: (Start)
G.f.: x*(1-3*x)/(1 - 4*x - 3*x^2).
a(n) = A015530(n) - 3*A015530(n-1) = 1 + 6*Sum_{k=0..n-2} A015530(k). (End)
a(n+1) = Sum_{k=0..n} 3^(n-k)*A122542(n,k), n>=0. - Philippe Deléham, Oct 27 2006
a(n) = upper left term in the 2 X 2 matrix [1,2; 3,3]^(n-1). - Gary W. Adamson, Mar 02 2008
G.f.: G(0)*(1-3*x)/(2-4*x), where G(k) = 1 + 1/(1 - x*(7*k-4)/(x*(7*k+3) - 2/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 16 2013
E.g.f.: exp(2*x)*( cosh(sqrt(7)*x) - (1/sqrt(7))*sinh(sqrt(7)*x) ). - G. C. Greubel, Oct 28 2024

More terms from Ray Chandler, Sep 19 2003

A057973 Number of polybricks: number of ways to arrange n 1 X 2 "bricks" in a wall (see illustrations).

Original entry on oeis.org

1, 2, 5, 16, 55, 225, 949, 4269, 19500, 91115, 429742, 2047660, 9820197, 47383255, 229725560, 1118568692, 5466616025, 26804560282, 131817042605, 649952289243

Offset: 1

Warren Power (wjpnply(AT)hotmail.com), Oct 21 2000

The tiling of bricks is topologically the same as that by regular hexagons and this sequence can also be seen as counting polyhexes where two polyhexes are equivalent iff they are related by a symmetry that is also a symmetry of the tiling by bricks.

Other references on polyforms are: www.mathpuzzle.com, Solomon W. Golomb, Ed Pegg, Eric Weisstein, David A. Klarner (Packing rectangles) and Michael Reid [These references should be expanded! - N. J. A. Sloane]

Brendan Owen and Livio Zucca, Polyform generation
Brendan Owen and Livio Zucca, The 16 polybricks of order 4
N. J. A. Sloane, The polybricks of orders 1, 2 and 3

More terms from Don Reble, Nov 01 2001

Corrected and extended by Joseph Myers, Sep 21 2002

A057707 a(n) is the number of free 'polyominoes' on the (4.8.8) grid with n octagons and n squares.

Original entry on oeis.org

1, 8, 80, 1296, 22601, 426305, 8340302, 167925635, 3453618607

Offset: 1

Warren Power (wjpnply(AT)hotmail.com), Oct 24 2000

This sequence enumerates a subset of the polyforms enumerated by A343577. - John Mason, Jan 15 2023

John Mason, Illustration for n = 2
Wikipedia, Truncated square tiling.
Livio Zucca, PolyMultiForms

Cf. A343577.

Better description from Larry Reeves (larryr(AT)acm.org), Nov 30 2000
a(6)-a(8) from John Mason, Jan 15 2023
a(9) from John Mason, Mar 01 2023

A075152 Number of possible permutations of a Rubik cube of size n X n X n.

Original entry on oeis.org

1, 3674160, 43252003274489856000, 7401196841564901869874093974498574336000000000, 282870942277741856536180333107150328293127731985672134721536000000000000000

Offset: 1

Warren Power, Sep 05 2002

nonn

More precisely, order of group of n X n X n Rubik cube, under assumptions not-s, not-m, not-i.
The three possible assumptions considered here are the following:
s (for n odd) indicates that we are working in the "supergroup" and so take account of twists of the face centers.
m (for n > 3) indicates that the pieces are marked so that we take account of the permutation of the identically-colored pieces on a face.
i (for n > 3) indicates that we are working in the theoretical invisible group and solve the pieces on the interior of the cube as well as the exterior. It is assumed that the M and S traits apply to the interior pieces as if they were on the exterior of a smaller cube.

Dan Hoey, posting to Cube Lovers List, Jun 24, 1987.
Rowley, Chris, The group of the Hungarian magic cube, in Algebraic structures and applications (Nedlands, 1980), pp. 33-43, Lecture Notes in Pure and Appl. Math., 74, Dekker, New York, 1982.

Robert Munafo, Table of n, a(n) for n = 1..27 (first 10 terms from Robert G. Wilson v)
Answers.com, Rubik's Cube.
Isaiah Bowers, How To Solve A Rubik's Cube.
Richard Carr, The Number of Possible Positions of an N x N x N Rubik Cube
Cube Lovers, Discussions on the mathematics of the cube
Cube Lovers Archive, Mailing List
Cube20.org, God's Number is 20
Christophe Goudey, Information
David Joyner, Adventures in Group Theory: Rubik's Cube, Merlin's Machine, and Other Mathematical Toys, 2008. See Example 4.5.1. p. 73.
Naren Ramesh, Generalising the configurations of an N X N X N Rubik's Cube, Parabola (2023) Vol. 59, Issue 3. See p. 22.
Jaap Scherphuis, Puzzle Pages
Eric Weisstein's World of Mathematics, Rubik's Cube
WikiHow, How to Solve a Rubik's Cube with the Layer Method
Wikipedia, Rubik's Cube
Wikipedia, Professor's Cube

See A007458, A054434, A074914, A080656-A080662 for other versions.

Cf. A079761, A079762, A152169 (sums give a(2)), A080601, A080602 (sums give a(3)).

f := proc(n) local A,B,C,D,E,F,G; if n mod 2 = 1 then A := (n-1)/2; F := 0; B := 1; C := 1; D := 0; E := (n+1)*(n-3)/4; G := (n-1)*(n-3)/4; else A := n/2; F := 1; B := 1; C := 0; D := 0; E := n*(n-2)/4; G := (n-2)^2/4; fi; (2^A*((8!/2)*3^7)^B*((12!/2)*2^11)^C*((4^6)/2)^D*(24!/2)^E)/(24^F*((24^6)/2)^G); end;

f[n_] := Block[{a, b, c, d, e, f, g}, If[OddQ@ n, a = (n - 1)/2; b = c = 1; d = f = 0; e = (n + 1) (n - 3)/4; g = (n - 1) (n - 3)/4, a = n/2; b = f = 1; c = d = 0; e = n (n - 2)/4; g = (n - 2)^2/4]; Ceiling[(2^a*((8!/2)*3^7)^b*((12!/2)*2^11)^c*((4^6)/2)^d*(24!/2)^e)/(24^f*((24^6)/2)^g)]]; Array[f, 10] (* Robert G. Wilson v, May 23 2009 *)
f[1]=1; f[2]=7!3^6; f[3]=8!3^7 12!2^10;f[n_]:=f[n-2]*24^6*(24!/24^6)^(n-2);Table[f[n],{n,1,10}] (* Herbert Kociemba, Dec 08 2016 *)
f[1]=1;f[n_]:=7!3^6(6*24!!)^(s=Mod[n,2])24!^(r=(n-s)/2-1)(24!/4!^6)^(r(r+s)); Array[f,5] (* Herbert Kociemba, Jul 03 2022 *)

A075152(n) := block( if n = 1 then return (1), [a:1,b:1,c:1,d:1,e:1,f:1,g:1], if mod(n, 2) = 1 then (  a : (n-1)/2,  f : 0,  b : 1,  c : 1,  d : 0,  e : (n+1)*(n-3)/4,  g : (n-1)*(n-3)/4 ) else (  a : n/2,  f : 1,   b : 1,   c : 0,   d : 0,   e : n*(n-2)/4, g : (n-2)^2/4  ), return ( (2^a * ((factorial(8)/2)*3^7)^b * ((factorial(12)/2)*2^11)^c * ((4^6)/2)^d * (factorial(24)/2)^e) / (24^f * ((24^6)/2)^g) ) )$ for i:1 thru 27 step 1 do ( sprint(i, A075152(i)), newline() )$ /* Robert Munafo, Nov 12 2014 */

A075152(n)=ceil(3674160*(11771943321600)^(n%2)*620448401733239439360000^floor((n-2)/2)*(3246670537110000)^floor(((n-2)/2)^2)) \\ Davis Smith, Mar 20 2020

a(1)=1; a(2)=7!*3^6; a(3)=8!*3^7*12!*2^10; a(n)=a(n-2)*24^6*(24!/24^6)^(n-2). - Herbert Kociemba, Dec 08 2016

a(n) = ceiling(3674160*11771943321600^(n mod 2)*620448401733239439360000^floor((n - 2)/2)*3246670537110000^floor(((n - 2)/2)^2)). - Davis Smith, Mar 20 2020

Entry revised by N. J. A. Sloane, Apr 01 2006

Offset changed to 1 by N. J. A. Sloane, Sep 02 2009

A048775 Number of (partially defined) monotone maps from intervals of 1..n to 1..n.

Original entry on oeis.org

1, 7, 31, 121, 456, 1709, 6427, 24301, 92368, 352705, 1352066, 5200287, 20058286, 77558745, 300540179, 1166803093, 4537567632, 17672631881, 68923264390, 269128937199, 1052049481838, 4116715363777, 16123801841526, 63205303218851, 247959266474026, 973469712824029

Offset: 1

Stephen C. Power (s.power(AT)lancaster.ac.uk)

More precisely, number of ways to pick a subinterval [i,i+1,...,j] of [1..n] and to map it to a nondecreasing sequence of the same length with symbols from [1..n]. If s is the length of the subinterval (1 <= s <= n), there are n+1-s ways to choose the subinterval and binomial(n+s-1,s) ways to choose the sequence, for a total of Sum_{s=1..n} (n+1-s)*binomial(n+s-1,s) = binomial(2*n+1, n+1)-(n+1) ways. - N. J. A. Sloane, Feb 02 2009
Arises in the classification of endomorphisms of certain finite-dimensional operator algebras.
Equals binomial transform of A163765 (using a different offset). - Gary W. Adamson, Aug 03 2009
From David Spivak, Dec 12 2012: (Start)
Number of morphisms of full subcategories of Simplex category.
A finite nonempty linear order of size m is isomorphic to [m]:={0,1,...,m}. The weakly monotone maps [k]->[m] form a category, often called the simplex category and denoted Delta. For m starting at -1, let D_m denote the full subcategory of Delta, spanned by objects {[0],[1],...,[m]}. The number of morphisms in D_m is a(n+1).
(End)
This sequence is the bisection of the 1st column of the triangle defined by T(n,k) = 1 if n=0 else T(n,k) = binomial(n-1,k2-1)-binomial(k2,k-1) where k2 = binomial(n,k) mod n. - Nikita Sadkov, Oct 08 2018

			a(2) = 7 because there are two maps with domain {1}, two with domain {2} and three maps with domain {1,2}.
When n=2, we are looking at the full subcategory of Delta spanned by [0],[1]. There is one monotone map [0]->[0], one monotone map [1]->[0], two monotone maps [0]->[1], and three monotone maps [1]->[1] (namely (0,0), (0,1), (1,1)). The total is 1+1+2+3=7. - _David Spivak_, Dec 12 2013

Harvey P. Dale, Table of n, a(n) for n = 1..1000
David Applegate and N. J. A. Sloane, The Gift Exchange Problem, arXiv:0907.0513 [math.CO], 2009.
Wikipedia, Simplex category

Cf. A001700, A144657, A305161.

List([1..26],n->Binomial(2*n+1,n+1)-(n+1)); # Muniru A Asiru, Oct 09 2018

[Binomial(2*n+1, n+1)-(n+1): n in [1..30]]; // Vincenzo Librandi, Oct 10 2018

seq(coeff(series(factorial(n)*(exp(2*x)*(BesselI(0,2*x)+BesselI(1,2*x))-exp(x)*(1+x)),x,n+1), x, n), n = 1 .. 26); # Muniru A Asiru, Oct 09 2018

Table[Binomial[2n+1,n+1]-(n+1),{n,30}] (* Harvey P. Dale, Feb 08 2013 *)
From Stefano Spezia, Oct 09 2018: (Start)
a[n_]:=(1/2)*Sum[Sum[(i+j) !/(i !*j !),{i,1,n}],{j,1,n}]; Array[a, 50] (* or *)
CoefficientList[Series[((1/(2*x))*(1/Sqrt[1-4*x]-1) - 1/(1-x)^2)/x, {x, 0, 50}], x] (* or *)
CoefficientList[Series[(Exp[2*x]*(BesselI[0,2*x] + BesselI[1,2*x]) - Exp[x]*(1 + x))/x, {x, 0, 50}], x]*Table[(k+1) !, {k, 0, 50}]
(End)

Vec((1/(2*x))*(1/sqrt(1-4*x)-1) - 1/(1-x)^2 + O(x^15)) \\ Stefano Spezia, Oct 09 2018

a(n) = binomial(2*n+1, n+1)-(n+1) = A001700(n)-n-1.
a(n) = (1/2)*Sum[Sum[(i+j)!/(i!*j!),{i,1,n}],{j,1,n}]. - Alexander Adamchuk, Jul 04 2006; corrected by N. J. A. Sloane, Jan 30 2009
G.f.: (1/(2*x))*(1/sqrt(1-4*x)-1) - 1/(1-x)^2. - N. J. A. Sloane, Feb 02 2009
a(n) = Sum_{k=0..n} (n-k+1)*C(n+k+1,n) = [x^n](1+x)^n*F(-n-2,-n-1;1;x/(1+x)). - Paul Barry, Oct 01 2010
Conjecture: (n+1)*a(n) + (-7*n-2)*a(n-1) + 3*(5*n-3)*a(n-2) + (-13*n+20)*a(n-3) + 2*(2*n-5)*a(n-4) = 0. - R. J. Mathar, Nov 30 2012
a(n) = (1/2) * Sum_{k=1..n} Sum_{i=1..n} C(k+i,i). - Wesley Ivan Hurt, Sep 19 2017
E.g.f.: exp(2*x)*(BesselI(0,2*x) + BesselI(1,2*x)) - exp(x)*(1 + x). - Ilya Gutkovskiy, Sep 19 2017

More terms from N. J. A. Sloane, Dec 15 2008

cp's OEIS Frontend

User: powers

A381681 a(n) is one of two integer components (with A000254) used in computing the inverse second moment of X+n, where X~Poisson(1).

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A343129 Numbers that are palindromes in base 3/2.

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A342802 Replace 2^k with (-3)^k in binary expansion of n.

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A124437 Experience Points thresholds for levels in the pen and paper role-playing game "Das Schwarze Auge" (DSA, a.k.a. "The Dark Eye").

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A086901 a(1) = a(2) = 1; for n>2, a(n) = 4*a(n-1) + 3*a(n-2).

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A057973 Number of polybricks: number of ways to arrange n 1 X 2 "bricks" in a wall (see illustrations).

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A057707 a(n) is the number of free 'polyominoes' on the (4.8.8) grid with n octagons and n squares.

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A075152 Number of possible permutations of a Rubik cube of size n X n X n.

A086901 a(1) = a(2) = 1; for n>2, a(n) = 4a(n-1) + 3a(n-2).