A008795 -id:A008795 - cp's OEIS frontend

A231056 The maximum number of X patterns that can be packed into an n X n array of coins.

0, 1, 1, 2, 4, 5, 8, 10, 13, 16, 20, 24, 29, 34, 40, 45, 51, 58, 65, 73, 80, 88, 97, 106, 116, 125, 135, 146, 157, 169, 180, 192, 205, 218, 232, 245, 259, 274, 289, 305, 320, 336, 353, 370, 388, 405, 423, 442, 461, 481, 500, 520, 541, 562, 584, 605, 627, 650, 673, 697, 720, 744, 769, 794

Offset: 2

Kival Ngaokrajang, Nov 03 2013

nonn

The X pattern (8c5s2 type) is a pattern in which 8 curves cover 5 coins, and is one of a total of 13 such distinct patterns that appear in a tightly-packed 3 X 3 square array of coins of identical size; each of the 8 curves is a circular arc lying along the edge of one of the 5 coins, and the 8 curves are joined end-to-end to form a continuous area.
a(n) is the maximum number of X patterns that can be packed into an n X n array of coins. The total coins left after packing X patterns into an n X n array of coins is A231064 and voids left is A231065.
a(n) is also the maximum number of "+" patterns (8c5s1 type) that can be packed into an n X n array of coins. See illustration in links.

Kival Ngaokrajang, Illustration of initial terms (X and +)

Cf. A008795, A230370 (3-curves); A074148, A227906, A229093, A229154 (4-curves); A001399, A230267, A230276 (5-curves); A229593, A228949, A229598, A002620, A230548, A230549, A230550 (6-curves).

Empirical g.f.: -x^3*(x^15 -2*x^14 +x^13 -x^12 +2*x^11 -2*x^10 +2*x^9 -x^8 +x^5 -x^4 +x^3 +x^2 -x +1) / ((x -1)^3*(x^4 +x^3 +x^2 +x +1)). - Colin Barker, Nov 27 2013

A231064 Coins left after packing X patterns into an n X n array of coins.

Original entry on oeis.org

4, 4, 11, 15, 16, 24, 24, 31, 35, 41, 44, 49, 51, 55, 56, 64, 69, 71, 75, 76, 84, 89, 91, 95, 96, 104, 109, 111, 115, 116, 124, 129, 131, 135, 136, 144, 149, 151, 155, 156, 164, 169, 171, 175, 176, 184, 189, 191, 195, 196, 204, 209, 211, 215, 216, 224, 229, 231, 235, 236, 244, 249, 251

Offset: 2

Kival Ngaokrajang, Nov 03 2013

nonn

The X pattern (8c5s2 type) is a pattern in which 8 curves cover 5 coins, and is one of a total of 13 such distinct patterns that appear in a tightly-packed 3 X 3 square array of coins of identical size; each of the 8 curves is a circular arc lying along the edge of one of the 5 coins, and the 8 curves are joined end-to-end to form a continuous area.
a(n) is the total number of coins left (the coins out side X patterns) after packing X patterns into an n X n array of coins. The maximum number of X patterns that can be packed into an n X n array of coins is A231056 and voids left is A231065.
a(n) is also the total number of coins left after packing "+" patterns (8c5s1 type) into an n X n array of coins. See illustration in links.

Kival Ngaokrajang, Illustration of initial terms (U)

Cf. A008795, A230370 (3-curves); A074148, A227906, A229093, A229154 (4-curves); A001399, A230267, A230276 (5-curves); A229593, A228949, A229598, A002620, A230548, A230549, A230550 (6-curves).

Empirical g.f.: x^2*(5*x^15 -5*x^14 -5*x^12 +5*x^11 -5*x^10 +5*x^9 +4*x^5 +x^4 +4*x^3 +7*x^2 +4) / ((x -1)^2*(x^4 +x^3 +x^2 +x +1)). - Colin Barker, Nov 27 2013

A231065 Voids left after packing X patterns into an of n X n array of coins.

Original entry on oeis.org

1, 0, 5, 8, 9, 16, 17, 24, 29, 36, 41, 48, 53, 60, 65, 76, 85, 92, 101, 108, 121, 132, 141, 152, 161, 176, 189, 200, 213, 224, 241, 256, 269, 284, 297, 316, 333, 348, 365, 380, 401, 420, 437, 456, 473, 496, 517, 536, 557, 576, 601, 624, 645, 668, 689, 716, 741, 764, 789, 812, 841, 868

Offset: 2

Kival Ngaokrajang, Nov 03 2013

nonn

The X pattern (8c5s2 type) is a pattern in which 8 curves cover 5 coins, and is one of a total of 13 such distinct patterns that appear in a tightly-packed 3 X 3 square array of coins of identical size; each of the 8 curves is a circular arc lying along the edge of one of the 5 coins, and the 8 curves are joined end-to-end to form a continuous area.
a(n) is the total number of voids (spaces among coins) left after packing X patterns into an n X n array of coins. The maximum number of X patterns that can be packed into an n X n array of coins is A231056 and coins left is A231064.
a(n) is also the total number of voids left after packing "+" patterns (8c5s1 type) into an n X n array of coins. See illustration in links.

Kival Ngaokrajang, Illustration of initial terms (V)

Cf. A008795, A230370 (3-curves); A074148, A227906, A229093, A229154 (4-curves); A001399, A230267, A230276 (5-curves); A229593, A228949, A229598, A002620, A230548, A230549, A230550 (6-curves).

Empirical g.f.: x^2*(4*x^16 -8*x^15 +4*x^14 -4*x^13 +8*x^12 -8*x^11 +8*x^10 -4*x^9 +4*x^6 -5*x^5 +2*x^4 +2*x^3 -6*x^2 +2*x -1) / ((x -1)^3*(x^4 +x^3 +x^2 +x +1)). - Colin Barker, Nov 27 2013

A317300 Sequence obtained by taking the general formula for generalized k-gonal numbers: m((k - 2)m - k + 4)/2, where m = 0, +1, -1, +2, -2, +3, -3, ... and k >= 5. Here k = 0.

Original entry on oeis.org

0, 1, -3, 0, -8, -3, -15, -8, -24, -15, -35, -24, -48, -35, -63, -48, -80, -63, -99, -80, -120, -99, -143, -120, -168, -143, -195, -168, -224, -195, -255, -224, -288, -255, -323, -288, -360, -323, -399, -360, -440, -399, -483, -440, -528, -483, -575, -528, -624, -575, -675, -624, -728, -675, -783

Offset: 0

Omar E. Pol, Jul 29 2018

Taking the same formula with k = 1 we have A317301.
Taking the same formula with k = 2 we have A001057 (canonical enumeration of integers).
Taking the same formula with k = 3 we have 0 together with A008795 (Molien series for 3-dimensional representation of dihedral group D_6 of order 6).
Taking the same formula with k = 4 we have A008794 (squares repeated) except the initial zero.
Taking the same formula with k >= 5 we have the generalized k-gonal numbers (see Crossrefs section).

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

Row 0 of A303301.
Cf. A001057, A008795, A008794, A174474, A317301.
Sequences of generalized k-gonal numbers: A001318 (k=5), A000217 (k=6), A085787 (k=7), A001082 (k=8), A118277 (k=9), A074377 (k=10), A195160 (k=11), A195162 (k=12), A195313 (k=13), A195818 (k=14), A277082 (k=15), A274978 (k=16), A303305 (k=17), A274979 (k=18), A303813 (k=19), A218864 (k=20), A303298 (k=21), A303299 (k=22), A303303 (k=23), A303814 (k=24), A303304 (k=25), A316724 (k=26), A316725 (k=27), A303812 (k=28), A303815 (k=29), A316729 (k=30).

concat(0, Vec(x*(1 - 4*x + x^2) / ((1 - x)^3*(1 + x)^2) + O(x^50))) \\ Colin Barker, Aug 01 2018

a(n) = -A174474(n+1).
From Colin Barker, Aug 01 2018: (Start)
G.f.: x*(1 - 4*x + x^2) / ((1 - x)^3*(1 + x)^2).
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n>4.
a(n) = -n*(n + 4) / 4 for n even.
a(n) = -(n - 3)*(n + 1) / 4 for n odd.
(End)

A317301 Sequence obtained by taking the general formula for generalized k-gonal numbers: m((k - 2)m - k + 4)/2, where m = 0, +1, -1, +2, -2, +3, -3, ... and k >= 5. Here k = 1.

Original entry on oeis.org

0, 1, -2, 1, -5, 0, -9, -2, -14, -5, -20, -9, -27, -14, -35, -20, -44, -27, -54, -35, -65, -44, -77, -54, -90, -65, -104, -77, -119, -90, -135, -104, -152, -119, -170, -135, -189, -152, -209, -170, -230, -189, -252, -209, -275, -230, -299, -252, -324, -275, -350, -299, -377, -324, -405, -350, -434

Offset: 0

Omar E. Pol, Jul 29 2018

Taking the same formula with k = 0 we have A317300.
Taking the same formula with k = 2 we have A001057 (canonical enumeration of integers).
Taking the same formula with k = 3 we have 0 together with A008795 (Molien series for 3-dimensional representation of dihedral group D_6 of order 6).
Taking the same formula with k = 4 we have A008794 (squares repeated) except the initial zero.
Taking the same formula with k >= 5 we have the generalized k-gonal numbers (see Crossrefs section).

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

Row 1 of A303301.
Cf. A000096, A001057, A008794, A008795, A317300.
Sequences of generalized k-gonal numbers: A001318 (k=5), A000217 (k=6), A085787 (k=7), A001082 (k=8), A118277 (k=9), A074377 (k=10), A195160 (k=11), A195162 (k=12), A195313 (k=13), A195818 (k=14), A277082 (k=15), A274978 (k=16), A303305 (k=17), A274979 (k=18), A303813 (k=19), A218864 (k=20), A303298 (k=21), A303299 (k=22), A303303 (k=23), A303814 (k=24), A303304 (k=25), A316724 (k=26), A316725 (k=27), A303812 (k=28), A303815 (k=29), A316729 (k=30).

/* By definition: */ k:=1; [0] cat [m*i*((k-2)*m*i-k+4)/2: i in [1,-1], m in [1..30]]; // Bruno Berselli, Jul 30 2018

Table[(-2 n (n + 1) - 5 (2 n + 1) (-1)^n + 5)/16, {n, 0, 60}] (* Bruno Berselli, Jul 30 2018 *)

concat(0, Vec(x*(1 - 3*x + x^2)/((1 + x)^2*(1 - x)^3) + O(x^50))) \\ Colin Barker, Aug 01 2018

From Bruno Berselli, Jul 30 2018: (Start)
O.g.f.: x*(1 - 3*x + x^2)/((1 + x)^2*(1 - x)^3).
E.g.f.: (-5*(1 + 2*x) + (5 - 2*x^2)*exp(2*x))*exp(-x)/16.
a(n) = a(-n+1) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5).
a(n) = (-2*n*(n + 1) - 5*(2*n + 1)*(-1)^n + 5)/16. Therefore:
a(n) = -n*(n + 6)/8 for even n;
a(n) = -(n - 5)*(n + 1)/8 for odd n. Also:
a(n) = a(n-5) for odd n > 3.
2*(2*n - 1)*a(n) + 2*(2*n + 1)*a(n-1) + n*(n^2 - 3) = 0. (End)

A338042 Draw n rays from each of two distinct points in the plane; a(n) is the number of vertices thus created. See Comments for details.

Original entry on oeis.org

2, 2, 4, 2, 8, 4, 14, 8, 22, 14, 32, 22, 44, 32, 58, 44, 74, 58, 92, 74, 112, 92, 134, 112, 158, 134, 184, 158, 212, 184, 242, 212, 274, 242, 308, 274, 344, 308, 382, 344, 422, 382, 464, 422, 508, 464, 554, 508, 602, 554, 652, 602, 704, 652, 758, 704, 814, 758

Offset: 1

Lars Blomberg, Oct 08 2020

nonn

The rays are evenly spaced around each point. The first ray of one point goes opposite to the direction to the other point. Should a ray hit the other point it terminates there, that is, it is converted to a line segment.

See A338041 for illustrations.

			For n=1:    <-----x     x----->   so a(1)=2.
For n=2:    <-----x<--->x----->   so a(2)=2.

Cf. A338041 (regions), A338043 (edges), A008795.

a(n)=if(n%2==1,(n^2 + 7)/4,(n^2 - 6*n + 16)/4)
vector(200, n, a(n))

a(n) = (n^2 + 7)/4, n odd; (n^2 - 6*n + 16)/4, n even (conjectured).
Conjectured by Stefano Spezia, Oct 08 2020 after Lars Blomberg: (Start)
G.f.: 2*x*(1 - x^2 - x^3 + 2*x^4)/((1 - x)^3*(1 + x)^2).
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n > 5. (End)
Hugo Pfoertner, Oct 08 2020: Apparently a(n)=2*(A008795(n-3)+1).

A008796 Molien series for 3-dimensional group [2,3]+ = 223; also for group H_{1,2} of order 384.

Original entry on oeis.org

1, 0, 2, 1, 4, 2, 7, 4, 10, 7, 14, 10, 19, 14, 24, 19, 30, 24, 37, 30, 44, 37, 52, 44, 61, 52, 70, 61, 80, 70, 91, 80, 102, 91, 114, 102, 127, 114, 140, 127, 154, 140, 169, 154, 184, 169, 200, 184, 217, 200, 234, 217, 252, 234, 271, 252, 290, 271, 310, 290, 331, 310, 352, 331, 374, 352, 397

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
E. Bannai, S. T. Dougherty, M. Harada and M. Oura, Type II Codes, Even Unimodular Lattices and Invariant Rings, IEEE Trans. Information Theory, Volume 45, Number 4, 1999, 1194-1205.
Index entries for linear recurrences with constant coefficients, signature (0,2,1,-1,-2,0,1).
Index entries for Molien series

Cf. A008795.

a:=[1,0,2,1,4,2,7];; for n in [8..70] do a[n]:=2*a[n-2]+a[n-3]-a[n-4]-2*a[n-5]+a[n-7]; od; a; # G. C. Greubel, Sep 11 2019

R:=PowerSeriesRing(Integers(), 70); Coefficients(R!( (1+x^4)/((1-x^2)^2*(1-x^3)) )); // G. C. Greubel, Sep 11 2019

seq(coeff(series((1+x^4)/((1-x^2)^2*(1-x^3)), x, n+1), x, n), n = 0..70); # G. C. Greubel, Sep 11 2019

LinearRecurrence[{0,2,1,-1,-2,0,1},{1,0,2,1,4,2,7},70] (* Harvey P. Dale, Apr 27 2014 *)
CoefficientList[Series[(1+x^4)/((1-x^2)^2*(1-x^3)), {x, 0, 70}], x] (* Vincenzo Librandi, Apr 28 2014 *)

a(n)=(9*(-1)^n*(2*n + 3) + 6*n^2 + 18*n + 24*!(n%3) + 21)/72 \\ Charles R Greathouse IV, Feb 10 2017

def A008796_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1+x^4)/((1-x^2)^2*(1-x^3))).list()
A008796_list(70) # G. C. Greubel, Sep 11 2019

G.f.: (1+x^4)/((1-x^2)^2*(1-x^3)).

a(n) = (1/72) * (9*(-1)^n*(2*n + 3) + 6*n^2 + 18*n + 29 - 8*A061347[n]). - Ralf Stephan, Apr 28 2014

Definition clarified by N. J. A. Sloane, Feb 02 2018

More terms added by G. C. Greubel, Sep 11 2019

A176447 a(2n) = -n, a(2n+1) = 2n+1.

Original entry on oeis.org

0, 1, -1, 3, -2, 5, -3, 7, -4, 9, -5, 11, -6, 13, -7, 15, -8, 17, -9, 19, -10, 21, -11, 23, -12, 25, -13, 27, -14, 29, -15, 31, -16, 33, -17, 35, -18, 37, -19, 39, -20, 41, -21, 43, -22, 45, -23, 47, -24, 49, -25, 51, -26, 53, -27, 55, -28, 57, -29, 59, -30, 61, -31, 63, -32, 65, -33, 67, -34, 69, -35

Offset: 0

Paul Curtz, Apr 18 2010

There is more complicated way of defining the sequence: consider the sequence of modified Bernoulli numbers EVB(n) = A176327(n)/A176289(n) and its inverse binomial transform IEVB(n) = A176328(n)/A176591(n). Then a(n) is the numerator of the difference EVB(n)-IEVB(n). The denominator of the difference is 1 if n=0, else A040001(n-1).
A particularity of EVB(n) is: its (forward) binomial transform is 1, 1, 7/6, 3/2, 59/30,.. = (-1)^n*IEVB(n).
Note that A026741 is related to the Rydberg-Ritz spectrum of the hydrogen atom.

			G.f. = x - x^2 + 3*x^3 - 2*x^4 + 5*x^5 - 3*x^6 + 7*x^7 - 4*x^8 + 9*x^9 - 5*x^10 + ...

R. J. Mathar, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Cf. A008795, A026741, A040001, A176289, A176327, A176328, A176591.

[n*(1-3*(-1)^n)/4: n in [0..60]]; // Vincenzo Librandi, Aug 04 2011

a[n_?EvenQ]:=-(n/2); a[n_?OddQ]:=n; Table[a[n], {n, 100}] (* Alonso del Arte, Dec 01 2010 *)
a[ n_] := n / If[ Mod[ n, 2] == 1, 1, -2]; (* Michael Somos, Jun 11 2013 *)
CoefficientList[Series[x (1 - x + x^2)/((x - 1)^2*(1 + x)^2), {x, 0, 70}], x]  (* Michael De Vlieger, Dec 10 2016 *)
LinearRecurrence[{0,2,0,-1},{0,1,-1,3},80] (* Harvey P. Dale, Nov 01 2017 *)

{a(n) = n / if( n%2, 1, -2)}; /* Michael Somos, Jun 11 2013 */

From R. J. Mathar, Dec 01 2010:  (Start)
a(n) = (-1)^n*A026741(n) = n*(1-3*(-1)^n)/4.
G.f.: x*(1-x+x^2) / ( (x-1)^2*(1+x)^2 ).
a(n) = +2*a(n-2) -a(n-4).  (End)
a(n) = -a(-n) for all n in Z. - Michael Somos, Jun 11 2013
From Michael Somos, Aug 30 2014: (Start)
Euler transform of length 6 sequence [ -1, 3, 1, 0, 0, -1].
0 = - 1 - a(n) - a(n+1) + a(n+2) + a(n+3) for all n in Z.
0 = 1 + a(n)*(-2 -a(n) + a(n+2)) - 2*a(n+1) - a(n+2) for all n in Z. (End)
From Michael Somos, May 04 2015: (Start)
a(n) is multiplicative with a(2^e) = -(2^(e-1)) if e>0, a(p^e) = p^e otherwise.
G.f.: (f(x) - 3 * f(-x)) / 4 where f(x) := x / (1 - x)^2.
G.f.: x * (1 - x) * (1 - x^6) / ((1 - x^2)^3 * (1 - x^3)). (End)
From Amiram Eldar, Sep 21 2023: (Start)
Dirichlet g.f.: zeta(s-1) * (1 - 3/2^s).
Sum_{k=0..n} a(k) = A008795(n-1), for n > 0.
Sum_{k=0..n} a(k) ~ n^2/8. (End)

A084861 Expansion of (1-3x+4x^2-3x^3+x^4)/(1-2x)^2.

Original entry on oeis.org

1, 1, 4, 9, 21, 48, 108, 240, 528, 1152, 2496, 5376, 11520, 24576, 52224, 110592, 233472, 491520, 1032192, 2162688, 4521984, 9437184, 19660800, 40894464, 84934656, 176160768, 364904448, 754974720, 1560281088, 3221225472

Offset: 0

Paul Barry, Jun 12 2003

Partial sums give A084860. Binomial transform of signed version of A008795.

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

CoefficientList[Series[(1-3x+4x^2-3x^3+x^4)/(1-2x)^2,{x,0,40}],x] (* or *) LinearRecurrence[{4,-4},{1,1,4,9,21},40] (* Harvey P. Dale, Nov 25 2020 *)

a(n) = 3(n+3)2^(n-4), n>2.

A111074 Let t(n) denote the triangular numbers (A000217). Sequence mixes t(n+2) and t(n).

Original entry on oeis.org

3, 0, 6, 1, 10, 3, 15, 6, 21, 10, 28, 15, 36, 21, 45, 28, 55, 36, 66, 45, 78, 55, 91, 66, 105, 78, 120, 91, 136, 105, 153, 120, 171, 136, 190, 153, 210, 171, 231, 190, 253, 210, 276, 231, 300, 253, 325, 276, 351, 300, 378, 325, 406, 351, 435, 378, 465, 406, 496, 435

Offset: 0

Stuart M. Ellerstein (ellerstein(AT)aol.com), Oct 10 2005

Cf. mix of t(n+1) and t(n): A008795.

Flatten[ Table[{(n + 2)(n + 3)/2, n(n + 1)/2}, {n, 0, 29}]] (* Robert G. Wilson v *)

G.f.: (3-3x+x^3)/((1-x)(1-x^2)^2); a(n)=a(n-1)+2a(n-2)-2a(n-3)-a(n-4)+a(n-5); - Paul Barry, Mar 22 2006

More terms from Robert G. Wilson v, Oct 11 2005

cp's OEIS Frontend

A231056 The maximum number of X patterns that can be packed into an n X n array of coins.

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A231064 Coins left after packing X patterns into an n X n array of coins.

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A231065 Voids left after packing X patterns into an of n X n array of coins.

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A317300 Sequence obtained by taking the general formula for generalized k-gonal numbers: m*((k - 2)*m - k + 4)/2, where m = 0, +1, -1, +2, -2, +3, -3, ... and k >= 5. Here k = 0.

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A317301 Sequence obtained by taking the general formula for generalized k-gonal numbers: m*((k - 2)*m - k + 4)/2, where m = 0, +1, -1, +2, -2, +3, -3, ... and k >= 5. Here k = 1.

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A338042 Draw n rays from each of two distinct points in the plane; a(n) is the number of vertices thus created. See Comments for details.

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A008796 Molien series for 3-dimensional group [2,3]+ = 223; also for group H_{1,2} of order 384.

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A176447 a(2n) = -n, a(2n+1) = 2n+1.

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A317300 Sequence obtained by taking the general formula for generalized k-gonal numbers: m((k - 2)m - k + 4)/2, where m = 0, +1, -1, +2, -2, +3, -3, ... and k >= 5. Here k = 0.

A317301 Sequence obtained by taking the general formula for generalized k-gonal numbers: m((k - 2)m - k + 4)/2, where m = 0, +1, -1, +2, -2, +3, -3, ... and k >= 5. Here k = 1.