A077957 -id:A077957 - cp's OEIS frontend

A351324 Number of tilings of a 7 X 3n rectangle with right trominoes.

1, 0, 520, 22656, 1795360, 115363072, 7876120608, 527256809600, 35522814546496, 2388257605782016, 160678147466414272, 10807663334085120512, 727010169682181839360, 48903265220016072792320, 3289569236212332037229184, 221278350342281369716796672

Offset: 0

Gerhard Kirchner, Feb 21 2022

See A351322 for algorithm.

This is the Hadamard sum of the following 4 sequences: 0, 0,0,0, 158208,.. (tilings which have both vertical and horizontal faults), 0,0,480,6144, 125952 ... (tilings which have horizontal faults but no vertical faults), 00,0,0,112192,.. (tilings which have vertical but no horizontal faults), 1, 0,40, 16512, 1399008 ,... (tilings which have neither horizontal nor vertical faults). - R. J. Mathar, Dec 08 2022

Cf. A077957, A000079, A046984, A084478, A351322, A351323, A236578 (straight trominoes), A233343 (mixed trominoes).

G.f.: (1 - 22*x - 1831*x^2 - 29454*x^3 - 270630*x^4 - 2070388*x^5 - 12125943*x^6 - 48147976*x^7 - 151548064*x^8 - 417242784*x^9 - 423562924*x^10 + 586224672*x^11 + 915719344*x^12 + 349980800*x^13 + 371621248*x^14 - 6541312*x^15 - 9691136*x^16 + 589824*x^17)/(1 - 22*x - 2351*x^2 - 40670*x^3 - 345038*x^4 - 3522884*x^5 - 28528327*x^6 - 145350120*x^7 - 623982088*x^8 - 2110011040*x^9 - 1354478796*x^10 + 9281598624*x^11 + 15001687984*x^12 + 3456230016*x^13 - 3194643904*x^14 - 1637793792*x^15 - 575934464*x^16 + 65175552*x^17).

a(n) = 22*a(n-1) + 2351*a(n-2) + 40670*a(n-3) + 345038*a(n-4) + 3522884*a(n-5) + 28528327*a(n-6) + 145350120*a(n-7) + 623982088*a(n-8) + 2110011040*a(n-9) + 1354478796*a(n-10) - 9281598624*a(n-11) - 15001687984*a(n-12) - 3456230016*a(n-13) + 3194643904*a(n-14) + 1637793792*a(n-15) + 575934464*a(n-16) - 65175552*a(n-17) for n>16.

A083879 a(0)=1, a(1)=4, a(n) = 8a(n-1) - 14a(n-2), n >= 2.

Original entry on oeis.org

1, 4, 18, 88, 452, 2384, 12744, 68576, 370192, 2001472, 10829088, 58612096, 317289536, 1717746944, 9299922048, 50350919168, 272608444672, 1475954689024, 7991119286784, 43265588647936, 234249039168512, 1268274072276992

Offset: 0

Paul Barry, May 08 2003

Binomial transform of A083878.
4th binomial transform of A077957. Inverse binomial transform of A083880. - Philippe Deléham, Nov 30 2008
From L. Edson Jeffery, Apr 26 2011: (Start)
Let G be the Gram matrix
  G =
  (4  1  0  1)
  (1  4  1  0)
  (0  1  4 -1)
  (1  0 -1  4)
of A028997. Then a(n) = (1/4)*Trace(G^n). (End)

Index entries for linear recurrences with constant coefficients, signature (8,-14).

Cf. A028997, A083880.

LinearRecurrence[{8,-14},{1,4},30] (* Harvey P. Dale, May 08 2013 *)

a(n) = 2^((n-2)/2)*(2*sqrt(2)-1)^n + 2^((n-2)/2)*(2*sqrt(2)+1)^n;
a(n) = Sum_{k=0..n} C(n, 2k)*5^(n-2k)2^k.
G.f.: (1-4x)/(1-8x+14x^2).
E.g.f.: exp(4x)cosh(x*sqrt(2)).
((4+sqrt(2))^n + (4-sqrt(2))^n)/2. Offset=0. a(3)=88. - Al Hakanson (hawkuu(AT)gmail.com), Oct 15 2008
a(n) = Sum_{k=0..n} A098158(n,k)*2^(3*k-n). - Philippe Deléham, Nov 30 2008

A097038 A Jacobsthal variant.

Original entry on oeis.org

0, 1, 1, 5, 7, 21, 35, 85, 155, 341, 651, 1365, 2667, 5461, 10795, 21845, 43435, 87381, 174251, 349525, 698027, 1398101, 2794155, 5592405, 11180715, 22369621, 44731051, 89478485, 178940587, 357913941, 715795115, 1431655765, 2863245995

Offset: 0

Paul Barry, Jul 19 2004

Convolution of A001045 and A077957.

Also interleaving of A002450(n+1) and A006095(n+1).

Index entries for linear recurrences with constant coefficients, signature (1,4,-2,-4).

Cf. A001045, A077957.

Cf. A002450, A006095.

concat(0, Vec(x/((1-2*x^2)*(1-x-2*x^2)) + O(x^50))) \\ Michel Marcus, Nov 13 2015

vector(50, n, n--; 2*2^n/3+(-1)^n/3-2^(n/2)*(1+(-1)^n)/2) \\ Altug Alkan, Nov 13 2015

G.f.: 1/(1-x-2*x^2) - 1/(1-2*x^2) = x/((1-2*x^2)*(1-x-2*x^2));
a(n) = 2*2^n/3+(-1)^n/3-2^(n/2)*(1+(-1)^n)/2;
a(n) = sum{k=0..floor((n+1)/2), binomial(n-k+1, k-1)2^k };
a(n) = sum{k=0..n, 2^(k/2)(1+(-1)^k)A001045(n-k)/2 };
a(n) = A001045(n+1)-A077957(n).

A131575 First differences of A131572.

Original entry on oeis.org

1, 1, 0, 2, 0, 4, 0, 8, 0, 16, 0, 32, 0, 64, 0, 128, 0, 256, 0, 512, 0, 1024, 0, 2048, 0, 4096, 0, 8192, 0, 16384, 0, 32768, 0, 65536, 0, 131072, 0, 262144, 0, 524288, 0, 1048576, 0, 2097152, 0, 4194304, 0, 8388608, 0

Offset: 0

Paul Curtz, Aug 28 2007

Index entries for linear recurrences with constant coefficients, signature (0, 2).

Differences[LinearRecurrence[{0,2},{0,1,2},50]] (* or *) LinearRecurrence[ {0, 2}, {1, 1, 0}, 50] (* or *) Join[{1},Riffle[2^Range[0,30],0]] (* Harvey P. Dale, Jul 10 2018 *)

a(n) = A131572(n+1) - A131572(n).
a(n) = A077957(n-1), n>0.
G.f.: (1 + x - 2x^2)/(1-2x^2). - Philippe Deléham, Oct 01 2011

Edited by R. J. Mathar, Jul 16 2008

More terms from Philippe Deléham, Oct 01 2011

A133851 Sloping binary representation of powers of 4 (A000302), slope = -1 .

Original entry on oeis.org

1, 0, 0, 4, 0, 0, 16, 0, 0, 64, 0, 0, 256, 0, 0, 1024, 0, 0, 4096, 0, 0, 16384, 0, 0, 65536, 0, 0, 262144, 0, 0, 1048576, 0, 0, 4194304, 0, 0, 16777216, 0, 0, 67108864, 0, 0, 268435456, 0, 0, 1073741824, 0, 0, 4294967296, 0, 0, 17179869184, 0, 0

Offset: 0

Philippe Deléham, Jan 06 2008

			When powers of 4 are written in binary (see A098608), under each other as:
0000000000001 (1)
0000000000100 (4)
0000000010000 (16)
0000001000000 (64)
0000100000000 (256)
0010000000000 (1024)
1000000000000 (4096)
and one collects their bits from the column=0 to NW-direction (from the least to the most significant end), one gets 1 (1), 00 (0), 000 (0), 0100 (4), 00000 (0), 000000 (0), 0010000 (16), etc. (see 0105033 for similar transformation done on nonnegative integers)

Cf. A037095, A077957, A105033, A000302, A098608, A102370(sloping binary numbers).

a(3n) = A000302(n), a(3n+1) = a(3n+2) = 0. - Alois P. Heinz, Dec 10 2020

A136489 Triangle T(n, k) = 3A007318(n, k) - 2A034851(n, k).

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 5, 5, 1, 1, 8, 10, 8, 1, 1, 9, 18, 18, 9, 1, 1, 12, 27, 40, 27, 12, 1, 1, 13, 39, 67, 67, 39, 13, 1, 1, 16, 52, 112, 134, 112, 52, 16, 1, 1, 17, 68, 164, 246, 246, 164, 68, 17, 1, 1, 20, 85, 240, 410, 504, 410, 240, 85, 20, 1

Offset: 0

Gary W. Adamson, Jan 01 2008

			First few rows of the triangle are:
  1;
  1,   1;
  1,   4,   1;
  1,   5,   5,   1;
  1,   8,  10,   8,   1;
  1,   9,  18,  18,   9,   1;
  1,  12,  27,  40,  27,  12,   1;
  1,  13,  39,  67,  67,  39,  13,   1;
  1,  16,  52, 112, 134, 112,  52,  16,   1;
  1,  17,  68, 164, 246, 246, 164,  68,  17,   1;
  ...

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A034851, A042948, A077957, A122746 (row sums).

A136489:= func< n,k | 2*Binomial(n,k) - Binomial(n mod 2, k mod 2)*Binomial(Floor(n/2), Floor(k/2)) >;
[A136489(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Aug 01 2023

T[n_, k_]:= 2*Binomial[n,k] -Binomial[Mod[n,2], Mod[k,2]]*Binomial[Floor[n/2], Floor[k/2]];
Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Aug 01 2023 *)

def A136489(n,k): return 2*binomial(n,k) - binomial(n%2, k%2)*binomial(n//2, k//2)
flatten([[A136489(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Aug 01 2023

T(n, k) = 3*A007318(n, k) - 2*A034851(n, k).
Sum_{k=0..n} T(n, k) = A122746(n).
From G. C. Greubel, Aug 01 2023: (Start)
T(n, k) = 2*A007318(n, k) - A051159(n, k).
T(n, k) = T(n-1, k) + T(n-1, k-1) if k is even.
T(n, n-k) = T(n, k).
T(n, n-1) = A042948(n).
Sum_{k=0..n} (-1)^k * T(n, k) = 2*[n=0] - A077957(n). (End)

A161161 Irregular triangle of differences T(n,k) = A083906(n,k) - A083906(n-1,k) of q-Binomial coefficients.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 1, 2, 3, 5, 2, 2, 1, 1, 2, 3, 5, 7, 5, 4, 3, 1, 1, 1, 2, 3, 5, 7, 11, 8, 9, 7, 6, 2, 2, 1, 1, 2, 3, 5, 7, 11, 15, 14, 15, 15, 13, 11, 7, 4, 3, 1, 1, 1, 2, 3, 5, 7, 11, 15, 22, 21, 25, 25, 27, 23, 22, 15, 13, 8, 6, 2, 2, 1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 32, 37, 42, 44

Offset: 1

Alford Arnold, Jun 04 2009

			The differences between 5 3 4 3 1 and 4 2 2 yield row four : 1 1 2 3 1.
Triangle begins:
  1;
  1, 1;
  1, 1, 2;
  1, 1, 2, 3, 1;
  1, 1, 2, 3, 5, 2,  2;
  1, 1, 2, 3, 5, 7,  5,  4,  3,  1;
  1, 1, 2, 3, 5, 7, 11,  8,  9,  7,  6,  2,  2;
  1, 1, 2, 3, 5, 7, 11, 15, 14, 15, 15, 13, 11,  7,  4,  3,  1;
  1, 1, 2, 3, 5, 7, 11, 15, 22, 21, 25, 25, 27, 23, 22, 15, 13, 8, 6, 2, 2;
  ...

Michael De Vlieger, Table of n, a(n) for n = 1..2390 (rows 1..30, flattened)
William Q. Erickson and Jan Kretschmann, The structure and normalized volume of Monge polytopes, arXiv:2311.07522 [math.CO], 2023. See p. 16.
M. Isachenkov, I. Kirsch, and V. Schomerus, Chiral Primaries in Strange Metals, arXiv preprint arXiv:1403.6857 [hep-th], 2014. See Eq. (4.6).

Cf. A000079 (row sums), A002865 (antidiagonal sums), A077957 (alternating row sums).

R:=PowerSeriesRing(Rationals(), 100);
qBinom:= func< n,k,x | n eq 0 or k eq 0 select 1 else (&*[(1-x^(n-j))/(1-x^(j+1)): j in [0..k-1]]) >;
A083906:= func< n,k | Coefficient(R!( (&+[qBinom(n,k,x): k in [0..n]]) ), k) >;
A161161:= func< n,k | A083906(n,k) - A083906(n-1,k) >;
[A161161(n,k): k in [0..Floor(n^2/4)], n in [1..12]]; // G. C. Greubel, Feb 13 2024

A161161 := proc(n,m)
     A083906(n,m)-A083906(n-1,m) ;
end proc:
for n from 0 to 10 do
     for k from 0 to A033638(n)-1 do
         printf("%d, ", A161161(n, k)) ;
     od:
od: # R. J. Mathar, Jul 13 2012

b[n_, k_] := b[n, k] = SeriesCoefficient[Sum[QBinomial[n, m, q], {m, 0, n}], {q, 0, k}];
T[n_, k_] := b[n, k] - b[n - 1, k];
Table[Table[T[n, k], {k, 0, n^2/4}], {n, 1, 10}] // Flatten (* Jean-François Alcover, Nov 25 2017 *)

def t(n, k): # t = A083906
    if k<0 or k> (n^2//4): return 0
    elif n<2 : return n+1
    else: return 2*t(n-1, k) - t(n-2, k) + t(n-2, k-n+1)
def A161161(n,k): return t(n, k) - t(n-1, k)
flatten([[A161161(n, k) for k in range(int(n^2//4)+1)] for n in range(1,13)]) # G. C. Greubel, Feb 13 2024

Sum_{k=0..floor(n^2/4)} T(n, k) = A000079(n-1) (row sums).
Sum_{k=0..(n+2 - ceiling(sqrt(4*n)))} T(n-k, k) = A002865(n+1) (antidiagonal sums).
Sum_{k=0..floor(n^2/4)} (-1)^k*T(n, k) = A077957(n-1). - G. C. Greubel, Feb 13 2024

A217730 Expansion of (1+2x-x^3)/(1-4x^2+2*x^4).

Original entry on oeis.org

1, 2, 4, 7, 14, 24, 48, 82, 164, 280, 560, 956, 1912, 3264, 6528, 11144, 22288, 38048, 76096, 129904, 259808, 443520, 887040, 1514272, 3028544, 5170048, 10340096, 17651648, 35303296, 60266496, 120532992, 205762688, 411525376, 702517760, 1405035520, 2398545664, 4797091328, 8189147136, 16378294272, 27959497216

Offset: 0

Philippe Deléham, Mar 22 2013

In general, a(n,j,m) = Sum_{r=1..m} (2^n*(1-(-1)^r)*cos(Pi*r/(m+1))^n*cot(Pi*r/(2*(m+1)))*sin(j*Pi*r/(m+1)))/(m+1) gives the number of paths of length n starting at the j-th node on the path graph P_m. Here we have the case m=7 and j=3. - Herbert Kociemba, Sep 17 2020

Shaun V. Ault and Charles Kicey, Counting paths in corridors using circular Pascal arrays, Discrete Math. 332 (2014), 45--54. MR3227977. See Fig. 5. - _N. J. A. Sloane_, Aug 04 2014
Index entries for linear recurrences with constant coefficients, signature (0,4,0,-2).

Cf. A007070, A077957, A204089, A216232.

First differences are in A062113.

m:=40; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1+2*x-x^3)/(1-4*x^2+2*x^4))); // Bruno Berselli, Mar 22 2013

CoefficientList[Series[(1 + 2 x - x^3)/(1 - 4 x^2 + 2 x^4), {x, 0, 40}], x] (* Bruno Berselli, Mar 22 2013 *)
a[n_,j_,m_]:=Sum[(2^(n+1)Cos[Pi r/(m+1)]^n Cot[Pi r/(2(m+1))] Sin[j Pi r/(m+1)])/(m+1),{r,1,m,2}]
Table[a[n,3,7],{n,0,40}]//Round (* Herbert Kociemba, Sep 17 2020 *)

makelist(coeff(taylor((1+2*x-x^3)/(1-4*x^2+2*x^4), x, 0, n), x, n), n, 0, 40); /* Bruno Berselli, Mar 22 2013 */

G.f.: (1+x)*(1+x-x^2)/(1-4*x^2+2*x^4).
a(n) = Sum_{k=0..n} A216232(n-k,k).
a(n) = 4*a(n-2) - 2*a(n-4) for n>=4, a(0)=1, a(1)=2, a(2)=4, a(3)=7.
a(2*n) = A007070(n), a(2*n-1) = a(2*n)/2 = A007070(n)/2.
a(n)*a(n+1)-a(n-1)*a(n+2) = (1-(-1)^n)*2^floor(n/2-1) for n>0. - Bruno Berselli, Mar 22 2013
a(n) = Sum_{r=1..7} (2^n*(1-(-1)^r)*cos(Pi*r/8)^n*cot(Pi*r/16)*sin(3*Pi*r/8))/8. - Herbert Kociemba, Sep 17 2020

A221337 T(n,k)=Number of nXk arrays of occupancy after each element stays put or moves to some king-move neighbor, with every occupancy equal to zero or two.

Original entry on oeis.org

0, 2, 2, 0, 6, 0, 4, 20, 20, 4, 0, 68, 0, 68, 0, 8, 232, 790, 790, 232, 8, 0, 792, 0, 10704, 0, 792, 0, 16, 2704, 34042, 142792, 142792, 34042, 2704, 16, 0, 9232, 0, 1937900, 0, 1937900, 0, 9232, 0, 32, 31520, 1470618, 26264018

Offset: 1

R. H. Hardin Jan 11 2013

Table starts
..0....2.....0.......4......0.......8........0......16.....0.32
..2....6....20......68....232.....792.....2704....9232.31520
..0...20.....0.....790......0...34042........0.1470618
..4...68...790...10704.142792.1937900.26264018
..0..232.....0..142792......0
..8..792.34042.1937900
..0.2704.....0
.16.9232
..0

			Some solutions for n=3 k=4
..0..2..2..2....2..0..0..0....2..0..2..0....0..0..2..0....0..2..2..0
..2..0..0..0....2..2..2..0....0..0..2..0....2..2..0..0....2..2..0..0
..0..2..0..2....0..2..0..2....2..0..2..2....2..0..2..2....0..0..2..2

Column 1 is A077957

Column 2 is A006012

A003143 a(2n) = floor( 172^n/14 ), a(2n+1) = floor( 122^n/7 ).

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 9, 13, 19, 27, 38, 54, 77, 109, 155, 219, 310, 438, 621, 877, 1243, 1755, 2486, 3510, 4973, 7021, 9947, 14043, 19894, 28086, 39789, 56173, 79579, 112347, 159158, 224694, 318317, 449389, 636635, 898779, 1273270, 1797558

Offset: 0

N. J. A. Sloane

D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 3, p. 207.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Index entries for linear recurrences with constant coefficients, signature (1,1,-1,1,-1,2,-2).

Cf. A010892, A049347, A077957.

[(17+7*(n mod 2))*2^(n div 2) div 14: n in [0..50]]; // Vincenzo Librandi, May 27 2016

A003143:=(1+z**3-z**4+z**5-z**6+z**7)/((z-1)*(z**2-z+1)*(z**2+z+1)*(2*z**2-1)); # [Conjectured (correctly) by Simon Plouffe in his 1992 dissertation.]

Flatten[Table[{Floor[17 2^n / 14], Floor[12 2^n / 7]}, {n, 0, 30}]] (* Vincenzo Librandi, May 27 2016 *)

PARI
```
a(n)=(17+7*(n%2))*2^(n\2)\14
```

[(((17 +7*(n%2))*2^(n//2))//14) for n in range(51)] # G. C. Greubel, Nov 04 2022

G.f.: (1 +x^3 -x^4 +x^5 -x^6 +x^7)/((1-x)(1-x+x^2)*(1+x+x^2)*(1-2*x^2)). - Simon Plouffe
a(n) = a(n-1) + a(n-2) - a(n-3) + a(n-4) - a(n-5) + 2*a(n-6) - 2*a(n-7) for n > 7. - Chai Wah Wu, May 25 2016
a(n) = (1/2)*[n=0] - 2/3 - (1/14)*(2*A010892(n) - 3*A010892(n-1)) + (1/42)*(4*A049347(n) - A049347(n-1)) + (1/14)*(17*A077957(n) + 24*A077957(n-1)). - G. C. Greubel, Nov 04 2022

More terms from Michael Somos, May 04 2000

cp's OEIS Frontend

A351324 Number of tilings of a 7 X 3n rectangle with right trominoes.

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A083879 a(0)=1, a(1)=4, a(n) = 8*a(n-1) - 14*a(n-2), n >= 2.

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A097038 A Jacobsthal variant.

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A131575 First differences of A131572.

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A133851 Sloping binary representation of powers of 4 (A000302), slope = -1 .

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A136489 Triangle T(n, k) = 3*A007318(n, k) - 2*A034851(n, k).

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A161161 Irregular triangle of differences T(n,k) = A083906(n,k) - A083906(n-1,k) of q-Binomial coefficients.

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A217730 Expansion of (1+2*x-x^3)/(1-4*x^2+2*x^4).

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Magma

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A083879 a(0)=1, a(1)=4, a(n) = 8a(n-1) - 14a(n-2), n >= 2.

A136489 Triangle T(n, k) = 3A007318(n, k) - 2A034851(n, k).

A217730 Expansion of (1+2x-x^3)/(1-4x^2+2*x^4).

A003143 a(2n) = floor( 172^n/14 ), a(2n+1) = floor( 122^n/7 ).