A077957 -id:A077957 - cp's OEIS frontend

A059474 Triangle read by rows: T(n,k) is coefficient of z^nw^k in 1/(1 - 2z - 2w + 2z*w) read by rows in order 00, 10, 01, 20, 11, 02, ...

1, 2, 2, 4, 6, 4, 8, 16, 16, 8, 16, 40, 52, 40, 16, 32, 96, 152, 152, 96, 32, 64, 224, 416, 504, 416, 224, 64, 128, 512, 1088, 1536, 1536, 1088, 512, 128, 256, 1152, 2752, 4416, 5136, 4416, 2752, 1152, 256, 512, 2560, 6784, 12160, 16032, 16032, 12160, 6784, 2560, 512

Offset: 0

N. J. A. Sloane, Feb 03 2001; revised Jun 12 2005

Pascal-like triangle: start with 1 at top; every subsequent entry is the sum of everything above you, plus 1.

			Triangle begins as:
   n\k [0]  [1]  [2]  [3]  [4]  [5]  [6] ...
  [0]   1;
  [1]   2,   2;
  [2]   4,   6,   4;
  [3]   8,  16,  16,   8;
  [4]  16,  40,  52,  40,  16;
  [5]  32,  96, 152, 152,  96,  32;
  [6]  64, 224, 416, 504, 416, 224,  64;
       ...

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

Cf. A000079, A001792, A007070, A057711, A077957, A084773.

See A059576 for a similar triangle.

A059474:= func< n,k | (&+[(-1)^j*2^(n-j)*Binomial(n-k,j)*Binomial(n-j,n-k): j in [0..n-k]]) >;
[A059474(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, May 21 2023

read transforms; SERIES2(1/(1-2*z-2*w+2*z*w),x,y,12): SERIES2TOLIST(%,x,y,12);
# Alternative
T := (n, k) -> 2^n*binomial(n, k)*hypergeom([-k, -n + k], [-n], 1/2):
for n from 0 to 10 do seq(simplify(T(n, k)), k = 0 .. n) end do; # Peter Luschny, Nov 26 2021

Table[(-1)^k*2^n*JacobiP[k, -n-1,0,0], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Oct 04 2017; May 21 2023 *)

def A059474(n,k): return 2^n*binomial(n, k)*simplify(hypergeometric([-k, k-n], [-n], 1/2))
flatten([[A059474(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, May 21 2023

G.f.: 1/(1 - 2*z - 2*w + 2*z*w).
T(n, k) = Sum_{j=0..n} (-1)^j*2^(n + k - j)*C(n, j)*C(n + k - j, n).
T(n, 0) = T(n, n) = A000079(n).
T(2*n, n) = A084773(n).
T(n, k) = 2^n*binomial(n, k)*hypergeom([-k, k - n], [-n], 1/2). - Peter Luschny, Nov 26 2021
From G. C. Greubel, May 21 2023: (Start)
T(n, n-k) = T(n, k).
Sum_{k=0..n} T(n, k) = A007070(n).
Sum_{k=0..n} (-1)^k * T(n, k) = A077957(n).
T(n, 1) = A057711(n+1) = 2*A001792(n) - [n=0].
T(n, 2) = 4*A049611(n-1). (End)

A083880 a(0)=1, a(1)=5, a(n) = 10a(n-1) - 23a(n-2), n >= 2.

Original entry on oeis.org

1, 5, 27, 155, 929, 5725, 35883, 227155, 1446241, 9237845, 59114907, 378678635, 2427143489, 15561826285, 99793962603, 640017621475, 4104915074881, 26328745454885, 168874407826587, 1083182932803515, 6947717948023649

Offset: 0

Paul Barry, May 08 2003

Binomial transform of A083879.

Inverse binomial transform of A147957. 5th binomial transform of A077957. - Philippe Deléham, Nov 30 2008

Index entries for linear recurrences with constant coefficients, signature (10,-23).

Cf. A083878, A006012.

[ n eq 1 select 1 else n eq 2 select 5 else 10*Self(n-1)-23*Self(n-2): n in [1..21] ]; // Klaus Brockhaus, Dec 16 2008

LinearRecurrence[{10,-23},{1,5},30] (* Harvey P. Dale, May 14 2018 *)

a(n)=if(n<0,0,polsym(23-10*x+x^2,n)[n+1]/2)

G.f.: (1-5x)/(1-10x+23x^2).
E.g.f.: exp(5x)cosh(x*sqrt(2)).
a(n) = ((5-sqrt(2))^n + (5+sqrt(2))^n)/2;
a(n) = Sum_{k=0..n} C(n, 2k)*5^(n-2k)*2^k.
a(n) = (Sum_{k=0..n} A098158(n,k)*5^(2k)*2^(n-k))/5^n. - Philippe Deléham, Nov 30 2008

Typo in definition corrected by Klaus Brockhaus, Dec 16 2008

A100213 Expansion of g.f.: x(4-7x+2x^2-8x^4+16x^5-16x^6)/((1-2x) (1-2x^2) (1-2x+2x^2) * (1+2*x^2)).

Original entry on oeis.org

4, 9, 14, 18, 32, 64, 128, 256, 544, 1104, 2144, 4128, 8192, 16384, 32768, 65536, 131584, 263424, 525824, 1049088, 2097152, 4194304, 8388608, 16777216, 33562624, 67129344, 134242304, 268443648, 536870912, 1073741824, 2147483648, 4294967296, 8590065664

Offset: 1

Creighton Dement, Nov 11 2004

The sequence can be created applying the pos operator (which sums over the positive coefficients) to the n-th power of the Floretion element (.5 'j + .5 'k + .5 j' + .5 k' + 1 'ii' + 1 e).

			a(5) = 32 because (.5 'j + .5 'k + .5 j' + .5 k' + 1 'ii' + 1 e)^5 = - 2 'j - 2 'k - 2 j' - 2 k' + 6 'ii' + 10 'jj' + 10 'kk' + 6 e,
and the sum of all positive coefficients is 6+10+10+6 = 32.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Creighton Dement, Floretion Online Multiplier.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,4,-16,24,-16).

Cf. A000079, A009116, A009545, A038503, A077957, A077966, A100212, A100215, A100216.

R:=PowerSeriesRing(Integers(), 50); Coefficients(R!( x*(4-7*x+2*x^2-8*x^4+16*x^5-16*x^6)/((1-4*x+6*x^2-4*x^3)*(1-4*x^4)) )); // G. C. Greubel, Mar 29 2024

Rest[CoefficientList[Series[x(4-7x+2x^2-8x^4+16x^5-16x^6)/((1-2x)(1-2x^2)(1-2x+2x^2)(1+2x^2)),{x,0,40}],x]] (* or *) LinearRecurrence[{4,-6,4,4,-16,24,-16},{4,9,14,18,32,64,128},40] (* Harvey P. Dale, Aug 23 2015 *)

def A100213_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x*(4-7*x+2*x^2-8*x^4+16*x^5-16*x^6)/((1-4*x+6*x^2-4*x^3)*(1-4*x^4)) ).list()
a=A100213_list(51); a[1:] # G. C. Greubel, Mar 29 2024

a(n) = A100215(n) - A100212(n).
a(n) = (-1)^n*A009116(n+3) + A100216 + A038503(n+1).
Equation above in Floretian Algebra operator speak: (pos) + (neg) = (ves) = (jes) + (les) + (tes)
a(n-1) = A000079(n+1) + (5*A077957(n) + 6*A077957(n-1))/4 + A009545(n)/2 + A009545(n+1) + A077966(n-1) - A077966(n)/4. - R. J. Mathar, May 07 2008
From G. C. Greubel, Mar 29 2024: (Start)
a(n) = (1/16)*( 2^(n+4) - 2*((1+5*i)*(1+i)^n + (1-5*i)*(1-i)^n) + (1 - (-1)^n)*2^((n+1)/2)*(5+i^(n+1)) + (1+(-1)^n)*2^(1+n/2)*(3-2*i^n) ).
a(2*n-1) = 2^(n-3)*( 2^(n+2) + 5 + (-1)^n - 6*cos(n*Pi/2) + 4*sin(n*Pi/2) ), for n >= 1.
a(2*n) = 2^(n-2)*( 2^(n+2) + 3 - 2*(-1)^n - cos(n*Pi/2) + 5*sin(n*Pi/2) ), n >= 1.
E.g.f.: -1 + exp(2*x) + (1/8)*(6*cosh(sqrt(2)*x) + 5*sqrt(2)* sinh(sqrt(2)*x) - (4*cos(sqrt(2)*x) + sqrt(2)*sin(sqrt(2)*x)) - 2*exp(x)*(cos(x) - 5*sin(x)) ). (End)

Replaced definition with generating function, changed offset to 1. - R. J. Mathar, Mar 12 2010

A103424 Expansion of e.g.f.: 1 + sinh(2*x).

Original entry on oeis.org

1, 2, 0, 8, 0, 32, 0, 128, 0, 512, 0, 2048, 0, 8192, 0, 32768, 0, 131072, 0, 524288, 0, 2097152, 0, 8388608, 0, 33554432, 0, 134217728, 0, 536870912, 0, 2147483648, 0, 8589934592, 0, 34359738368, 0, 137438953472, 0, 549755813888, 0, 2199023255552

Offset: 0

Paul Barry, Feb 05 2005

Binomial transform is A103425.

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

Cf. A077957, A103425, A199572.

With[{nn=50},CoefficientList[Series[1+Sinh[2x],{x,0,nn}],x] Range[ 0,nn-1]!] (* Harvey P. Dale, Jun 29 2014 *)
CoefficientList[Series[(1 + 2 x - 4 x^2)/(1 - 4 x^2), {x, 0, 50}], x] (* Vincenzo Librandi, Jun 30 2014 *)

G.f.: (1+2*x-4*x^2)/(1-4*x^2).
E.g.f.: 1 + sinh(2*x).
a(n) = 0^n+(2^n-(-2)^n)/2.
a(n) = Sum_{k=0..n} binomial(n, k)*(-1)^(k(n-k)).
a(n+1) = 2*A199572(n) = 2*A077957(n)^2. [Ralf Stephan, Jul 17 2013]

A108084 Triangle, read by rows, where T(0,0) = 1, T(n,k) = 2^n*T(n-1,k) + T(n-1,k-1).

Original entry on oeis.org

1, 2, 1, 8, 6, 1, 64, 56, 14, 1, 1024, 960, 280, 30, 1, 32768, 31744, 9920, 1240, 62, 1, 2097152, 2064384, 666624, 89280, 5208, 126, 1, 268435456, 266338304, 87392256, 12094464, 755904, 21336, 254, 1, 68719476736, 68451041280, 22638755840, 3183575040, 205605888, 6217920, 86360, 510, 1

Offset: 0

Gerald McGarvey, Jun 05 2005

Triangle T(n,k), 0 <= k <= n, read by rows given by [2, 2, 8, 12, 32, 56, 128, 240, 512, ...] DELTA [1, 0, 2, 0, 4, 0, 8, 0, 16, 0, 32, 0, ...] = A014236 (first zero omitted) DELTA A077957 where DELTA is the operator defined in A084938. - Philippe Deléham, Aug 23 2006

			Triangle begins:
      1;
      2,     1;
      8,     6,    1;
     64,    56,   14,    1;
   1024,   960,  280,   30,  1;
  32768, 31744, 9920, 1240, 62, 1;

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

Cf. A023531 (q=0), A007318 (q=1), this sequence (q=2), A173007 (q=3), A173008 (q=4).

Cf. A006125, A022166, A028362.

function T(n,k,q)
  if k lt 0 or k gt n then return 0;
  elif k eq n then return 1;
  else return q^n*T(n-1,k,q) + T(n-1,k-1,q);
  end if; return T; end function;
[T(n,k,2): k in [0..n], n in [0..10]]; // G. C. Greubel, Feb 20 2021

T[n_, k_, q_]:= T[n,k,q]= If[k<0 || k>n, 0, If[k==n, 1, q^n*T[n-1,k,q] +T[n-1,k-1,q] ]];
Table[T[n,k,2], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 20 2021 *)

def T(n, k, q):
    if (k<0 or k>n): return 0
    elif (k==n): return 1
    else: return q^n*T(n-1,k,q) + T(n-1,k-1,q)
flatten([[T(n,k,2) for k in (0..n)] for n in (0..10)]) # G. C. Greubel, Feb 20 2021

Sum_{k=0..n} T(n, k) = A028362(n).
T(n,0) = 2^(n*(n+1)/2) = A006125(n+1). - Philippe Deléham, Nov 05 2006
T(n,k) = 2^binomial(n+1-k,2) * A022166(n,k) for 0 <= k <= n. - Werner Schulte, Mar 25 2019

A147957 a(n) = ((6 + sqrt(2))^n + (6 - sqrt(2))^n)/2.

Original entry on oeis.org

1, 6, 38, 252, 1732, 12216, 87704, 637104, 4663312, 34298208, 253025888, 1870171584, 13839178816, 102484311936, 759279663488, 5626889356032, 41707163713792, 309171726460416, 2292017151256064, 16992367115418624, 125979822242317312, 934017384983574528, 6924894663564105728

Offset: 0

Al Hakanson (hawkuu(AT)blogspot.com), Nov 17 2008

6th binomial transform of A077957. Binomial transform of A083880. Inverse binomial transform of A147958. - Philippe Deléham, Nov 30 2008

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (12,-34).

Cf. A077957, A083880, A098158, A147958.

Z:= PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((6+r2)^n+(6-r2)^n)/2: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Nov 19 2008

LinearRecurrence[{12, -34}, {1, 6}, 50] (* G. C. Greubel, Aug 17 2018 *)

my(x='x+O('x^50)); Vec((1-6*x)/(1-12*x+34*x^2)) \\ G. C. Greubel, Aug 17 2018

From Philippe Deléham, Nov 19 2008: (Start)
a(n) = 12*a(n-1) - 34*a(n-2), n > 1; a(0)=1, a(1)=6.
G.f.: (1 - 6*x)/(1 - 12*x + 34*x^2).
a(n) = (Sum_{k=0..n} A098158(n,k)*6^(2k)*2^(n-k))/6^n. (End)
E.g.f.: exp(6*x)*cosh(sqrt(2)*x). - Ilya Gutkovskiy, Aug 11 2017

Extended beyond a(6) by Klaus Brockhaus, Nov 19 2008

A147958 a(n) = ((7 + sqrt(2))^n + (7 - sqrt(2))^n)/2.

Original entry on oeis.org

1, 7, 51, 385, 2993, 23807, 192627, 1577849, 13036417, 108350935, 904201491, 7566326929, 63431106929, 532418131343, 4472591813139, 37592633210825, 316085049734017, 2658336935367463, 22360719757645683, 188108240644768801

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Nov 17 2008

nonn

7th binomial transform of A077957. Binomial transform of A147957. Inverse binomial transform of A147959. - Philippe Deléham, Nov 30 2008

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (14, -47).

Cf. A077957, A098158, A147957, A147959.

Z:= PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((7+r2)^n+(7-r2)^n)/2: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Nov 19 2008

LinearRecurrence[{14, -47}, {1, 7}, 50] (* G. C. Greubel, Aug 17 2018 *)

x='x+O('x^30); Vec((1-7*x)/(1-14*x+47*x^2)) \\ G. C. Greubel, Aug 17 2018

From Philippe Deléham, Nov 19 2008: (Start)
a(n) = 14*a(n-1) - 47*a(n-2), n > 1; a(0)=1, a(1)=7.
G.f.: (1 - 7*x)/(1 - 14*x + 47*x^2).
a(n) = (Sum_{k=0..n} A098158(n,k)*7^(2k)*2^(n-k))/7^n. (End)
E.g.f.: exp(7*x)*cosh(sqrt(2)*x). - Ilya Gutkovskiy, Aug 11 2017

Extended beyond a(6) by Klaus Brockhaus, Nov 19 2008

A147959 a(n) = ((8 + sqrt(2))^n + (8 - sqrt(2))^n)/2.

Original entry on oeis.org

1, 8, 66, 560, 4868, 43168, 388872, 3545536, 32618512, 302072960, 2810819616, 26244590336, 245642629184, 2303117466112, 21620036448384, 203127300275200, 1909594544603392, 17959620096591872, 168959059780059648

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Nov 17 2008

Binomial transform of A147958. Inverse binomial transform of A147960. 8th binomial transform of A077957. - Philippe Deléham, Nov 30 2008

			a(3) = ((8 + sqrt(2))^3 + (8 - sqrt(2))^3)/2
     = (8^3 + 3*8^2*sqrt(2) + 3*8*2 + 2*sqrt(2)
      + 8^3 - 3*8^2*sqrt(2) + 3*8*2 - 2*sqrt(2))/2
     =  8^3                 + 3*8*2
     =  560.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (16,-62).

Cf. A077957, A147958, A147960.

Z:= PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((8+r2)^n+(8-r2)^n)/2: n in [0..18] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Nov 19 2008

LinearRecurrence[{16, -62}, {1, 8}, 50] (* G. C. Greubel, Aug 17 2018 *)

x='x+O('x^30); Vec((1-8*x)/(1-16*x+62*x^2)) \\ G. C. Greubel, Aug 17 2018

From Philippe Deléham, Nov 19 2008: (Start)
a(n) = 16*a(n-1) - 62*a(n-2), n > 1; a(0)=1, a(1)=8.
G.f.: (1 - 8*x)/(1 - 16*x + 62*x^2).
a(n) = (Sum_{k=0..n} A098158(n,k)*8^(2k)*2^(n-k))/8^n. (End)
E.g.f.: exp(8*x)*cosh(sqrt(2)*x). - Ilya Gutkovskiy, Aug 11 2017

Extended beyond a(6) by Klaus Brockhaus, Nov 19 2008

A147960 a(n) = ((9 + sqrt(2))^n + (9 - sqrt(2))^n)/2.

Original entry on oeis.org

1, 9, 83, 783, 7537, 73809, 733139, 7365591, 74662657, 762046137, 7818480563, 80531005311, 831898131121, 8612216940609, 89299952572403, 927034007995143, 9631915890692737, 100138799400852969, 1041577033850627219

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Nov 17 2008

nonn

Binomial transform of A147959. 9th binomial transform of A077957. - Philippe Deléham, Nov 30 2008

Hankel transform is := [1, 2, 0, 0, 0, 0, 0, 0, 0, 0, ...]. - Philippe Deléham, Dec 04 2008

G. C. Greubel, Table of n, a(n) for n = 0..980
Index entries for linear recurrences with constant coefficients, signature (18, -79).

Cf. A077957, A098158, A147959.

Z:= PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((9+r2)^n+(9-r2)^n)/2: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Nov 19 2008

LinearRecurrence[{18, -79}, {1, 9}, 50] (* G. C. Greubel, Aug 17 2018 *)

x='x+O('x^30); Vec((1-9*x)/(1-18*x+79*x^2)) \\ G. C. Greubel, Aug 17 2018

From Philippe Deléham, Nov 19 2008: (Start)
a(n) = 18*a(n-1) - 79*a(n-2), n > 1; a(0)=1, a(1)=9.
G.f.: (1 - 9*x)/(1 - 18*x + 79*x^2).
a(n) = (Sum_{k=0..n} A098158(n,k)*9^(2k)*2^(n-k))/9^n. (End)
E.g.f.: exp(9*x)*cosh(sqrt(2)*x). - Ilya Gutkovskiy, Aug 11 2017

Extended beyond a(6) by Klaus Brockhaus, Nov 19 2008

A351323 Number of tilings of a 6 X n rectangle with right trominoes.

Original entry on oeis.org

1, 0, 4, 8, 18, 72, 162, 520, 1514, 4312, 13242, 39088, 118586, 361712, 1103946, 3403624, 10513130, 32614696, 101530170, 316770752, 990771834, 3104283168, 9741133578, 30606719000, 96263812906, 303028237848, 954563802106, 3008665176560, 9487377712634, 29928407213328

Offset: 0

Gerhard Kirchner, Feb 21 2022

nonn

See A351322 for algorithm. The subsequence 1,8,162,... for 6 X 3n rectangles also has a depending recurrence with 11 parameters.

The sequence is the Hadamard sum of the following 4 sequences: 0, 0, 0, 0, 16, 0, 128, 0, 256, 768, 1024,0, 13440, 0, 16384, .. (tilings which have both horizontal and vertical faults), 0, 0, 4, 8, 0, 0, 16, 0, 0, 128, 0, 0, 1536, 0, 0,.. (tilings which have horizontal faults but no vertical faults), 0, 0, 0, 0, 0, 64, 16, 480, 1140, 3200, 11208, 36032, 95924, 333856, 1003096,.. (tilings which have vertical faults but no horizontal faults), 1, 0, 0, 0, 2, 8, 2, 40, 118, 216, 1010, 3056, 7686, 27856, 84466,... (tilings which have neither vertical nor horizontal faults). - R. J. Mathar, Dec 08 2022

			For a 6 X 2 rectangle there are 4 tilings:
   ___   ___   ___   ___
  |  _| |  _| |_  | |_  |
  |_| | |_| | | |_| | |_|
  |___| |___| |___| |___|
  |  _| |_  | |  _| |_  |
  |_| | | |_| |_| | | |_|
  |___| |___| |___| |___|

Index entries for linear recurrences with constant coefficients, signature (2,8,2,-43,-42,36,102,0,-44,-8,-8).

Cf. A077957, A000079, A046984, A084478, A351322, A351324, A236576 (straight trominoes), A233290 (mixed trominoes).

G.f.: (1 - x)*(1 - x - 5*x^2 - 7*x^3 + 6*x^4 + 12*x^5 + 6*x^6)/(1 - 2*x - 8*x^2 - 2*x^3 + 43*x^4 + 42*x^5 - 36*x^6 - 102*x^7 + 44*x^9 + 8*x^10 + 8*x^11).

a(n) = Sum_{i=0..10} b(i)*a(n-11+i) for n>10 where {b(i)} = {-8,-8,-44,0,102,36,-42,-43,2,8,2}.

cp's OEIS Frontend

A059474 Triangle read by rows: T(n,k) is coefficient of z^n*w^k in 1/(1 - 2*z - 2*w + 2*z*w) read by rows in order 00, 10, 01, 20, 11, 02, ...

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A083880 a(0)=1, a(1)=5, a(n) = 10*a(n-1) - 23*a(n-2), n >= 2.

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A100213 Expansion of g.f.: x*(4-7*x+2*x^2-8*x^4+16*x^5-16*x^6)/((1-2*x) * (1-2*x^2) * (1-2*x+2*x^2) * (1+2*x^2)).

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A103424 Expansion of e.g.f.: 1 + sinh(2*x).

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A108084 Triangle, read by rows, where T(0,0) = 1, T(n,k) = 2^n*T(n-1,k) + T(n-1,k-1).

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A147957 a(n) = ((6 + sqrt(2))^n + (6 - sqrt(2))^n)/2.

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A147958 a(n) = ((7 + sqrt(2))^n + (7 - sqrt(2))^n)/2.

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A059474 Triangle read by rows: T(n,k) is coefficient of z^nw^k in 1/(1 - 2z - 2w + 2z*w) read by rows in order 00, 10, 01, 20, 11, 02, ...

A083880 a(0)=1, a(1)=5, a(n) = 10a(n-1) - 23a(n-2), n >= 2.

A100213 Expansion of g.f.: x(4-7x+2x^2-8x^4+16x^5-16x^6)/((1-2x) (1-2x^2) (1-2x+2x^2) * (1+2*x^2)).