A098158 -id:A098158 - cp's OEIS frontend

A201701 Riordan triangle ((1-x)/(1-2x), x^2/(1-2x)).

1, 1, 0, 2, 1, 0, 4, 3, 0, 0, 8, 8, 1, 0, 0, 16, 20, 5, 0, 0, 0, 32, 48, 18, 1, 0, 0, 0, 64, 112, 56, 7, 0, 0, 0, 0, 128, 256, 160, 32, 1, 0, 0, 0, 0, 256, 576, 432, 120, 9, 0, 0, 0, 0, 0, 512, 1280, 1120, 400, 50, 1, 0, 0, 0, 0, 0

Offset: 0

Philippe Deléham, Dec 03 2011

Triangle T(n,k), read by rows, given by (1,1,0,0,0,0,0,0,0,...) DELTA (0,1,-1,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.
Skewed version of triangle in A200139.
Triangle without zeros: A207537.
For the version with negative odd numbered columns, which is Riordan ((1-x)/(1-2*x), -x^2/(1-2*x)) see comments on A028297 and A039991. - Wolfdieter Lang, Aug 06 2014
This is an example of a stretched Riordan array in the terminology of Section 2 of Corsani et al. - Peter Bala, Jul 14 2015

			The triangle T(n,k) begins:
  n\k      0     1     2     3     4    5   6  7 8 9 10 11 ...
  0:       1
  1:       1     0
  2:       2     1     0
  3:       4     3     0     0
  4:       8     8     1     0     0
  5:      16    20     5     0     0    0
  6:      32    48    18     1     0    0   0
  7:      64   112    56     7     0    0   0  0
  8:     128   256   160    32     1    0   0  0 0
  9:     256   576   432   120     9    0   0  0 0 0
  10:    512  1280  1120   400    50    1   0  0 0 0  0
  11:   1024  2816  2816  1232   220   11   0  0 0 0  0  0
  ...  reformatted and extended. - _Wolfdieter Lang_, Aug 06 2014

C. Corsani, D. Merlini, and R. Sprugnoli, Left-inversion of combinatorial sums, Discrete Mathematics, 180 (1998) 107-122.

Columns include A011782, A001792, A001793, A001794, A006974, A006975, A006976.
Diagonals sums are in A052980.
Cf. A028297, A081265, A124182, A131577, A039991 (zero-columns deleted, unsigned and zeros appended).
Cf. A098158, A200139, A207537.
Cf. A028297 (signed version, zeros deleted). Cf. A034839.

(* The function RiordanArray is defined in A256893. *)
RiordanArray[(1 - #)/(1 - 2 #)&, #^2/(1 - 2 #)&, 11] // Flatten (* Jean-François Alcover, Jul 16 2019 *)

T(n,k) = 2*T(n-1,k) + T(n-2,k-1) with T(0,0) = T(1,0) = 1, T(1,1) = 0 and T(n,k) = 0 for k<0 or for n

Sum_{k=0..n} T(n,k)^2 = A002002(n) for n>0.

Sum_{k=0..n} T(n,k)*x^k = A138229(n), A006495(n), A138230(n), A087455(n), A146559(n), A000012(n), A011782(n), A001333(n), A026150(n), A046717(n), A084057(n), A002533(n), A083098(n), A084058(n), A003665(n), A002535(n), A133294(n), A090042(n), A125816(n), A133343(n), A133345(n), A120612(n), A133356(n), A125818(n) for x = -6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17 respectively.

G.f.: (1-x)/(1-2*x-y*x^2). - Philippe Deléham, Mar 03 2012

From Peter Bala, Jul 14 2015: (Start)

Factorizes as A034839 * A007318 = (1/(1 - x), x^2/(1 - x)^2) * (1/(1 - x), x/(1 - x)) as a product of Riordan arrays.

T(n,k) = Sum_{i = k..floor(n/2)} binomial(n,2*i) *binomial(i,k). (End)

Name changed, keyword:easy added, crossrefs A028297 and A039991 added, and g.f. corrected by Wolfdieter Lang, Aug 06 2014

A202064 Triangle T(n,k), read by rows, given by (2, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 2, 0, 3, 1, 0, 4, 4, 0, 0, 5, 10, 1, 0, 0, 6, 20, 6, 0, 0, 0, 7, 35, 21, 1, 0, 0, 0, 8, 56, 56, 8, 0, 0, 0, 0, 9, 84, 126, 36, 1, 0, 0, 0, 0, 10, 120, 252, 120, 10, 0, 0, 0, 0, 0, 11, 165, 462, 330, 55, 1, 0, 0, 0, 0, 0

Offset: 0

Philippe Deléham, Dec 10 2011

Riordan array (x/(1-x)^2, x^2/(1-x)^2).
Mirror image of triangle in A119900.
A203322*A130595 as infinite lower triangular matrices. - Philippe Deléham, Jan 05 2011
From Gus Wiseman, Jul 07 2025: (Start)
Also the number of subsets of {1..n} containing n with k maximal runs (sequences of consecutive elements increasing by 1). For example, row n = 5 counts the following subsets:
  {5}          {1,5}      {1,3,5}
  {4,5}        {2,5}
  {3,4,5}      {3,5}
  {2,3,4,5}    {1,2,5}
  {1,2,3,4,5}  {1,4,5}
               {2,3,5}
               {2,4,5}
               {1,2,3,5}
               {1,2,4,5}
               {1,3,4,5}
For anti-runs instead of runs we have A053538.
Without requiring n see A210039, A202023, reverse A098158, A109446.
(End)

			Triangle begins :
1
2, 0
3, 1, 0
4, 4, 0, 0
5, 10, 1, 0, 0
6, 20, 6, 0, 0, 0
7, 35, 21, 1, 0, 0, 0
8, 56, 56, 8, 0, 0, 0, 0

Cf. A007318, A005314 (antidiagonal sums), A119900, A084938, A130595, A203322.
Column k = 1 is A000027.
Row sums are A000079.
Column k = 2 is A000292.
Without zeros we have A034867.
Last nonzero term in each row appears to be A124625.
A034839 counts subsets by number of maximal runs, for anti-runs A384893.
A116674 counts strict partitions by number of maximal runs, for anti-runs A384905.
Cf. A000045, A000071, A001629, A010027, A053538, A208342, A210034, A245563, A268193, A384177, A384890.
Cf. A098158, A109446, A202023, A210039.

Table[Length[Select[Subsets[Range[n]],MemberQ[#,n]&&Length[Split[#,#2==#1+1&]]==k&]],{n,12},{k,n}] (* Gus Wiseman, Jul 07 2025 *)

G.f.: 1/((1-x)^2-y*x^2).
Sum_{k, 0<=k<=n} T(n,k)*x^k = A000027(n+1), A000079(n), A000129(n+1), A002605(n+1), A015518(n+1), A063727(n), A002532(n+1), A083099(n+1), A015519(n+1), A003683(n+1), A002534(n+1), A083102(n), A015520(n+1), A091914(n) for x = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 10, 11, 12, 13 respectively.
T(n,k) = binomial(n+1,2k+1).
T(n,k) = 2*T(n-1,k) + T(n-2,k-1) - T(n-2,k), T(0,0) = 1, T(1,0) = 2, T(1,1) = 0 and T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Mar 15 2012

A133345 a(n) = 2a(n-1) + 14a(n-2) for n>1, a(0)=1, a(1)=1.

Original entry on oeis.org

1, 1, 16, 46, 316, 1276, 6976, 31816, 161296, 768016, 3794176, 18340576, 89799616, 436367296, 2129929216, 10369000576, 50557010176, 246280028416, 1200358199296, 5848636796416, 28502288382976, 138885491915776

Offset: 0

Philippe Deléham, Dec 21 2007

Binomial transform of A001024 (powers of 15), with interpolated zeros.

a(n) is the number of compositions of n when there are 1 type of 1 and 15 types of other natural numbers. - Milan Janjic, Aug 13 2010

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,14).

Cf. A001024, A098158.

[n le 2 select 1 else 2*(Self(n-1) +7*Self(n-2)): n in [1..41]]; // G. C. Greubel, Oct 15 2022

LinearRecurrence[{2,14},{1,1},30] (* Harvey P. Dale, Jan 07 2016 *)

Vec((1-x)/(1-2*x-14*x^2)+O(x^99)) \\ Charles R Greathouse IV, Jan 12 2012

A133345=BinaryRecurrenceSequence(2,14,1,1)
[A133345(n) for n in range(41)] # G. C. Greubel, Oct 15 2022

G.f.: (1-x)/(1-2*x-14*x^2).
a(n) = Sum_{k=0..n} A098158(n,k)*15^(n-k). - Philippe Deléham, Dec 26 2007
If p[1]=1, and p[i]=15, (i>1), and if A is Hessenberg matrix of order n defined by: A[i,j]=p[j-i+1], (i<=j), A[i,j]=-1, (i=j+1), and A[i,j]=0 otherwise. Then, for n>=1, a(n)=det A. - Milan Janjic, Apr 29 2010
a(n) = (b*i)^(n-1)*(b*i*ChebyshevU(n, -i/b) - ChebyshevU(n-1, -i/b)), with b = sqrt(14). - G. C. Greubel, Oct 15 2022

A133356 a(n) = 2a(n-1) + 16a(n-2) for n>1, a(0)=1, a(1)=1.

Original entry on oeis.org

1, 1, 18, 52, 392, 1616, 9504, 44864, 241792, 1201408, 6271488, 31765504, 163874816, 835997696, 4293992448, 21963948032, 112631775232, 576686718976, 2955481841664, 15137951186944, 77563611840512, 397334442672128

Offset: 0

Philippe Deléham, Dec 21 2007

Binomial transform of A001026 (powers of 17), with interpolated zeros .

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,16).

First differences of A161007.

Cf. A001026, A098158, A161007.

[n le 2 select 1 else 2*(Self(n-1) +8*Self(n-2)): n in [1..41]]; // G. C. Greubel, Oct 15 2022

LinearRecurrence[{2,16},{1,1},30] (* Harvey P. Dale, Dec 12 2012 *)

Vec((1-x)/(1-2*x-16*x^2)+O(x^99)) \\ Charles R Greathouse IV, Jan 12 2012

A133356=BinaryRecurrenceSequence(2,16,1,1)
[A133356(n) for n in range(41)] # G. C. Greubel, Oct 15 2022

G.f.: (1-x)/(1-2*x-16*x^2).
a(n) = Sum_{k=0..n} A098158(n,k)*17^(n-k). - Philippe Deléham, Dec 26 2007
If p[1]=1, and p[i]=17, (i>1), and if A is Hessenberg matrix of order n defined by: A[i,j] = p[j-i+1], (i<=j), A[i,j] = -1, (i=j+1), and A[i,j]=0 otherwise. Then, for n>=1, a(n)=det A. - Milan Janjic, Apr 29 2010
a(n) = (4*i)^(n-1)*(4*i*ChebyshevU(n, -i/4) - ChebyshevU(n-1, -i/4)) = A161007(n) - A161007(n-1). - G. C. Greubel, Oct 15 2022

A138229 Expansion of (1-x)/(1-2x+6x^2).

Original entry on oeis.org

1, 1, -4, -14, -4, 76, 176, -104, -1264, -1904, 3776, 18976, 15296, -83264, -258304, -17024, 1515776, 3133696, -2827264, -24456704, -31949824, 82840576, 357380096, 217716736, -1708847104, -4723994624, 805093376

Offset: 0

Paul Barry, Mar 06 2008

Binomial transform of [1, 0, -5, 0, 25, 0, -125, 0, 625, 0, ...]=: powers of -5 with interpolated zeros. - Philippe Deléham, Dec 02 2008

Michael De Vlieger, Table of n, a(n) for n = 0..2571
Beata Bajorska-Harapińska, Barbara Smoleń, Roman Wituła, On Quaternion Equivalents for Quasi-Fibonacci Numbers, Shortly Quaternaccis, Advances in Applied Clifford Algebras (2019) Vol. 29, 54.
Index entries for linear recurrences with constant coefficients, signature (2,-6).

Cf. A088139.

CoefficientList[Series[(1-x)/(1-2x+6x^2),{x,0,30}],x] (* or *) LinearRecurrence[ {2,-6},{1,1},30] (* Harvey P. Dale, Feb 29 2012 *)
TrigExpand@Table[6^(n/2) Cos[n ArcTan[Sqrt[5]]], {n, 0, 20}] (* or *)
Table[Sum[(-5)^k Binomial[n, 2 k], {k, 0, n/2}], {n, 0, 20}] (* Vladimir Reshetnikov, Sep 20 2016 *)

[lucas_number2(n,2,6)/2 for n in range(0,28)] # Zerinvary Lajos, Jul 08 2008

From Philippe Deléham, Nov 14 2008: (Start)
a(n) = 2*a(n-1) - 6*a(n-2), a(0)=1, a(1)=1.
a(n) = Sum_{k=0..n} A098158(n,k)*(-5)^(n-k). (End)
a(n) = Sum_{k=0..n} A124182(n,k)*(-6)^(n-k). - Philippe Deléham, Nov 15 2008
G.f.: G(0)/2, where G(k) = 1 + 1/(1 - x*(5*k+1)/(x*(5*k+6) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 26 2013
a(n) = real part of the quaternion (1 + i + 2*j)^n. - Peter Bala, Mar 29 2015

A145302 a(n) = ((7 + sqrt(7))^n + (7 - sqrt(7))^n)/2.

Original entry on oeis.org

1, 7, 56, 490, 4508, 42532, 406112, 3899224, 37532432, 361686640, 3487250816, 33630672544, 324364881344, 3128620091968, 30177356271104, 291080943932800, 2807684251672832, 27082179878242048, 261227779725129728

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Oct 06 2008

nonn

Binomial transform is A152265, inverse binomial transform is A146966.

Index entries for linear recurrences with constant coefficients, signature (14, -42).

Cf. A098158, A152265, A146966.

Z:= PolynomialRing(Integers()); N:=NumberField(x^2-7); S:=[ ((7+r7)^n+(7-r7)^n)/2: n in [0..18] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Oct 20 2008, Jul 09 2009

From R. J. Mathar, Oct 10 2008: (Start)
a(n) = 14*a(n-1) - 42*a(n-2).
G.f.: (1-7x)/(1-14x+42x^2). (End)
a(n) = Sum_{k=0..n} 7^k*A098158(n,k). - Philippe Deléham, Oct 14 2008

More terms from R. J. Mathar, Oct 10 2008

Edited by Klaus Brockhaus, Jul 09 2009

A145303 a(n) = ((8 + sqrt(8))^n + (8 - sqrt(8))^n)/2.

Original entry on oeis.org

1, 8, 72, 704, 7232, 76288, 815616, 8777728, 94769152, 1024753664, 11088986112, 120037572608, 1299617939456, 14071782965248, 152369922834432, 1649898919297024, 17865667030024192, 193456332999753728

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Oct 06 2008

nonn

Binomial transform is A152267, inverse binomial transform is A147689.

Index entries for linear recurrences with constant coefficients, signature (16, -56).

Cf. A081180, A098158, A152267, A147689.

Z:= PolynomialRing(Integers()); N:=NumberField(x^2-8); S:=[ ((8+r8)^n+(8-r8)^n)/2: n in [0..17] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Oct 20 2008

From R. J. Mathar, Oct 10 2008: (Start)
a(n) = 16*a(n-1) - 56*a(n-2).
G.f.: (1-8x)/(1-16x+56x^2).
a(n) = 2^n*A081180(n+1) - 2^(n+2)*A081180(n). (End)
a(n) = Sum_{k=0..n} 8^k*A098158(n,k). - Philippe Deléham, Oct 14 2008

More terms from R. J. Mathar, Oct 10 2008

Edited by Klaus Brockhaus, Jul 09 2009

A081335 a(n) = (6^n + 2^n)/2.

Original entry on oeis.org

1, 4, 20, 112, 656, 3904, 23360, 140032, 839936, 5039104, 30233600, 181399552, 1088393216, 6530351104, 39182090240, 235092508672, 1410554986496, 8463329787904, 50779978465280, 304679870267392, 1828079220555776

Offset: 0

Paul Barry, Mar 18 2003

Binomial transform of A034478. 4th binomial transform of (1, 0, 4, 0, 16, 0, 64, ...).

Case k=4 of the family of recurrences a(n) = 2*k*a(n-1) - (k^2-4)*a(n-2), a(0)=1, a(1)=k.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (8,-12).

Cf. A081336.

List([0..30], n-> 2^(n-1)*(3^n + 1)); # G. C. Greubel, Aug 02 2019

[(6^n+2^n)/2: n in [0..30]]; // Vincenzo Librandi, Aug 08 2013

LinearRecurrence[{8, -12}, {1, 4}, 30] (* Harvey P. Dale, May 03 2013 *)
CoefficientList[Series[(1-4x)/((1-2x)(1-6x)), {x,0,30}], x] (* Vincenzo Librandi, Aug 08 2013 *)

a(n)=(6^n+2^n)/2 \\ Charles R Greathouse IV, Oct 07 2015

[2^(n-1)*(3^n + 1) for n in (0..30)] # G. C. Greubel, Aug 02 2019

a(n) = 8*a(n-1) - 12*a(n-2), a(0)=1, a(1)=4.
G.f.: (1-4*x)/((1-2*x)*(1-6*x)).
E.g.f.: exp(4*x)*cosh(2*x).
a(n) = Sum_{k=0..floor(n/2)} binomial(n,2*k) * 4^(n-k) = Sum_{k=0..n} binomial(n,k) * 4^(n-k/2) * (1+(-1)^k)/2. - Paul Barry, Nov 22 2003
a(n) = Sum_{k=0..n} 4^k*A098158(n,k). - Philippe Deléham, Dec 04 2006

A119275 Inverse of triangle related to Padé approximation of exp(x).

Original entry on oeis.org

1, -2, 1, 0, -6, 1, 0, 12, -12, 1, 0, 0, 60, -20, 1, 0, 0, -120, 180, -30, 1, 0, 0, 0, -840, 420, -42, 1, 0, 0, 0, 1680, -3360, 840, -56, 1, 0, 0, 0, 0, 15120, -10080, 1512, -72, 1, 0, 0, 0, 0, -30240, 75600, -25200, 2520, -90, 1, 0, 0, 0, 0, 0, -332640, 277200, -55440, 3960, -110, 1

Offset: 0

Paul Barry, May 12 2006

Inverse of A119274.
Row sums are (-1)^(n+1)*A000321(n+1).
Bell polynomials of the second kind B(n,k)(1,-2). - Vladimir Kruchinin, Mar 25 2011
Also the inverse Bell transform of the quadruple factorial numbers Product_{k=0..n-1} (4*k+2) (A001813) giving unsigned values and adding 1,0,0,0,... as column 0. For the definition of the Bell transform see A264428 and for cross-references A265604. - Peter Luschny, Dec 31 2015

			Triangle begins
1,
-2, 1,
0, -6, 1,
0, 12, -12, 1,
0, 0, 60, -20, 1,
0, 0, -120, 180, -30, 1,
0, 0, 0, -840, 420, -42, 1,
0, 0, 0, 1680, -3360, 840, -56, 1,
0, 0, 0, 0, 15120, -10080, 1512, -72, 1
Row 4: D(x^4) = (1 - x*(d/dx)^2 + x^2/2!*(d/dx)^4 - ...)(x^4) = x^4 - 12*x^3 + 12*x^2.

Eric Weisstein's World of Mathematics, Bell Polynomial

Cf. A059344 (unsigned row reverse).

Cf. A034839, A001813, A001815, A086645, A098158, A130561, A231846.

# The function BellMatrix is defined in A264428.
# Adds (1,0,0,0, ..) as column 0.
BellMatrix(n -> `if`(n<2,(n+1)*(-1)^n,0), 9); # Peter Luschny, Jan 27 2016

Table[(-1)^(n - k) (n - k)!*Binomial[n + 1, k + 1] Binomial[k + 1, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Michael De Vlieger, Oct 12 2016 *)
BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
rows = 12;
M = BellMatrix[If[#<2, (#+1) (-1)^#, 0]&, rows];
Table[M[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 24 2018, after Peter Luschny *)

# uses[inverse_bell_matrix from A265605]
# Unsigned values and an additional first column (1,0,0, ...).
multifact_4_2 = lambda n: prod(4*k + 2 for k in (0..n-1))
inverse_bell_matrix(multifact_4_2, 9) # Peter Luschny, Dec 31 2015

T(n,k) = [k<=n]*(-1)^(n-k)*(n-k)!*C(n+1,k+1)*C(k+1,n-k).
From Peter Bala, May 07 2012: (Start)
E.g.f.: exp(x*(t-t^2)) - 1 = x*t + (-2*x+x^2)*t^2/2! + (-6*x^2+x^3)*t^3/3! + (12*x^2-12*x^3+x^4)*t^4/4! + .... Cf. A059344. Let D denote the operator sum {k >= 0} (-1)^k/k!*x^k*(d/dx)^(2*k). The n-th row polynomial R(n,x) = D(x^n) and satisfies the recurrence equation R(n+1,x) = x*R(n,x)-2*n*x*R(n-1,x). The e.g.f. equals D(exp(x*t)).
(End)
From Tom Copeland, Oct 11 2016: (Start)
With initial index n = 1 and unsigned, these are the partition row polynomials of A130561 and A231846 with c_1 = c_2 = x and c_n = 0 otherwise. The first nonzero, unsigned element of each diagonal is given by A001813 (for each row, A001815) and dividing along the corresponding diagonal by this element generates A098158 with its first column removed (cf. A034839 and A086645).
The n-th polynomial is generated by (x - 2y d/dx)^n acting on 1 and then evaluated at y = x, e.g., (x - 2y d/dx)^2 1 = (x - 2y d/dx) x = x^2 - 2y evaluated at y = x gives p_2(x) = -2x + x^2.
(End)

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A145301 a(n) = 12*a(n-1) - 30*a(n-2) with a(0)=1 and a(1)=6.

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A201701 Riordan triangle ((1-x)/(1-2*x), x^2/(1-2*x)).

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A202064 Triangle T(n,k), read by rows, given by (2, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

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A133345 a(n) = 2*a(n-1) + 14*a(n-2) for n>1, a(0)=1, a(1)=1.

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A133356 a(n) = 2*a(n-1) + 16*a(n-2) for n>1, a(0)=1, a(1)=1.

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A138229 Expansion of (1-x)/(1-2x+6x^2).

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A145302 a(n) = ((7 + sqrt(7))^n + (7 - sqrt(7))^n)/2.

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A145301 a(n) = 12a(n-1) - 30a(n-2) with a(0)=1 and a(1)=6.

A201701 Riordan triangle ((1-x)/(1-2x), x^2/(1-2x)).

A133345 a(n) = 2a(n-1) + 14a(n-2) for n>1, a(0)=1, a(1)=1.

A133356 a(n) = 2a(n-1) + 16a(n-2) for n>1, a(0)=1, a(1)=1.