A106854 -id:A106854 - cp's OEIS frontend

A019669 Decimal expansion of Pi/2.

1, 5, 7, 0, 7, 9, 6, 3, 2, 6, 7, 9, 4, 8, 9, 6, 6, 1, 9, 2, 3, 1, 3, 2, 1, 6, 9, 1, 6, 3, 9, 7, 5, 1, 4, 4, 2, 0, 9, 8, 5, 8, 4, 6, 9, 9, 6, 8, 7, 5, 5, 2, 9, 1, 0, 4, 8, 7, 4, 7, 2, 2, 9, 6, 1, 5, 3, 9, 0, 8, 2, 0, 3, 1, 4, 3, 1, 0, 4, 4, 9, 9, 3, 1, 4, 0, 1, 7, 4, 1, 2, 6, 7, 1, 0, 5, 8, 5, 3

Offset: 1

N. J. A. Sloane

With offset 2, decimal expansion of 5*Pi. - Omar E. Pol, Oct 03 2013
Decimal expansion of the number of radians in a quadrant. - John W. Nicholson, Oct 07 2013
Not the same as A085679. First differing term occurs at 10^-49, as list -49, or 51st counting term (a(-49)= 5 and A085679(-49) = 4). - John W. Nicholson, Oct 07 2013
5*Pi is also the surface area of a sphere whose diameter equals the square root of 5. More generally x*Pi is also the surface area of a sphere whose diameter equals the square root of x. - Omar E. Pol, Dec 22 2013
Pi/2 is also the radius of a sphere whose surface area equals the volume of the circumscribed cube. - Omar E. Pol, Dec 27 2013

			Pi/2 = 1.570796326794896619231321691639751442098584699...
5*Pi = 15.70796326794896619231321691639751442098584699...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Sections 1.4.1 and 1.4.2, pp. 20-21.

Harry J. Smith, Table of n, a(n) for n = 1..20000
David H. Bailey and Richard E. Crandall, On the Random Character of Fundamental Constant Expansions, Experimental Mathematics, Vol. 10 (2001), Issue 2, p. 185 (preprint draft).
Richard J. Mathar, Chebyshev approximation of x^m (-log x)^l in the interval 0 <= x <= 1, arXiv:2408.15212 [math.CA], 2024. See p. 2.
Michael Penn, A nice sum from the Harvard MIT math trust, YouTube video, 2022.
L. D. Servi, Nested Square Roots of 2, The American Mathematical Monthly 110:4 (Apr. 2003), pp. 326-330.
Johan Wästlund, An Elementary Proof of the Wallis Product Formula for pi, The American Mathematical Monthly 114:10 (Dec. 2007), pp. 914-917.
Eric W. Weisstein and Jonathan Sondow, Wallis Formula, MathWorld.
Wikipedia, Viète's formula.
Index entries for transcendental numbers.

Cf. A053300 (continued fraction), A060294 (2/Pi).

Cf. A000796, A019692, A122952, A019694 (Pi through 4*Pi), A106854.

Digits:=100: evalf(Pi/2); # Wesley Ivan Hurt, Oct 26 2016

RealDigits[N[Pi/2, 200]] (* Vladimir Joseph Stephan Orlovsky, Dec 02 2009 *)

default(realprecision, 20080); x=Pi/2; for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b019669.txt", n, " ", d)); \\ Harry J. Smith, May 31 2009

Pi/2 = log(i)/i, where i = sqrt(-1). - Eric Desbiaux, Jun 27 2009
Pi/2 = Product_{n>=1} (n/(n+1))^((-1)^n) = 2 * 2/3 * 4/3 * 4/5 * 6/5 * 6/7 * 8/7 * 8/9 * 10/9 * ... (Wallis formula). - William Keith and Alonso del Arte, Jun 24 2012
Equals Sum_{k>1} 2^k/binomial(2*k,k). - Bruno Berselli, Sep 11 2015
The previous result is the particular case n = 1 of the more general identity: Pi/2 = 4^(n-1) * n!/(2*n)! * Sum_{k >= 2} 2^(k+1)*(k + n - 1)!*(k + 2*n - 2)!/(2*k + 2*n - 2)! valid for n = 0,1,2,... . - Peter Bala, Oct 26 2016
Pi/2 = Product_{n>=1} (4*n^2)/(4*n^2-1). - Fred Daniel Kline, Oct 29 2016
Pi/2 = lim_{n->oo} F(2^(n+3))/2, with one half of the area of a regular 2^(n+3)-gon, for n >= 0, inscribed in the unit circle, written as iterated square roots of 2 as F(2^(n+3))/2 = 2^n*sqrt(2 + sq2(n)), with sq2(n) = sqrt(2 + sq2(n-1)), n >= 1, with input sq2(0) = 0 (2 appears n times in sq2(n)). Viète's infinite product formula works with the partial product F(2^(n+2))/2 = Product_{j=1..n} (2/sq2(j)), n >= 1, which corresponds to the above given formula. - Wolfdieter Lang, Jul 06 2018
Pi/2 = Integral_{x = 0..oo} sin(x)^2/x^2 dx = 1/2 + Sum_{n >= 1} sin(n)^2/n^2, by the Abel-Plana formula. - Peter Bala, Nov 05 2019
From Amiram Eldar, Aug 15 2020: (Start)
Equals Sum_{k>=0} k!/(2*k + 1)!!.
Equals Sum_{k>=0} (-1)^k/(k + 1/2).
Equals Integral_{x=0..oo} 1/(x^2 + 1) dx.
Equals Integral_{x=0..oo} sin(x)/x dx.
Equals Integral_{x=0..oo} exp(x/2)/(exp(x) + 1) dx.
Equals Product_{p prime > 2} p/(p + (-1)^((p-1)/2)). (End)
Pi/2 = Integral_{x = 0..oo} 1/(1 - x^2 + x^4) dx = (1 + 2/3 + 1/5) - (1/7 + 2/9 + 1/11) + (1/13 + 2/15 + 1/17) - .... - Peter Bala, Jul 22 2022
Equals arcsin(9/10) + sqrt(19)*Sum_{k >= 1} A106854(k-1)/(k*10^k) (see Bailey and Crandall, 2001). - Paolo Xausa, Jul 15 2024
Equals 2F1(1/2,1/2 ; 3/2; 1). - R. J. Mathar, Aug 20 2024
Pi/2 = [1;1,1/2,1/3,...,1/n,...] by Wallis's approximation. - Thomas Ordowski, Oct 19 2024
From Stefano Spezia, Oct 21 2024: (Start)
Equals Sum_{k>=0} 2^k/((2*k + 1)*binomial(2*k,k)) (see Finch).
Equals Limit_{n->oo} 2^(4*n)/((2*n + 1)*binomial(2*n,n)^2) (see Finch). (End)
Equals Integral_{x=-oo..oo} sech((2*x^3 + x^2 - 5*x)/(x^2 - 1)) dx. - Kritsada Moomuang, May 29 2025

A109466 Riordan array (1, x(1-x)).

Original entry on oeis.org

1, 0, 1, 0, -1, 1, 0, 0, -2, 1, 0, 0, 1, -3, 1, 0, 0, 0, 3, -4, 1, 0, 0, 0, -1, 6, -5, 1, 0, 0, 0, 0, -4, 10, -6, 1, 0, 0, 0, 0, 1, -10, 15, -7, 1, 0, 0, 0, 0, 0, 5, -20, 21, -8, 1, 0, 0, 0, 0, 0, -1, 15, -35, 28, -9, 1, 0, 0, 0, 0, 0, 0, -6, 35, -56, 36, -10, 1, 0, 0, 0, 0, 0, 0, 1, -21, 70, -84, 45, -11, 1, 0, 0, 0, 0

Offset: 0

Philippe Deléham, Aug 28 2005

Inverse is Riordan array (1, xc(x)) (A106566).
Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, -1, 1, 0, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.
Modulo 2, this sequence gives A106344. - Philippe Deléham, Dec 18 2008
Coefficient array of the polynomials Chebyshev_U(n, sqrt(x)/2)*(sqrt(x))^n. - Paul Barry, Sep 28 2009

			Rows begin:
  1;
  0,  1;
  0, -1,  1;
  0,  0, -2,  1;
  0,  0,  1, -3,  1;
  0,  0,  0,  3, -4,   1;
  0,  0,  0, -1,  6,  -5,   1;
  0,  0,  0,  0, -4,  10,  -6,   1;
  0,  0,  0,  0,  1, -10,  15,  -7,  1;
  0,  0,  0,  0,  0,   5, -20,  21, -8,  1;
  0,  0,  0,  0,  0,  -1,  15, -35, 28, -9, 1;
From _Paul Barry_, Sep 28 2009: (Start)
Production array is
  0,    1,
  0,   -1,    1,
  0,   -1,   -1,   1,
  0,   -2,   -1,  -1,   1,
  0,   -5,   -2,  -1,  -1,  1,
  0,  -14,   -5,  -2,  -1, -1,  1,
  0,  -42,  -14,  -5,  -2, -1, -1,  1,
  0, -132,  -42, -14,  -5, -2, -1, -1,  1,
  0, -429, -132, -42, -14, -5, -2, -1, -1, 1 (End)

Paul Barry, Embedding structures associated with Riordan arrays and moment matrices, arXiv preprint arXiv:1312.0583 [math.CO], 2013.
Tom Copeland, Addendum to Elliptic Lie Triad

Cf. A026729 (unsigned version), A000108, A030528, A124644.

/* As triangle */ [[(-1)^(n-k)*Binomial(k, n-k): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Jan 14 2016

(* The function RiordanArray is defined in A256893. *)
RiordanArray[1&, #(1-#)&, 13] // Flatten (* Jean-François Alcover, Jul 16 2019 *)

Number triangle T(n, k) = (-1)^(n-k)*binomial(k, n-k).
T(n, k)*2^(n-k) = A110509(n, k); T(n, k)*3^(n-k) = A110517(n, k).
Sum_{k=0..n} T(n,k)*A000108(k)=1. - Philippe Deléham, Jun 11 2007
From Philippe Deléham, Oct 30 2008: (Start)
Sum_{k=0..n} T(n,k)*A144706(k) = A082505(n+1).
Sum_{k=0..n} T(n,k)*A002450(k) = A100335(n).
Sum_{k=0..n} T(n,k)*A001906(k) = A100334(n).
Sum_{k=0..n} T(n,k)*A015565(k) = A099322(n).
Sum_{k=0..n} T(n,k)*A003462(k) = A106233(n). (End)
Sum_{k=0..n} T(n,k)*x^(n-k) = A053404(n), A015447(n), A015446(n), A015445(n), A015443(n), A015442(n), A015441(n), A015440(n), A006131(n), A006130(n), A001045(n+1), A000045(n+1), A000012(n), A010892(n), A107920(n+1), A106852(n), A106853(n), A106854(n), A145934(n), A145976(n), A145978(n), A146078(n), A146080(n), A146083(n), A146084(n) for x = -12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,7,8,9,10,11,12 respectively. - Philippe Deléham, Oct 27 2008
Sum_{k=0..n} T(n,k)*x^k = A000007(n), A010892(n), A099087(n), A057083(n), A001787(n+1), A030191(n), A030192(n), A030240(n), A057084(n), A057085(n+1), A057086(n) for x = 0,1,2,3,4,5,6,7,8,9,10 respectively. - Philippe Deléham, Oct 28 2008
G.f.: 1/(1-y*x+y*x^2). - Philippe Deléham, Dec 15 2011
T(n,k) = T(n-1,k-1) - T(n-2,k-1), T(n,0) = 0^n. - Philippe Deléham, Feb 15 2012
Sum_{k=0..n} T(n,k)*x^(n-k) = F(n+1,-x) where F(n,x)is the n-th Fibonacci polynomial in x defined in A011973. - Philippe Deléham, Feb 22 2013
Sum_{k=0..n} T(n,k)^2 = A051286(n). - Philippe Deléham, Feb 26 2013
Sum_{k=0..n} T(n,k)*T(n+1,k) = -A110320(n). - Philippe Deléham, Feb 26 2013
For T(0,0) = 0, the signed triangle below has the o.g.f. G(x,t) = [t*x(1-x)]/[1-t*x(1-x)] = L[t*Cinv(x)] where L(x) = x/(1-x) and Cinv(x)=x(1-x) with the inverses Linv(x) = x/(1+x) and C(x)= [1-sqrt(1-4*x)]/2, an o.g.f. for the shifted Catalan numbers A000108, so the inverse o.g.f. is Ginv(x,t) = C[Linv(x)/t] = [1-sqrt[1-4*x/(t(1+x))]]/2 (cf. A124644 and A030528). - Tom Copeland, Jan 19 2016

A190958 a(n) = 2a(n-1) - 10a(n-2), with a(0) = 0, a(1) = 1.

Original entry on oeis.org

0, 1, 2, -6, -32, -4, 312, 664, -1792, -10224, -2528, 97184, 219648, -532544, -3261568, -1197696, 30220288, 72417536, -157367808, -1038910976, -504143872, 9380822016, 23803082752, -46202054656, -330434936832, -198849327104, 2906650714112, 7801794699264

Offset: 0

Vladimir Joseph Stephan Orlovsky, May 24 2011

For the difference equation a(n) = c*a(n-1) - d*a(n-2), with a(0) = 0, a(1) = 1, the solution is a(n) = d^((n-1)/2) * ChebyshevU(n-1, c/(2*sqrt(d))) and has the alternate form a(n) = ( ((c + sqrt(c^2 - 4*d))/2)^n - ((c - sqrt(c^2 - 4*d))/2)^n )/sqrt(c^2 - 4*d). In the case c^2 = 4*d then the solution is a(n) = n*d^((n-1)/2). The generating function is x/(1 - c*x + d^2) and the exponential generating function takes the form (2/sqrt(c^2 - 4*d))*exp(c*x/2)*sinh(sqrt(c^2 - 4*d)*x/2) for c^2 > 4*d, (2/sqrt(4*d - c^2))*exp(c*x/2)*sin(sqrt(4*d - c^2)*x/2) for 4*d > c^2, and x*exp(sqrt(d)*x) if c^2 = 4*d. - G. C. Greubel, Jun 10 2022

Vincenzo Librandi, Table of n, a(n) for n = 0..190
Index entries for linear recurrences with constant coefficients, signature (2,-10).

Sequences of the form a(n) = c*a(n-1) - d*a(n-2), with a(0)=0, a(1)=1:
c/d...1.......2.......3.......4.......5.......6.......7.......8.......9......10
.A128834,A107920,A106852,A106853,A106854,A145934,A145976,A145978,A146078,A146080
.A000027,A009545,A088137,A088138,A045873,A088139,A089181,A090591,A127357,A190958
.A001906,A000225,A057083,A049072,A190959,A190960,A190961,A190962,A190963,A190964
.A001353,A007070,A003462,A001787,A099456,A190965,A168175,A190966,A190967,A190968
.A004254,A107839,A116415,A002450,A030191,A001047,A099450,A190969,A190970,A190971
.A001109,A154244,A138395,A084326,A003463,A030192,A081179,A006516,A027471,A106392
.A004187,A186446,A190972,A190973,A190974,A003464,A030240,A152268,A099459,A016127
.A001090,A190975,A190976,A099156,A190977,A190978,A023000,A057084,A154245,A154235
.A018913,A190979,A190980,A190981,A190982,A190983,A190984,A023001,A057085,A178869
A004189,A190869,A190985,A190986,A190987,A190988,A190989,A190990,A002452,A057086

I:=[0,1]; [n le 2 select I[n] else 2*Self(n-1)-10*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Sep 17 2011

Mathematica
```
LinearRecurrence[{2,-10}, {0,1}, 50]
```

a(n)=([0,1; -10,2]^n*[0;1])[1,1] \\ Charles R Greathouse IV, Apr 08 2016

[lucas_number1(n,2,10) for n in (0..50)] # G. C. Greubel, Jun 10 2022

G.f.: x / ( 1 - 2*x + 10*x^2 ). - R. J. Mathar, Jun 01 2011
E.g.f.: (1/3)*exp(x)*sin(3*x). - Franck Maminirina Ramaharo, Nov 13 2018
a(n) = 10^((n-1)/2) * ChebyshevU(n-1, 1/sqrt(10)). - G. C. Greubel, Jun 10 2022
a(n) = (1/3)*10^(n/2)*sin(n*arctan(3)) = Sum_{k=0..floor(n/2)} (-1)^k*3^(2*k)*binomial(n,2*k+1). - Gerry Martens, Oct 15 2022

A145934 Expansion of 1/(1-x(1-6x)).

Original entry on oeis.org

1, 1, -5, -11, 19, 85, -29, -539, -365, 2869, 5059, -12155, -42509, 30421, 285475, 102949, -1609901, -2227595, 7431811, 20797381, -23793485, -148577771, -5816861, 885649765, 920550931, -4393347659, -9916653245, 16443432709

Offset: 0

Philippe Deléham, Oct 25 2008

Row sums of Riordan array (1, x(1-6x)).

For positive n, a(n) equals the determinant of the n X n tridiagonal matrix with 1's along the main diagonal, 3's along the superdiagonal, and 2's along the subdiagonal (see Mathematica code below). - John M. Campbell, Jul 08 2011

Seiichi Manyama, Table of n, a(n) for n = 0..2569
Taras Goy and Mark Shattuck, Determinants of Toeplitz-Hessenberg Matrices with Generalized Leonardo Number Entries, Ann. Math. Silesianae (2023). See p. 17.
Index entries for linear recurrences with constant coefficients, signature (1,-6).

Cf. A010892, A107920, A106852, A106853, A106854.

I:=[1,1]; [n le 2 select I[n] else Self(n-1) - 6*Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 14 2018

Table[Det[Array[KroneckerDelta[#1, #2] + KroneckerDelta[#1, #2 + 1]*2 + KroneckerDelta[#1, #2 - 1]*3 &, {n, n}]], {n, 1, 40}] (* John M. Campbell, Jul 08 2011 *)
LinearRecurrence[{1,-6}, {1,1}, 30] (* G. C. Greubel, Jan 14 2018 *)

Vec(1/(1-x*(1-6*x))+O(x^99)) \\ Charles R Greathouse IV, Sep 23 2012

[lucas_number1(n,1,6) for n in range(1, 29)] # Zerinvary Lajos, Apr 22 2009

a(n) = Sum_{k=0..n} A109466(n,k)*6^(n-k).

a(n) = a(n-1) - 6*a(n-2); a(0)=1, a(1)=1. - Philippe Deléham, Oct 25 2008

A145976 Expansion of 1/(1-x(1-7x)).

Original entry on oeis.org

1, 1, -6, -13, 29, 120, -83, -923, -342, 6119, 8513, -34320, -93911, 146329, 803706, -220597, -5846539, -4302360, 36623413, 66739933, -189623958, -656803489, 670564217, 5268188640, 574239121, -36303081359, -40322755206, 213798814307

Offset: 0

Philippe Deléham, Oct 26 2008

Row sums of Riordan array (1,x(1-7x)).

G. C. Greubel, Table of n, a(n) for n = 0..1500
Taras Goy and Mark Shattuck, Determinants of Toeplitz-Hessenberg Matrices with Generalized Leonardo Number Entries, Ann. Math. Silesianae (2023). See p. 17.
Index entries for linear recurrences with constant coefficients, signature (1,-7).

Cf. A010892, A107920, A106852, A106853, A106854, A145934.

I:=[1,1]; [n le 2 select I[n] else Self(n-1) - 7*Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 19 2018

Join[{a=1,b=1},Table[c=b-7*a;a=b;b=c,{n,80}]] (* Vladimir Joseph Stephan Orlovsky, Jan 22 2011 *)
CoefficientList[Series[1/(1-x(1-7x)),{x,0,50}],x] (* or *) LinearRecurrence[{1,-7},{1,1},50] (* Harvey P. Dale, May 11 2011 *)

Vec(1/(1-x*(1-7*x)) + O(x^40)) \\ Michel Marcus, Jan 29 2016

[lucas_number1(n,1,7) for n in range(1, 29)] # Zerinvary Lajos, Apr 22 2009

a(n) = a(n-1) - 7*a(n-2), a(0)=1, a(1)=1.

a(n) = Sum_{k=0..n} A109466(n,k)*7^(n-k).

Corrected by Zerinvary Lajos, Apr 22 2009

Corrected by D. S. McNeil, Aug 20 2010

A145978 Expansion of 1/(1-x(1-8x)).

Original entry on oeis.org

1, 1, -7, -15, 41, 161, -167, -1455, -119, 11521, 12473, -79695, -179479, 458081, 1893913, -1770735, -16922039, -2756159, 132620153, 154669425, -906291799, -2143647199, 5106687193, 22255864785, -18597632759, -196644551039, -47863488967, 1525292919345

Offset: 0

Philippe Deléham, Oct 26 2008

Row sums of Riordan array (1,1(1-8x)).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,-8).

Cf. A010892, A107920, A106852, A106853, A106854, A145934, A145976.

I:=[1,1]; [n le 2 select I[n] else Self(n-1) - 8*Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 19 2018

LinearRecurrence[{1,-8},{1,1},50] (* G. C. Greubel, Jan 29 2016 *)

Vec(1/(1-x*(1-8*x)) + O(x^40)) \\ Michel Marcus, Jan 29 2016

[lucas_number1(n,1,8) for n in range(1, 27)] # Zerinvary Lajos, Apr 22 2009

a(n) = a(n-1) - 8*a(n-2), a(0)=1, a(1)=1.

a(n) = Sum_{k=0..n} A109466(n,k)*8^(n-k).

A146078 Expansion of 1/(1-x(1-9x)).

Original entry on oeis.org

1, 1, -8, -17, 55, 208, -287, -2159, 424, 19855, 16039, -162656, -307007, 1156897, 3919960, -6492113, -41771753, 16657264, 392603041, 242687665, -3290739704, -5474928689, 24141728647, 73416086848, -143859470975, -804604252607

Offset: 0

Philippe Deléham, Oct 27 2008

Row sums of Riordan array (1, x(1-9x)).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,-9).

Cf. A010892, A107920, A106852, A106853, A106854, A145934, A145976, A145978.

I:=[1,1]; [n le 2 select I[n] else Self(n-1) - 9*Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 19 2018

LinearRecurrence[{1, -9}, {1, 1}, 100] (* G. C. Greubel, Jan 30 2016 *)

x='x+O('x^30); Vec(1/(1-x+9*x^2)) \\ G. C. Greubel, Jan 19 2018

[lucas_number1(n,1,9) for n in range(1, 27)] # Zerinvary Lajos, Apr 22 2009

a(n) = a(n-1) - 9*a(n-2), a(0)=1, a(1)=1.
a(n) = Sum_{k=0..n} A109466(n,k)*9^(n-k).
From G. C. Greubel, Jan 31 2016: (Start)
G.f.: 1/(1-x+9*x^2).
E.g.f.: exp(x/2)*(cos(sqrt(35)*x/2) + (1/sqrt(35))*sin(sqrt(35)*x/2)). (End)
a(n) = Product_{k=1..n} (1 + 6*cos(k*Pi/(n+1))). - Peter Luschny, Nov 28 2019
a(n) = 3^n * U(n, 1/6), where U(n, x) is the Chebyshev polynomial of the second kind. - Federico Provvedi, Mar 28 2022

A146083 Expansion of 1/(1 - x(1 - 11x)).

Original entry on oeis.org

1, 1, -10, -21, 89, 320, -659, -4179, 3070, 49039, 15269, -524160, -692119, 5073641, 12686950, -43123101, -182679551, 291674560, 2301149621, -907270539, -26219916370, -16239940441, 272179139629, 450818484480, -2543152051439

Offset: 0

Philippe Deléham, Oct 27 2008

Row sums of Riordan array (1,x(1-11x)).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,-11).

Cf. A010892, A107920, A106852, A106853, A106854, A145934, A145976, A145978, A146078, A146080.

[n le 2 select 1 else Self(n-1)-11*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Jan 31 2016

LinearRecurrence[{1,-11},{1,1},30] (* Harvey P. Dale, Jan 07 2016 *)

Vec(1/(1-x*(1-11*x))+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

[lucas_number1(n,1,11) for n in range(1, 26)] # Zerinvary Lajos, Apr 22 2009

a(n) = a(n-1)-11*a(n-2) ; a(0)=1, a(1)=1.
a(n) = Sum_{k=0..n} A109466(n,k)*11^(n-k).
From G. C. Greubel, Jan 31 2016: (Start)
G.f.: 1/(1-x+11*x^2).
E.g.f.: exp(x/2)*(cos(sqrt(43)*x/2) + (1/sqrt(43))*sin(sqrt(43)*x/2)).
(End)

A146084 Expansion of 1/(1-x(1-12x)).

Original entry on oeis.org

1, 1, -11, -23, 109, 385, -923, -5543, 5533, 72049, 5653, -858935, -926771, 9380449, 20501701, -92063687, -338084099, 766680145, 4823689333, -4376472407, -62260744403, -9743075519, 737385857317, 854302763545, -7994327524259

Offset: 0

Philippe Deléham, Oct 27 2008

Row sums of Riordan array (1,x(1-12x)).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1, -12).

Cf. A010892, A107920, A106852, A106853, A106854, A145934, A145976, A145978, A146078, A146080, A146083

LinearRecurrence[{1, -12}, {1, 1}, 100] (* G. C. Greubel, Jan 30 2016 *)

Vec(1/(1-x*(1-12*x))+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

[lucas_number1(n,1,12) for n in range(1, 26)] # Zerinvary Lajos, Apr 22 2009

a(n) = a(n-1)-12*a(n-2), a(0)=1, a(1)=1.

a(n) = Sum_{k, 0<=k<=n} A109466(n,k)*12^(n-k).

A128100 Triangle read by rows: T(n,k) is the number of ways to tile a 2 X n rectangle with k pieces of 2 X 2 tiles and n-2k pieces of 1 X 2 tiles (0 <= k <= floor(n/2)).

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 5, 5, 1, 8, 10, 3, 13, 20, 9, 1, 21, 38, 22, 4, 34, 71, 51, 14, 1, 55, 130, 111, 40, 5, 89, 235, 233, 105, 20, 1, 144, 420, 474, 256, 65, 6, 233, 744, 942, 594, 190, 27, 1, 377, 1308, 1836, 1324, 511, 98, 7, 610, 2285, 3522, 2860, 1295, 315, 35, 1, 987, 3970

Offset: 0

Emeric Deutsch, Feb 18 2007

Row sums are the Jacobsthal numbers (A001045). Column 0 yields the Fibonacci numbers (A000045); the other columns yield convolved Fibonacci numbers (A001629, A001628, A001872, A001873, etc.). Sum_{k=0..floor(n/2)} k*T(n,k) = A073371(n-2).
Triangle T(n,k), with zeros omitted, given by (1, 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. - Philippe Deléham, Jan 24 2012
Riordan array (1/(1-x-x^2), x^2/(1-x-x^2)), with zeros omitted. - Philippe Deléham, Feb 06 2012
Diagonal sums are A000073(n+2) (tribonacci numbers). - Philippe Deléham, Feb 16 2014
Number of induced subgraphs of the Fibonacci cube Gamma(n-1) that are isomorphic to the hypercube Q_k. Example: row n=4 is 5, 5, 1; indeed, the Fibonacci cube Gamma(3) is a square with an additional pendant edge attached to one of its vertices; it has 5 vertices (i.e., Q_0's), 5 edges (i.e., Q_1's) and 1 square (i.e., Q_2). - Emeric Deutsch, Aug 12 2014
Row n gives the coefficients of the polynomial p(n,x) defined as the numerator of the rational function given by f(n,x) = 1 + (x + 1)/f(n-1,x), where f(x,0) = 1. Conjecture: for n > 2, p(n,x) is irreducible if and only if n is a (prime - 2). - Clark Kimberling, Oct 22 2014

			Triangle starts:
   1;
   1;
   2,  1;
   3,  2;
   5,  5,  1;
   8, 10,  3;
  13, 20,  9,  1;
  21, 38, 22,  4;
From _Philippe Deléham_, Jan 24 2012: (Start)
Triangle (1, 1, -1, 0, 0, ...) DELTA (0, 1, -1, 0, 0, 0, ...) begins:
   1;
   1,  0;
   2,  1,  0;
   3,  2,  0,  0;
   5,  5,  1,  0,  0;
   8, 10,  3,  0,  0,  0;
  13, 20,  9,  1,  0,  0,  0;
  21, 38, 22,  4,  0,  0,  0,  0; (End)
From _Clark Kimberling_, Oct 22 2014: (Start)
Here are the first 4 polynomials p(n,x) as in Comment and generated by Mathematica program:
  1
  2 +  x
  3 + 2x
  5 + 5x + x^2. (End)

C.-P. Chou and H. A. Witek, An Algorithm and FORTRAN Program for Automatic Computation of the Zhang-Zhang Polynomial of Benzenoids, MATCH: Commun. Math. Comput. Chem, 68 (2012) 3-30. See Eq. (9). - From _N. J. A. Sloane_, Dec 23 2012
S. Klavzar, M. Mollard, Cube polynomial of Fibonacci and Lucas cubes, preprint.
S. Klavzar, M. Mollard, Cube polynomial of Fibonacci and Lucas cubes, Acta Appl. Math. 117, 2012, 93-105. - _Emeric Deutsch_, Aug 12 2014

Cf. A001045, A000045, A001629, A001628, A001872, A001873, A073371.

Cf. A109466, A119473.

G:=1/(1-z-(1+t)*z^2): Gser:=simplify(series(G,z=0,19)): for n from 0 to 16 do P[n]:=sort(coeff(Gser,z,n)) od: for n from 0 to 16 do seq(coeff(P[n],t,j),j=0..floor(n/2)) od; # yields sequence in triangular form

p[x_, n_] := 1 + (x + 1)/p[x, n - 1]; p[x_, 1] = 1;
Numerator[Table[Factor[p[x, n]], {n, 1, 20}]]  (* Clark Kimberling, Oct 22 2014 *)

G.f.: 1/(1-z-(1+t)z^2).
Sum_{k=0..n} T(n,k)*x^k = A053404(n), A015447(n), A015446(n), A015445(n), A015443(n), A015442(n), A015441(n), A015440(n), A006131(n), A006130(n), A001045(n+1), A000045(n+1), A000012(n), A010892(n), A107920(n+1), A106852(n), A106853(n), A106854(n), A145934(n), A145976(n), A145978(n), A146078(n), A146080(n), A146083(n), A146084(n) for x = 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, -1, -2, -3, -4, -5, -6, -7, -8, -9, -10, -11, -12, and -13, respectively. - Philippe Deléham, Jan 24 2012
T(n,k) = T(n-1,k) + T(n-2,k) + T(n-2,k-1). - Philippe Deléham, Jan 24 2012
G.f.: T(0)/2, where T(k) = 1 + 1/(1 - (2*k+1+ x*(1+y))*x/((2*k+2+ x*(1+y))*x + 1/T(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Nov 06 2013
T(n,k) = Sum_{i=k..floor(n/2)} binomial(n-i,i)*binomial(i,k). See Corollary 3.3 in the Klavzar et al. link. - Emeric Deutsch, Aug 12 2014

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