A056854 -id:A056854 - cp's OEIS frontend

A299741 Array read by antidiagonals upwards: a(i,0) = 2, i >= 0; a(i,1) = i+2, i >= 0; a(i,j) = (i+2) * a(i,j-1) - a(i,j-2), for i >= 0, j > 1.

2, 2, 2, 2, 3, 2, 2, 4, 7, 2, 2, 5, 14, 18, 2, 2, 6, 23, 52, 47, 2, 2, 7, 34, 110, 194, 123, 2, 2, 8, 47, 198, 527, 724, 322, 2, 2, 9, 62, 322, 1154, 2525, 2702, 843, 2, 2, 10, 79, 488, 2207, 6726, 12098, 10084, 2207, 2, 2, 11, 98, 702, 3842, 15127, 39202, 57965, 37634, 5778, 2

Offset: 0

William W. Collier, Feb 18 2018

Note the similarity in form of the recursive steps in the array definition above and the polynomial definition under FORMULA.

			i\j |0  1   2    3      4       5        6          7           8            9
----+-------------------------------------------------------------------------
   0|2  2   2    2      2       2        2          2           2            2
   1|2  3   7   18     47     123      322        843        2207         5778
   2|2  4  14   52    194     724     2702      10084       37634       140452
   3|2  5  23  110    527    2525    12098      57965      277727      1330670
   4|2  6  34  198   1154    6726    39202     228486     1331714      7761798
   5|2  7  47  322   2207   15127   103682     710647     4870847     33385282
   6|2  8  62  488   3842   30248   238142    1874888    14760962    116212808
   7|2  9  79  702   6239   55449   492802    4379769    38925119    345946302
   8|2 10  98  970   9602   95050   940898    9313930    92198402    912670090
   9|2 11 119 1298  14159  154451  1684802   18378371   200477279   2186871698
  10|2 12 142 1692  20162  240252  2862862   34114092   406506242   4843960812
  11|2 13 167 2158  27887  360373  4656962   60180133   777684767  10049721838
  12|2 14 194 2702  37634  524174  7300802  101687054  1416317954  19726764302
  13|2 15 223 3330  49727  742575 11088898  165590895  2472774527  36926027010
  14|2 16 254 4048  64514 1028176 16386302  261152656  4162056194  66331746448
  15|2 17 287 4862  82367 1395377 23639042  400468337  6784322687 114933017342
  16|2 18 322 5778 103682 1860498 33385282  599074578 10749957122 192900153618
  17|2 19 359 6802 128879 2441899 46267202  876634939 16609796639 314709501202
  18|2 20 398 7940 158402 3160100 63043598 1257711860 25091193602 500566160180
  19|2 21 439 9198 192719 4037901 84603202 1772629341 37140612959 778180242798

William W. Collier, a(i,j) = f(i+2,j)
William W. Collier, Experimental Mathematics on Wisteria Tables, Talk to Poughkeepsie ACM Chapter.
OEIS Wiki, The (1,2) Pascal Triangle.

The array first appeared in A298675.
Rows 1 through 29 of the array appear in these OEIS entries: A005248, A003500, A003501, A003499, A056854, A086903, A056918, A087799, A057076, A087800, A078363, A067902, A078365, A090727, A078367, A087215, A078369, A090728, A090729, A090730, A090731, A090732, A090733, A090247, A090248, A090249, A090251. Also entries occur for rows 45, 121, and 320:  A087265, A065705, A089775. Each of these entries asserts that a(i,j)=f(i+2,j) is true for that row.
A few of the columns appear in the OEIS: A008865 (for column 2), A058794 and A007754 (for column 3), and A230586 (for column 5).
Main diagonal gives A343261.

A:= proc(i, j) option remember; `if`(min(i, j)=0, 2,
      `if`(j=1, i+2, (i+2)*A(i, j-1)-A(i, j-2)))
    end:
seq(seq(A(d-k, k), k=0..d), d=0..12);  # Alois P. Heinz, Mar 05 2019

a[, 0] = a[0, ] = 2; a[i_, 1] := i + 2;
a[i_, j_] := a[i, j] =(i + 2) a[i, j - 1] - a[i, j - 2];
Table[a[i - j, j], {i, 0, 10}, {j, 0, i}] // Flatten (* Jean-François Alcover, Dec 07 2019 *)

Let k be an integer, and let r1 and r2 be the roots of x + 1/x = k. Then f(k,n) = r1^n + r2^n is an integer, for integer n >= 0. Theorem: a(i,j) = f(i+2,j), for i,j >= 0. Proof: See the Collier link.
Define polynomials recursively by:
    p[0](n) = 2, for n >= 0 ( [ and ] demark subscripts).
    p[1](n) = n + 2, for n >= 0.
    p[j](n) = p[j-1](n) * p[1](n) - p[j-2](n), for j > 1, n >= 0. The coefficients of these polynomials occur as the even numbered, upward diagonals in the OEIS Wiki link. Conjecture: a(i,j) = p[j](i), i,j >= 0.

Edited by N. J. A. Sloane, Apr 04 2018

A060964 Table by antidiagonals where T(n,k) = n*T(n,k-1) - T(n,k-2) with T(n,0) = 2 and T(n,1) = n.

Original entry on oeis.org

2, 0, 2, -2, 1, 2, 0, -1, 2, 2, 2, -2, 2, 3, 2, 0, -1, 2, 7, 4, 2, -2, 1, 2, 18, 14, 5, 2, 0, 2, 2, 47, 52, 23, 6, 2, 2, 1, 2, 123, 194, 110, 34, 7, 2, 0, -1, 2, 322, 724, 527, 198, 47, 8, 2, -2, -2, 2, 843, 2702, 2525, 1154, 322, 62, 9, 2, 0, -1, 2, 2207, 10084, 12098, 6726, 2207, 488, 79, 10, 2

Offset: 0

Henry Bottomley, May 09 2001

			Square array begins as:
  2, 0, -2,   0,   2,    0,    -2, ...
  2, 1, -1,  -2,  -1,    1,     2, ...
  2, 2,  2,   2,   2,    2,     2, ...
  2, 3,  7,  18,  47,  123,   322, ...
  2, 4, 14,  52, 194,  724,  2702, ...
  2, 5, 23, 110, 527, 2525, 12098, ...

G. C. Greubel, Antidiagonals n = 0..100, flattened

Rows include A003499, A003500, A003501, A005248, A007395, A056854, A056918, A057079.

Columns include A001477, A007395, A008865, A058794.

T:= function(n,k)
    if k=0 then return 2;
    elif k=1 then return n;
    else return n*T(n,k-1) - T(n,k-2);
    fi; end;
Flat(List([0..12], n-> List([0..n], k-> T(k,n-k) ))); # G. C. Greubel, Jan 15 2020

function T(n,k)
  if k eq 0 then return 2;
  elif k eq 1 then return n;
  else return n*T(n, k-1) - T(n, k-2);
  end if; return T; end function;
[T(k,n-k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jan 15 2020

seq(seq( simplify(k*ChebyshevU(n-k-1, k/2) -2*ChebyshevU(n-k-2, k/2)), k=0..n), n=0..12); # G. C. Greubel, Jan 15 2020

Table[k*ChebyshevU[n-k-1, k/2] -2*ChebyshevU[n-k-2, k/2], {n,0,12}, {k,0,n} ]//Flatten

T(n,k) = n*polchebyshev(k-1,2,n/2) -2*polchebyshev(k-2,2,n/2);
for(n=0,12, for(k=0,n, print1(T(k,n-k), ", "))) \\ G. C. Greubel, Jan 15 2020

[[k*chebyshev_U(n-k-1, k/2) -2*chebyshev_U(n-k-2, k/2) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Jan 15 2020

For all m, T(n, k) = T(n, |m|)*T(n, |k - m|) - T(n, |k - 2m|).
T(n, 2k) = T(n, k)^2 - 2.
T(n, 2k + 1) = T(n, k)*T(n, k + 1) - n.
T(n, 3k) = T(n, k)^3 - 3*T(n, k).
T(n, 4k) = T(n, k)^4 - 4*T(n, k)^2 + 2.
T(n, 5k) = T(n, k)^5 - 5*T(n, k)^3 + 5*T(n, k) etc.
T(n, -k) = T(n, k).
T(-n, k) = T(-n, -k) = (-1)^k * T(n, k).
T(n, k) = ( n*( ((n + sqrt(n^2 -4))/2)^k - ((n - sqrt(n^2 -4))/2)^k ) - 2*( ((n + sqrt(n^2 -4))/2)^(k-1) - ((n - sqrt(n^2 -4))/2)^(k-1) ) )/sqrt(n^2 -4).
T(n, k) = n*ChebyshevU(k-1, n/2) - 2*ChebyshevU(k-2, n/2). - G. C. Greubel, Jan 15 2020

A081076 a(n) = Lucas(4n) + 3, or 5Fibonacci(2n-1)Fibonacci(2n+1).

Original entry on oeis.org

5, 10, 50, 325, 2210, 15130, 103685, 710650, 4870850, 33385285, 228826130, 1568397610, 10749957125, 73681302250, 505019158610, 3461452808005, 23725150497410, 162614600673850, 1114577054219525, 7639424778862810

Offset: 0

R. K. Guy, Mar 04 2003

Hugh C. Williams, Edouard Lucas and Primality Testing, John Wiley and Sons, 1998, p. 75.

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

Cf. A000032 (Lucas numbers), A000045 (Fibonacci numbers), A056854.

List([0..30], n-> Lucas(1, -1, 4*n)[2] +3 ); # G. C. Greubel, May 26 2020

[Lucas(4*n) +3: n in [0..30]]; // G. C. Greubel, May 26 2020

luc := proc(n) option remember: if n=0 then RETURN(2) fi: if n=1 then RETURN(1) fi: luc(n-1)+luc(n-2): end: for n from 0 to 30 do printf(`%d,`,luc(4*n)+3) od: # James Sellers, Mar 05 2003

Table[LucasL[4 n] + 3, {n,0,30}] (* Wesley Ivan Hurt, Nov 20 2014 *)

Vec(-5*(2*x^2-6*x+1)/((x-1)*(x^2-7*x+1)) + O(x^30)) \\ Michel Marcus, Dec 23 2014

[lucas_number2(4*n,1,-1) + 3 for n in (0..30)] # G. C. Greubel, May 26 2020

a(n) = 8*a(n-1) - 8*a(n-2) + a(n-3).
G.f.: 5*(1 -6*x +2*x^2)/((1-x)*(1-7*x+x^2)). - Colin Barker, Jun 22 2012
Sum_{n>=0} 1/a(n) = phi/5, where phi = A001622 = (1 + sqrt(5))/2. - Diego Rattaggi, Apr 22 2020
From G. C. Greubel, May 26 2020: (Start)
a(n) = 5*A064170(n+1).
a(n) = Lucas(n)^4 - 4*(-1)^n*Lucas(n)^2 + 5.
E.g.f.: 3*exp(x) + 2*exp(7*x/2)*cosh(3*sqrt(5)*x/2). (End)

More terms from James Sellers, Mar 05 2003

A081004 a(n) = Fibonacci(4n+2) + 1, or Fibonacci(2n+2)*Lucas(2n).

Original entry on oeis.org

2, 9, 56, 378, 2585, 17712, 121394, 832041, 5702888, 39088170, 267914297, 1836311904, 12586269026, 86267571273, 591286729880, 4052739537882, 27777890035289, 190392490709136, 1304969544928658, 8944394323791465, 61305790721611592, 420196140727489674

Offset: 0

R. K. Guy, Mar 01 2003

Hugh C. Williams, Edouard Lucas and Primality Testing, John Wiley and Sons, 1998, p. 75.

Nathaniel Johnston, Table of n, a(n) for n = 0..600
Index entries for linear recurrences with constant coefficients, signature (8,-8,1).

Cf. A000045 (Fibonacci numbers), A000032 (Lucas numbers), A056854 (first differences).

List([0..30], n-> Fibonacci(4*n+2)+1); # G. C. Greubel, Jul 15 2019

[Fibonacci(4*n+2)+1: n in [0..30]]; // Vincenzo Librandi, Apr 15 2011

with(combinat): for n from 0 to 30 do printf(`%d,`,fibonacci(4*n+2)+1) od: # James Sellers, Mar 03 2003

Table[Fibonacci[4n+2] +1, {n,0,30}] (* Wesley Ivan Hurt, Nov 20 2014 *)

vector(30, n, n--; fibonacci(4*n+2)+1) \\ G. C. Greubel, Jul 15 2019

[fibonacci(4*n+2)+1 for n in (0..30)] # G. C. Greubel, Jul 15 2019

a(n) = A033890(n)+1.
a(n) = 8*a(n-1) - 8*a(n-2) + a(n-3).
G.f.: (2-7*x)/((1-x)*(1-7*x+x^2)). - Colin Barker, Jun 24 2012

More terms from James Sellers, Mar 03 2003

A081078 a(n) = Lucas(4n) - 3, or Lucas(2n-1)*Lucas(2n+1).

Original entry on oeis.org

4, 44, 319, 2204, 15124, 103679, 710644, 4870844, 33385279, 228826124, 1568397604, 10749957119, 73681302244, 505019158604, 3461452807999, 23725150497404, 162614600673844, 1114577054219519, 7639424778862804, 52361396397820124, 358890350005878079

Offset: 1

R. K. Guy, Mar 04 2003

Hugh C. Williams, Edouard Lucas and Primality Testing, John Wiley and Sons, 1998, p. 75.

Amiram Eldar, Table of n, a(n) for n = 1..1196
Index entries for linear recurrences with constant coefficients, signature (8,-8,1).

Cf. A000032 (Lucas numbers), A056854 (Lucas(4n)), A244847.

luc := proc(n) option remember: if n=0 then RETURN(2) fi: if n=1 then RETURN(1) fi: luc(n-1)+luc(n-2): end: for n from 1 to 40 do printf(`%d,`,luc(4*n)-3) od: # James Sellers, Mar 05 2003

LinearRecurrence[{8,-8,1},{4,44,319},30] (* Harvey P. Dale, Jun 08 2014 *)
Table[LucasL[4*n] - 3, {n, 1, 20}] (* Amiram Eldar, Oct 27 2020 *)

Vec(x*(-4-12*x+x^2) / ( (x-1)*(x^2-7*x+1)) + O(x^30)) \\ Michel Marcus, Dec 23 2014

a(n) = 8a(n-1) - 8a(n-2) + a(n-3).
From R. J. Mathar, Sep 03 2010: (Start)
G.f.: x*(-4-12*x+x^2) / ( (x-1)*(x^2-7*x+1)).
a(n) = A056854(n)-3. (End)
From Peter Bala, Nov 30 2013: (Start)
a(n) = Lucas(2*n)^2 - 5.
Sum_{n>=1} 1/a(n) = (5 - sqrt(5))/10. (End)
Sum_{n>=1} 1/a(n) = A244847. - Amiram Eldar, Oct 27 2020

A098305 Unsigned member r=-5 of the family of Chebyshev sequences S_r(n) defined in A092184.

Original entry on oeis.org

0, 1, 5, 36, 245, 1681, 11520, 78961, 541205, 3709476, 25425125, 174266401, 1194439680, 8186811361, 56113239845, 384605867556, 2636127833045, 18068288963761, 123841894913280, 848824975429201, 5817932933091125, 39876705556208676, 273319005960369605, 1873356336166378561

Offset: 0

Wolfdieter Lang, Oct 18 2004

((-1)^(n+1))*a(n) = S_{-5}(n), n>=0, defined in A092184.

Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (6,6,-1).

Cf. A000032 (Lucas), A056854, A092184.

a(n) = 2*(T(n, 7/2)-(-1)^n)/9, with twice the Chebyshev polynomials of the first kind evaluated at x=7/2: 2*T(n, 7/2) = A056854(n) = ((7+sqrt(45))^n + (7-sqrt(45))^n)/2^n.
a(n) = 7*a(n-1)-a(n-2)+2*(-1)^(n+1), n>=2, a(0)=0, a(1)=1.
a(n) = 6*a(n-1) + 6*a(n-2) - a(n-3), n>=3, a(0)=0, a(1)=1, a(2)=5.
G.f.: x*(1-x)/((1+x)*(1-7*x+x^2)) = x*(1-x)/(1-6*x-6*x^2+x^3) (from the Stephan link, see A092184).
a(n) = (Lucas(4*n) - 2*(-1)^n)/9. - Greg Dresden, Oct 10 2020

More terms from Michel Marcus, Oct 11 2020

A193127 Numbers of spanning trees of the antiprism graphs.

Original entry on oeis.org

2, 36, 384, 3528, 30250, 248832, 1989806, 15586704, 120187008, 915304500, 6900949462, 51599794176, 383142771674, 2828107288188, 20768716848000, 151840963183392, 1105779284582146, 8024954790380544, 58059628319357318, 418891171182561000, 3014678940049375872, 21646865272061272716

Offset: 1

Eric W. Weisstein, Jul 16 2011

Antiprism graphs are defined for n>=3; extended to n=1 using closed form.

Stefano Spezia, Table of n, a(n) for n = 1..1100
Zbigniew R. Bogdanowicz, The number of spanning trees in a superprism, Discrete Math. Lett. 13 (2024) 66-73. See p. 66.
Eric Weisstein's World of Mathematics, Antiprism Graph
Eric Weisstein's World of Mathematics, Spanning Tree
Index entries for linear recurrences with constant coefficients, signature (16,-80,130,-80,16,-1).

Cf. A056854.

Table[2 n (GoldenRatio^(4 n) + GoldenRatio^(-4 n) - 2)/5, {n, 20}] // Round
LinearRecurrence[{16, -80, 130, -80, 16, -1}, {2, 36, 384, 3528, 30250, 248832}, 20]
CoefficientList[Series[(2 (1 + 2 x - 16 x^2 + 2 x^3 + x^4))/((-1 + x)^2 (1 - 7 x + x^2)^2), {x, 0, 20}], x] (* Eric W. Weisstein, Mar 28 2018 *)
Table[2 n (LucasL[4 n] - 2)/5, {n, 20}] (* Eric W. Weisstein, Mar 28 2018 *)

a(n)=my(x=quadgen(5)^n); real(2*n*(x^4+x^-4-2)/5) \\ Charles R Greathouse IV, Dec 17 2013

a(n) = 2/5*n*(phi^(4*n) + phi^(-4*n) - 2), where phi is the golden ratio.
a(n) = +16*a(n-1)-80*a(n-2)+130*a(n-3)-80*a(n-4)+16*a(n-5)-a(n-6).
O.g.f.: (2*x*(1 + 2*x - 16*x^2 + 2*x^3 + x^4))/((-1 + x)^2*(1 - 7*x + x^2)^2).
5*a(n) = 2*n*(A056854(n) - 2). - Eric W. Weisstein, Mar 28 2018

A131373 A046854 + A065941 - A000012.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 5, 2, 1, 1, 3, 6, 6, 3, 1, 1, 3, 10, 7, 10, 3, 1, 1, 4, 11, 14, 14, 11, 4, 1, 1, 4, 16, 15, 29, 15, 16, 4, 1, 1, 5, 17, 26, 35, 35, 26, 17, 5, 1

Offset: 0

Gary W. Adamson, Jul 03 2007

Row sums = A131269: (1, 2, 3, 6, 11, 20, 35, 60,...).

			First few rows of the triangle are:
1;
1, 1;
1, 1, 1;
1, 2, 2, 1;
1, 2, 5, 2, 1;
1, 3, 6, 6, 3, 1;
1, 3, 10, 7, 10, 3, 1;
1, 4, 11, 14, 14, 11, 4, 1;
1, 4, 16, 15, 29, 15, 16, 4, 1;
...

Cf. A056854, A065941, A000012, A131269.

A046854 + A065941 - A000012 as infinite lower triangular matrices.

cp's OEIS Frontend

A299741 Array read by antidiagonals upwards: a(i,0) = 2, i >= 0; a(i,1) = i+2, i >= 0; a(i,j) = (i+2) * a(i,j-1) - a(i,j-2), for i >= 0, j > 1.

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A060964 Table by antidiagonals where T(n,k) = n*T(n,k-1) - T(n,k-2) with T(n,0) = 2 and T(n,1) = n.

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A081076 a(n) = Lucas(4n) + 3, or 5*Fibonacci(2n-1)*Fibonacci(2n+1).

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A081004 a(n) = Fibonacci(4n+2) + 1, or Fibonacci(2n+2)*Lucas(2n).

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A081078 a(n) = Lucas(4n) - 3, or Lucas(2n-1)*Lucas(2n+1).

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A098305 Unsigned member r=-5 of the family of Chebyshev sequences S_r(n) defined in A092184.

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A081076 a(n) = Lucas(4n) + 3, or 5Fibonacci(2n-1)Fibonacci(2n+1).