A123011 -id:A123011 - cp's OEIS frontend

A153518 Triangular T(n,k) = T(n-1, k) + T(n-1, k-1) + 5*T(n-2, k-1), read by rows.

2, 5, 5, 2, 46, 2, 2, 123, 123, 2, 2, 135, 476, 135, 2, 2, 147, 1226, 1226, 147, 2, 2, 159, 2048, 4832, 2048, 159, 2, 2, 171, 2942, 13010, 13010, 2942, 171, 2, 2, 183, 3908, 26192, 50180, 26192, 3908, 183, 2, 2, 195, 4946, 44810, 141422, 141422, 44810, 4946, 195, 2

Offset: 1

Roger L. Bagula, Dec 28 2008

			Triangle begins as:
  2;
  5,   5;
  2,  46,    2;
  2, 123,  123,     2;
  2, 135,  476,   135,      2;
  2, 147, 1226,  1226,    147,      2;
  2, 159, 2048,  4832,   2048,    159,     2;
  2, 171, 2942, 13010,  13010,   2942,   171,    2;
  2, 183, 3908, 26192,  50180,  26192,  3908,  183,   2;
  2, 195, 4946, 44810, 141422, 141422, 44810, 4946, 195, 2;

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Sequences with variable (p,q,j): A153516 (0,1,2), this sequences (0,1,3), A153520 (0,1,4), A153521 (0,1,5), A153648 (1,0,3), A153649 (1,1,4), A153650 (1,4,5), A153651 (1,5,6), A153652 (2,1,7), A153653 (2,1,8), A153654 (2,1,9), A153655 (2,1,10), A153656 (2,3,9), A153657 (2,7,10).

Cf. A123011.

f:= func< n,j | Round(((3-(-1)^n)/2)*NthPrime(j)^(n-1) - 2^((3-(-1)^n)/2)) >;
function T(n,k,p,q,j)
  if n eq 2 then return NthPrime(j);
  elif (n eq 3 and k eq 2 or n eq 4 and k eq 2 or n eq 4 and k eq 3) then return f(n,j);
  elif (k eq 1 or k eq n) then return 2;
  else return T(n-1,k,p,q,j) + T(n-1,k-1,p,q,j) + (p*j+q)*NthPrime(j)*T(n-2,k-1,p,q,j);
  end if; return T;
end function;
[T(n,k,0,1,3): k in [1..n], n in [1..12]]; // G. C. Greubel, Mar 04 2021

A153518 := proc(n,k) option remember ; if n =1 then 2; elif n = 2 then 5; elif k=1 or k=n then 2; elif n = 3 then 46 ; elif n = 4 then 123 ; else procname(n-1,k-1)+procname(n-1,k)+5*procname(n-2,k-1) ; end: end: for n from 1 to 13 do for k from 1 to n do printf("%d,",A153518(n,k)) ; od: od: # R. J. Mathar, Jan 22 2009

T[n_, k_, p_, q_, j_]:= T[n,k,p,q,j]= If[n==2, Prime[j], If[n==3 && k==2 || n==4 && 2<=k<=3, ((3-(-1)^n)/2)*Prime[j]^(n-1) -2^((3-(-1)^n)/2), If[k==1 || k==n, 2, T[n-1,k,p,q,j] + T[n-1,k-1,p,q,j] + (p*j+q)*Prime[j]*T[n-2,k-1,p,q,j] ]]];
Table[T[n,k,0,1,3], {n,12}, {k,n}]//Flatten (* modified by G. C. Greubel, Mar 04 2021 *)

@CachedFunction
def f(n,j): return ((3-(-1)^n)/2)*nth_prime(j)^(n-1) - 2^((3-(-1)^n)/2)
def T(n,k,p,q,j):
    if (n==2): return nth_prime(j)
    elif (n==3 and k==2 or n==4 and 2<=k<=3): return f(n,j)
    elif (k==1 or k==n): return 2
    else: return T(n-1,k,p,q,j) + T(n-1,k-1,p,q,j) + (p*j+q)*nth_prime(j)*T(n-2,k-1,p,q,j)
flatten([[T(n,k,0,1,3) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Mar 04 2021

T(n,k) = T(n-1, k) + T(n-1, k-1) + 5*T(n-2, k-1).
Recurrence row sums: s(n) = 2*s(n-1) + 5*s(n-2), n > 4, with s(1) = 2, s(2) = 10, s(3) = 50, s(4) = 250. - R. J. Mathar, Jan 22 2009
From G. C. Greubel, Mar 04 2021: (Start)
T(n,k,p,q,j) =  T(n-1,k,p,q,j) + T(n-1,k-1,p,q,j) + (p*j+q)*prime(j)*T(n-2,k-1,p,q,j) with T(2,k,p,q,j) = prime(j), T(3,2,p,q,j) = 2*prime(j)^2 -4, T(4,2,p,q, j) = T(4,3,p,q,j) = prime(j)^2 -2, T(n,1,p,q,j) = T(n,n,p,q,j) = 2 and (p, q, j) = (0,1,3).
Sum_{k=0..n} T(n,k,0,1,3) = 4*(-5)^n*[n<2] + 50*(i*sqrt(5))^(n-2)*(ChebyshevU(n-2, -i/sqrt(5)) - (3*i/sqrt(5))*ChebyshevU(n-3, -i/sqrt(5))) = 4*(-5)^n*[n<2] + 50*A123011(n-2). (End)

More terms from R. J. Mathar, Jan 22 2009

Edited by G. C. Greubel, Mar 04 2021

A164549 a(n) = 4a(n-1) + 2a(n-2) for n > 1; a(0) = 1, a(1) = 6.

Original entry on oeis.org

1, 6, 26, 116, 516, 2296, 10216, 45456, 202256, 899936, 4004256, 17816896, 79276096, 352738176, 1569504896, 6983495936, 31072993536, 138258966016, 615181851136, 2737245336576, 12179345048576, 54191870867456

Offset: 0

Klaus Brockhaus, Aug 15 2009

nonn

Binomial transform of A123011. Inverse binomial transform of A164550.
INVERT transform of the sequence (1, 5, 5*3, 5*3^2, 5*3^3, 5*3^4, ...); i.e., of (1, 5, 15, 45, 135, 405, ...). The sequence can also be obtained by extracting the upper left terms in matrix powers of [(1,5); (1,3)]. - Gary W. Adamson, Jul 31 2016
The sequence is A090017 (1, 4, 18, 80, 356, ...) convolved with (1, 2, 0, 0, 0, ...). Also, the upper left terms extracted from matrix powers of [(1,5); (1,3)]. - Gary W. Adamson, Aug 20 2016

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,2).

Cf. A123011, A164550.

Cf. A084057, A108306.

[ n le 2 select 5*n-4 else 4*Self(n-1)+2*Self(n-2): n in [1..22] ];

LinearRecurrence[{4,2},{1,6},30] (* Harvey P. Dale, Mar 16 2013 *)
CoefficientList[Series[(1 +2x)/(1 -4x -2x^2), {x, 0, 24}], x] (* Michael De Vlieger, Aug 02 2016 *)

Vec((1+2*x)/(1-4*x-2*x^2) + O(x^30)) \\ Michel Marcus, Feb 04 2016

[(i*sqrt(2))^n*(chebyshev_U(n, -i*sqrt(2)) - sqrt(2)*i*chebyshev_U(n-1, -i*sqrt(2))) for n in (0..30)] # G. C. Greubel, Jul 16 2021

a(n) = ((3+2*sqrt(6))*(2+sqrt(6))^n + (3-2*sqrt(6))*(2-sqrt(6))^n)/6.
G.f.: (1+2*x)/(1-4*x-2*x^2).
a(n) = (i*sqrt(2))^n*(ChebyshevU(n, -i*sqrt(2)) - sqrt(2)*i*ChebyshevU(n-1, -i*sqrt(2))). - G. C. Greubel, Jul 16 2021

A154235 a(n) = ( (4 + sqrt(6))^n - (4 - sqrt(6))^n )/(2*sqrt(6)).

Original entry on oeis.org

1, 8, 54, 352, 2276, 14688, 94744, 611072, 3941136, 25418368, 163935584, 1057300992, 6819052096, 43979406848, 283644733824, 1829363802112, 11798463078656, 76094066608128, 490767902078464, 3165202550546432

Offset: 1

Al Hakanson (hawkuu(AT)gmail.com), Jan 05 2009

Lim_{n -> infinity} a(n)/a(n-1) = 4 + sqrt(6) = 6.4494897427....
Binomial transform of A164550, second binomial transform of A164549, third binomial transform of A123011, fourth binomial transform of A164532.
Binomial transform is A164551, second binomial transform is A164552, third binomial transform is A164553.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (8,-10).

Cf. A010464 (decimal expansion of square root of 6), A123011, A164532, A164549, A164550, A164551, A164552, A164553.

a:=[1,8];; for n in [3..30] do a[n]:=8*a[n-1]-10*a[n-2]; od; a; # G. C. Greubel, May 21 2019

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-6); S:=[ ((4+r)^n-(4-r)^n)/(2*r): n in [1..20] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jan 07 2009

LinearRecurrence[{8, -10}, {1, 8}, 30] (* or *) Table[Simplify[((4 + Sqrt[6])^n -(4-Sqrt[6])^n)/(2*Sqrt[6])], {n, 30}] (* G. C. Greubel, Sep 06 2016 *)

a(n)=([0,1; -10,8]^(n-1)*[1;8])[1,1] \\ Charles R Greathouse IV, Sep 07 2016

my(x='x+O('x^30)); Vec(x/(1-8*x+10*x^2)) \\ G. C. Greubel, May 21 2019

[lucas_number1(n,8,10) for n in range(1, 21)] # Zerinvary Lajos, Apr 23 2009

From Philippe Deléham, Jan 06 2009: (Start)
a(n) = 8*a(n-1) - 10*a(n-2) for n > 1, where a(0)=0, a(1)=1.
G.f.: x/(1 - 8*x + 10*x^2). (End)

Extended beyond a(7) by Klaus Brockhaus, Jan 07 2009

Edited by Klaus Brockhaus, Oct 04 2009

A164532 a(n) = 6*a(n-2) for n > 2; a(1) = 1, a(2) = 4.

Original entry on oeis.org

1, 4, 6, 24, 36, 144, 216, 864, 1296, 5184, 7776, 31104, 46656, 186624, 279936, 1119744, 1679616, 6718464, 10077696, 40310784, 60466176, 241864704, 362797056, 1451188224, 2176782336, 8707129344, 13060694016, 52242776064, 78364164096

Offset: 1

Klaus Brockhaus, Aug 15 2009

nonn

Interleaving of A000400 and A067411 without initial term 1.

Binomial transform is apparently A123011. Fourth binomial transform is A154235.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,6).

Cf. A000400 (powers of 6), A067411, A123011, A154235.

[ n le 2 select 3*n-2 else 6*Self(n-2): n in [1..29] ];

LinearRecurrence[{0,6}, {1,4}, 40] (* G. C. Greubel, Jul 16 2021 *)

[((1 - (-1)^n)*sqrt(6)/2 + 2*(1 + (-1)^n))*6^(n/2 -1) for n in (1..40)] # G. C. Greubel, Jul 16 2021

a(n) = (5 - (-1)^n)*6^(1/4*(2*n - 5 + (-1)^n)).
G.f.: x*(1+4*x)/(1-6*x^2).
a(n+3) = a(n+2)*a(n+1)/a(n). - Reinhard Zumkeller, Mar 04 2011
a(n) = ((1-(-1)^n)*sqrt(6)/2 + 2*(1+(-1)^n))*6^(n/2 -1). - G. C. Greubel, Jul 16 2021

cp's OEIS Frontend

A153518 Triangular T(n,k) = T(n-1, k) + T(n-1, k-1) + 5*T(n-2, k-1), read by rows.

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A164549 a(n) = 4*a(n-1) + 2*a(n-2) for n > 1; a(0) = 1, a(1) = 6.

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A154235 a(n) = ( (4 + sqrt(6))^n - (4 - sqrt(6))^n )/(2*sqrt(6)).

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A164532 a(n) = 6*a(n-2) for n > 2; a(1) = 1, a(2) = 4.

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Crossrefs

Programs

Magma

Mathematica

Sage

Formula

A164549 a(n) = 4a(n-1) + 2a(n-2) for n > 1; a(0) = 1, a(1) = 6.