A383720 -id:A383720 - cp's OEIS frontend

A084158 a(n) = A000129(n) * A000129(n+1)/2.

0, 1, 5, 30, 174, 1015, 5915, 34476, 200940, 1171165, 6826049, 39785130, 231884730, 1351523251, 7877254775, 45912005400, 267594777624, 1559656660345, 9090345184445, 52982414446326, 308804141493510, 1799842434514735, 10490250465594899, 61141660359054660, 356359711688733060

Offset: 0

Paul Barry, May 18 2003

May be called Pell triangles.

Vincenzo Librandi, Table of n, a(n) for n = 0..500
Paul Barry, Symmetric Third-Order Recurring Sequences, Chebyshev Polynomials, and Riordan Arrays, JIS 12 (2009) 09.8.6.
Sergio Falcon, On the Sequences of Products of Two k-Fibonacci Numbers, American Review of Mathematics and Statistics, March 2014, Vol. 2, No. 1, pp. 111-120.
Ronald Orozco López, Generating Functions of Generalized Simplicial Polytopic Numbers and (s,t)-Derivatives of Partial Theta Function, arXiv:2408.08943 [math.CO], 2024. See p. 11.
Ronald Orozco López, Simplicial d-Polytopic Numbers Defined on Generalized Fibonacci Polynomials, arXiv:2501.11490 [math.CO], 2025. See p. 6.
Index entries for linear recurrences with constant coefficients, signature (5,5,-1).

Cf. A000129, A001652, A001654, A002203, A002315, A003499.

Cf. A041011, A046729, A079291, A084159, A084175, A163960, A383720.

[Floor(((Sqrt(2)+1)^(2*n+1)-(Sqrt(2)-1)^(2*n+1)-2*(-1)^n)/16): n in [0..35]]; // Vincenzo Librandi, Jul 05 2011

with(combinat): a:=n->fibonacci(n,2)*fibonacci(n-1,2)/2: seq(a(n), n=1..22); # Zerinvary Lajos, Apr 04 2008

LinearRecurrence[{5,5,-1},{0,1,5},30] (* Harvey P. Dale, Sep 07 2011 *)

Pell(n)=([2, 1; 1, 0]^n)[2, 1];
a(n)=Pell(n)*Pell(n+1)/2 \\ Charles R Greathouse IV, Mar 21 2016

a(n)=([0,1,0; 0,0,1; -1,5,5]^n*[0;1;5])[1,1] \\ Charles R Greathouse IV, Mar 21 2016

[(lucas_number2(2*n+1,2,-1) -2*(-1)^n)/16 for n in (0..30)] # G. C. Greubel, Aug 18 2022

a(n) = ((sqrt(2)+1)^(2*n+1) - (sqrt(2)-1)^(2*n+1) - 2*(-1)^n)/16.
a(n) = 5*a(n-1) + 5*a(n-2) - a(n-3). - Mohamed Bouhamida, Sep 02 2006; corrected by Antonio Alberto Olivares, Mar 29 2008
a(n) = (-1/8)*(-1)^n + (( sqrt(2)+1)/16)*(3+2*sqrt(2))^n + ((-sqrt(2)+1)/16)*(3-2*sqrt(2))^n. - Antonio Alberto Olivares, Mar 30 2008
sqrt(a(n) - a(n-1)) = A000129(n). - Antonio Alberto Olivares, Mar 30 2008
O.g.f.: x/((1+x)(1-6*x+x^2)). - R. J. Mathar, May 18 2008
a(n) = A041011(n)*A041011(n+1). - R. K. Guy, May 18 2008
From Mohamed Bouhamida, Aug 30 2008: (Start)
a(n) = 6*a(n-1) - a(n-2) - (-1)^n.
a(n) = 7*(a(n-1) - a(n-2)) + a(n-3) - 2*(-1)^n. (End)
In general, for n>k+1, a(n+k) = A003499(k+1)*a(n-1) - a(n-k-2) - (-1)^n A000129(k+1)^2. - Charlie Marion, Jan 04 2012
For n>0, a(2n-1)*a(2n+1) = oblong(a(2n)); a(2n)*a(2n+2) = oblong(a(2n+1)-1). - Charlie Marion, Jan 09 2012
a(n) = A046729(n)/4. - Wolfdieter Lang, Mar 07 2012
a(n) = sum of squares of first n Pell numbers A000129 (A079291). - N. J. A. Sloane, Jun 18 2012
a(n) = (A002315(n) - (-1)^n)/8. - Adam Mohamed, Sep 05 2024
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*(sqrt(2)-1) (A163960). - Amiram Eldar, Dec 02 2024
G.f.: x * exp( Sum_{k>=1} Pell(3*k)/Pell(k) * x^k/k ). - Seiichi Manyama, May 07 2025

A383742 Square array A(n,k), n>=0, k>=0, read by antidiagonals downwards, where column k is the expansion of g.f. x/(1 - A002203(k)x + (-1)^kx^2).

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 6, 5, 4, 0, 1, 14, 35, 12, 5, 0, 1, 34, 197, 204, 29, 6, 0, 1, 82, 1155, 2772, 1189, 70, 7, 0, 1, 198, 6725, 39236, 39005, 6930, 169, 8, 0, 1, 478, 39203, 551532, 1332869, 548842, 40391, 408, 9, 0, 1, 1154, 228485, 7761996, 45232349, 45278310, 7722793, 235416, 985, 10

Offset: 0

Seiichi Manyama, May 07 2025

			Square array begins:
  0,  0,    0,     0,       0,        0, ...
  1,  1,    1,     1,       1,        1, ...
  2,  2,    6,    14,      34,       82, ...
  3,  5,   35,   197,    1155,     6725, ...
  4, 12,  204,  2772,   39236,   551532, ...
  5, 29, 1189, 39005, 1332869, 45232349, ...

Columns k=0..6 give A001477, A000129, A001109, A041085(n-1), A091761, A292423, A097731(n-1).
Rows n=0..5 give A000004, A000012, A002203, A383720, A383740, A383741.
Main diagonal gives A380083.
Cf. A028412.

A[n_, k_] := Fibonacci[k*n, 2]/Fibonacci[k, 2]; A[n_, 0] := n; Table[A[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, May 08 2025 *)

pell(n) = ([2, 1; 1, 0]^n)[2, 1];
a(n, k) = if(k==0, n, pell(k*n)/pell(k));

A(0,k) = 0, A(1,k) = 1; A(n,k) = A002203(k) * A(n-1,k) - (-1)^k * A(n-2,k) for n > 1.

A(n,k) = Pell(k*n)/Pell(k) for k > 0.

cp's OEIS Frontend

A084158 a(n) = A000129(n) * A000129(n+1)/2.

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A383742 Square array A(n,k), n>=0, k>=0, read by antidiagonals downwards, where column k is the expansion of g.f. x/(1 - A002203(k)x + (-1)^kx^2).

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cp's OEIS Frontend

A084158 a(n) = A000129(n) * A000129(n+1)/2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

PARI

SageMath

Formula

A383742 Square array A(n,k), n>=0, k>=0, read by antidiagonals downwards, where column k is the expansion of g.f. x/(1 - A002203(k)*x + (-1)^k*x^2).

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Author

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Examples

Crossrefs

Programs

Mathematica

PARI

Formula

A383742 Square array A(n,k), n>=0, k>=0, read by antidiagonals downwards, where column k is the expansion of g.f. x/(1 - A002203(k)x + (-1)^kx^2).