A006646 -id:A006646 - cp's OEIS frontend

A006668 Exponential self-convolution of Pell numbers (divided by 2).

0, 0, 1, 6, 32, 160, 784, 3808, 18432, 89088, 430336, 2078208, 10035200, 48455680, 233967616, 1129701376, 5454692352, 26337607680, 127169265664, 614027624448, 2964787822592, 14315262312448, 69120201588736

Offset: 0

N. J. A. Sloane

Binomial transform of A084150. - Paul Barry, May 16 2003

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-4,-8).

Cf. A006646, A002203.

[Floor(((2+Sqrt(8))^n+(2-Sqrt(8))^n-2^(n+1))/16): n in [0..30] ]; // Vincenzo Librandi, Aug 20 2011

LinearRecurrence[{6,-4,-8},{0,0,1},30] (* Harvey P. Dale, Jul 15 2014 *)
Table[2^(n-4)*(LucasL[n, 2] - 2), {n, 0, 20}] (* Vladimir Reshetnikov, Oct 07 2016 *)

a(n) = ((2+sqrt(8))^n+(2-sqrt(8))^n-2^(n+1))/16; E.g.f. : exp(2x)(sinh(sqrt(2)x))^2/4=(exp(x)sinh(sqrt(2)x)/sqrt(2))^2/2. - Paul Barry, May 16 2003
G.f.: x^2/((1-2*x)*(1-4*x-4*x^2)). - Bruno Berselli, Aug 20 2011
a(n) = A006646(n)/2 = 2^(n-4)*(A002203(n) - 2). - Vladimir Reshetnikov, Oct 07 2016

A084150 A Pell related sequence.

Original entry on oeis.org

0, 0, 1, 3, 14, 50, 199, 749, 2892, 11028, 42301, 161799, 619706, 2372006, 9081955, 34767953, 133109592, 509594856, 1950956857, 7469077707, 28594853414, 109473250778, 419110475455, 1604533706357, 6142840740900, 23517417426300

Offset: 0

Paul Barry, May 16 2003

Binomial transform of expansion of (sinh(sqrt(2)x))^2/4 = (0, 0, 1, 0, 8, 0, 64, ...). Inverse binomial transform of A006668.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,5,-7).

Cf. A006646, A006668, A015519, A084058.

[n le 3 select Floor((n-1)/2) else 3*Self(n-1) +5*Self(n-2) -7*Self(n-3): n in [1..41]]; // G. C. Greubel, Oct 11 2022

LinearRecurrence[{3,5,-7}, {0,0,1}, 41] (* G. C. Greubel, Oct 11 2022 *)

A084058 = BinaryRecurrenceSequence(2,7,1,1)
def A084150(n): return (A084058(n) - 1)/8
[A084150(n) for n in range(41)] # G. C. Greubel, Oct 11 2022

a(n) = ( (1+sqrt(8))^n + (1-sqrt(8))^n - 2 )/16.
E.g.f.: (1/4)*exp(x)*( sinh(sqrt(2)*x) )^2.
G.f.: x^2 / ( (1-x)*(1-2*x-7*x^2) ). - R. J. Mathar, Feb 05 2011
a(n) = (A015519(n) - A015519(n-1) - 1)/8 = (A084058(n) - 1)/8. - G. C. Greubel, Oct 11 2022

cp's OEIS Frontend

A006668 Exponential self-convolution of Pell numbers (divided by 2).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Formula