A264960 -id:A264960 - cp's OEIS frontend

A013776 a(n) = 2^(4*n+1).

2, 32, 512, 8192, 131072, 2097152, 33554432, 536870912, 8589934592, 137438953472, 2199023255552, 35184372088832, 562949953421312, 9007199254740992, 144115188075855872, 2305843009213693952

Offset: 0

N. J. A. Sloane

a(n) ~ -Pi*E(2*n)/B(2*n), E(n) Euler number, B(n) Bernoulli number. - Peter Luschny, Oct 28 2012
Equivalently, powers of 2 with final digit 2. - Muniru A Asiru, Mar 15 2019
As phi(a(n)) = (2^n)^4 is a perfect biquadrate (where phi is the Euler totient A000010), this is a subsequence of A078164 and A307690. - Bernard Schott, Mar 28 2022

			G.f. = 2 + 32*x + 512*x^2 + 8192*x^3 + 131072*x^4 + 2097152*x^5 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Tanya Khovanova, Recursive Sequences
Index to divisibility sequences
Index entries for linear recurrences with constant coefficients, signature (16).

Subsequence of A307690.
Cf. A000010, A004171, A016813, A264960, A307690.
Intersection of A000079 and A078164.

List([0..20],n->2^(4*n+1)); # Muniru A Asiru, Mar 15 2019

[2^(4*n+1): n in [0..20]]; // Vincenzo Librandi, Jun 27 2011

[2^(4*n+1)$n=0..20]; # Muniru A Asiru, Apr 10 2019

2^(4*Range[0,20]+1) (* G. C. Greubel, Mar 15 2019 *)
NestList[16#&,2,20] (* Harvey P. Dale, Jul 28 2019 *)

a(n)=2<<(4*n) \\ Charles R Greathouse IV, Apr 07 2012

[2^(4*n+1) for n in (0..20)] # G. C. Greubel, Mar 15 2019

From Philippe Deléham, Nov 23 2008: (Start)
a(n) = 16*a(n-1), n > 0, a(0) = 2.
G.f.: 2/(1 - 16*x). (End)
From Peter Bala, Nov 29 2015: (Start)
a(n) = Sum_{k = 0..n} binomial(2*k,k)*binomial(4*n + 2 - 2*k, 2*n + 1 - k).
Bisection of A264960. (End)
a(n) = A000079(A016813(n)). - Michel Marcus, Nov 30 2015
a(n) = Sum_{k = 0..2*n} binomial(4*n + 2, 2*k + 1) = A004171(2*n). - Peter Bala, Nov 25 2016
E.g.f.: 2*exp(16*x). - G. C. Greubel, Jun 30 2019
From Bernard Schott, Apr 15 2022: (Start)
Sum_{n>=0} 1/a(n) = 8/15.
Sum_{n>=0} (-1)^n/a(n) = 8/17. (End)

Wrong comment deleted by Kevin Ryde, Apr 16 2022

A112830 Table of number of domino tilings of generalized Aztec pillows of type (1, ..., 1, 3, 1, ..., 1)_n.

Original entry on oeis.org

1, 1, 5, 1, 10, 25, 1, 17, 65, 113, 1, 26, 146, 346, 481, 1, 37, 292, 932, 1637, 1985, 1, 50, 533, 2248, 5013, 7218, 8065, 1, 65, 905, 4937, 13897, 24201, 30529, 32513, 1, 82, 1450, 10018, 35218, 74530, 108970, 126034, 130561, 1, 101, 2216, 19016, 82436

Offset: 0

Christopher Hanusa (chanusa(AT)math.binghamton.edu), Sep 21 2005

The number of tilings of a generalized Aztec pillow of type (k 1's followed by a 3 followed by n-k-1 1's) is entry (n,k+1).

			The number of tilings of a generalized Aztec pillow of type (1,1,3,1)_n is entry (4,3) = 346.

C. Hanusa, A Gessel-Viennot-Type Method for Cycle Systems with Applications to Aztec Pillows, PhD Thesis, 2005, University of Washington, Seattle, USA.

A092440 (main diagonal), A092441 (first subdiagonal), A002522 (column k = 1), A066455 (column k = 2). Cf. A264960.

matrix(11,11,[seq([seq(((2^n-sum(binomial(n,j),j=0..k))^2+(binomial(n-1,k))^2)/2,n=k+1..k+11)],k=0..10)]);

T(2*n,n) = A264960(n). - Peter Bala, Nov 29 2015

cp's OEIS Frontend

A013776 a(n) = 2^(4*n+1).

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