A160156 -id:A160156 - cp's OEIS frontend

A034299 Alternating sum transform (PSumSIGN) of A000975.

1, 1, 4, 6, 15, 27, 58, 112, 229, 453, 912, 1818, 3643, 7279, 14566, 29124, 58257, 116505, 233020, 466030, 932071, 1864131, 3728274, 7456536, 14913085, 29826157, 59652328, 119304642, 238609299

Offset: 0

N. J. A. Sloane

			G.f. = 1 + x + 4*x^2 + 6*x^3 + 15*x^4 + 27*x^5 + 58*x^6 + 112*x^7 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
N. J. A. Sloane, Transforms
Index entries for linear recurrences with constant coefficients, signature (1,3,-1,-2).

Cf. A160156.

[(2^(n+5)+(6*n+13)*(-1)^n-9)/36: n in [0..50]]; // G. C. Greubel, Oct 12 2017

CoefficientList[Series[(1/(1-x^2))/(1-x-2x^2),{x,0,40}],x] (* Vincenzo Librandi, Apr 04 2012 *)
Table[(2^(n + 5) + (6 n + 13) (-1)^n - 9)/36, {n, 0, 28}] (* Bruno Berselli, Apr 04 2012 *)
LinearRecurrence[{1,3,-1,-2},{1,1,4,6},30] (* Harvey P. Dale, Jun 11 2019 *)

{a(n) = (32 * 2^n - 9 + (6*n + 13) * (-1)^n) / 36}; /* Michael Somos, Jan 23 2014 */

a(n) = sum{k=0..floor(n/2), A001045(n-2k+1)}. - Paul Barry, Nov 24 2003
G.f.: (1/(1-x^2))/(1-x-2x^2); a(n) = sum{k=0..n+1, A001045(k)*(1-(-1)^floor((n+k)/2))}; - Paul Barry, Apr 16 2005
a(n) = sum_{k, 0<=k<=n} A126258(n,k). - Philippe Deléham, Mar 13 2007
a(n) = 2*a(n-1)+A001057(n+1), with a(0)=1. - Bruno Berselli, Nov 09 2010
a(n) = (2^(n+5)+(6n+13)(-1)^n-9)/36. - Bruno Berselli, Apr 04 2012
a(n) = a(n-1) + 2*a(n-2) + (1 + (-1)^n) / 2. - Michael Somos, Jan 23 2014
A160156(n) = a(2*n). - Michael Somos, Oct 16 2020

A364214 Numbers whose canonical representation as a sum of distinct Jacobsthal numbers (A280049) is palindromic.

Original entry on oeis.org

1, 2, 4, 5, 6, 10, 12, 15, 18, 21, 22, 30, 34, 42, 44, 49, 58, 63, 66, 71, 80, 85, 86, 102, 110, 126, 130, 146, 154, 170, 172, 183, 198, 209, 218, 229, 244, 255, 258, 269, 284, 295, 304, 315, 330, 341, 342, 374, 390, 422, 430, 462, 478, 510, 514, 546, 562, 594

Offset: 1

Amiram Eldar, Jul 14 2023

The even-indexed Jacobsthal numbers A001045(2*n) = A002450(n) = (4^n-1)/3, for n >= 1, are terms since their representation is 2*n-1 1's.
A001045(2*n+1) - 1 = A020988(n) = (2/3)*(4^n-1) is a term for n >= 1, since its representation is 2*n 1's.
A001045(n) + 1 = A128209(n) is a term for n >= 0, since its representation for n = 0 is 1 and its representation for n >= 1 is n-1 0's between 2 1's.
A160156(n) is a term for n >= 0 since its representation is n 0's interleaved with n+1 1's.

			The first 10 terms are:
   n  a(n)  A280049(a(n))
  --  ----  -------------
   1     1              1
   2     2             11
   3     4            101
   4     5            111
   5     6           1001
   6    10           1111
   7    12          10001
   8    15          10101
   9    18          11011
  10    21          11111

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001045, A002450, A003159, A020988, A280049.

Similar sequences: A002113, A006995, A014190, A094202, A331191, A351712, A351717, A352087, A352105, A352319, A352341.

Position[Select[Range[1000], EvenQ[IntegerExponent[#, 2]] &], _?(PalindromeQ[IntegerDigits[#, 2]] &)] // Flatten

s(n) = if(n < 2, n > 0, n = s(n-1); until(valuation(n, 2)%2 == 0, n++); n); \\ A003159
is(n) = {my(d = binary(s(n))); d == Vecrev(d);}

A350717 a(n) = 4*a(n-1) - n - 1, for n > 0, a(0) = 1.

Original entry on oeis.org

1, 2, 5, 16, 59, 230, 913, 3644, 14567, 58258, 233021, 932072, 3728275, 14913086, 59652329, 238609300, 954437183, 3817748714, 15270994837, 61083979328, 244335917291, 977343669142, 3909374676545, 15637498706156, 62549994824599, 250199979298370, 1000799917193453, 4003199668773784

Offset: 0

Paul Curtz, Feb 03 2022

Last digit (using 0 to 9) is of period 10: repeat [1, 2, 5, 6, 9, 0, 3, 4, 7, 8].

Index entries for linear recurrences with constant coefficients, signature (6,-9,4).

Cf. A007583 (first differences), A014825, A160156.

LinearRecurrence[{6, -9, 4}, {1, 2, 5}, 28] (* Amiram Eldar, Feb 03 2022 *)

a(n) = if (n, 4*a(n-1) - n - 1, 1); \\ Michel Marcus, Feb 03 2022

print([(2**(2*n+1) + 3*n + 7)//9 for n in range(30)])
# Gennady Eremin, Feb 05 2022

a(n) = (2^(2*n+1) + 3*n + 7)/9.
a(n) = 6*a(n-1) - 9*a(n-2) + 4*a(n-3), n >= 3.
a(n) = a(n-1) + A007583(n-1).
a(n) = 2*a(n-1) + A014825(n-1).
G.f.: (-2*x^2 + 4*x - 1)/((x - 1)^2*(4*x - 1)). - Thomas Scheuerle, Feb 03 2022
a(n) = -1 + 5*a(n-1) - 4*a(n-2), n >= 2.
a(n) = 1 + A160156(n-1), n >= 1.

More terms from Michel Marcus, Feb 03 2022

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A034299 Alternating sum transform (PSumSIGN) of A000975.

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A364214 Numbers whose canonical representation as a sum of distinct Jacobsthal numbers (A280049) is palindromic.

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A350717 a(n) = 4*a(n-1) - n - 1, for n > 0, a(0) = 1.

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