A168277 -id:A168277 - cp's OEIS frontend

A168276 a(n) = 2*n - (-1)^n - 1.

2, 2, 6, 6, 10, 10, 14, 14, 18, 18, 22, 22, 26, 26, 30, 30, 34, 34, 38, 38, 42, 42, 46, 46, 50, 50, 54, 54, 58, 58, 62, 62, 66, 66, 70, 70, 74, 74, 78, 78, 82, 82, 86, 86, 90, 90, 94, 94, 98, 98, 102, 102, 106, 106, 110, 110, 114, 114, 118, 118, 122, 122, 126, 126, 130, 130

Offset: 1

Vincenzo Librandi, Nov 22 2009

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A039722, A168277.

Cf. A063210. - R. J. Mathar, Nov 25 2009

[2*n-1-(-1)^n: n in [1..70]]; // Vincenzo Librandi, Sep 16 2013

CoefficientList[Series[2 (1 + x^2) / ((1 + x) (1 - x)^2), {x, 0, 80}], x] (* Vincenzo Librandi, Sep 16 2013 *)
Table[2 n - 1 - (-1)^n, {n, 70}] (* Bruno Berselli, Sep 17 2013 *)
LinearRecurrence[{1,1,-1},{2,2,6},70] (* Harvey P. Dale, Oct 22 2014 *)

a(n) = 4*n - a(n-1) - 4, with n>1, a(1)=2.
from R. J. Mathar, Nov 25 2009: (Start)
a(n) = 2*n - (-1)^n - 1.
a(n) = 2*A109613(n-1).
G.f.: 2*x*(1 + x^2)/((1+x)*(1-x)^2). (End)
a(n) = a(n-1) + a(n-2) - a(n-3). - Vincenzo Librandi, Sep 16 2013
a(n) = A168277(n) + 1. - Vincenzo Librandi, Sep 17 2013
E.g.f.: (-1 + 2*exp(x) + (2*x -1)*exp(2*x))*exp(-x). - G. C. Greubel, Jul 16 2016
Sum_{n>=1} 1/a(n)^2 = Pi^2/16. - Amiram Eldar, Aug 21 2022

Previous definition replaced with closed-form expression by Bruno Berselli, Sep 17 2013

A186949 a(n) = 2^n - 2*(binomial(1,n) - binomial(0,n)).

Original entry on oeis.org

1, 0, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33554432, 67108864, 134217728, 268435456, 536870912, 1073741824

Offset: 0

Paul Barry, Mar 01 2011

Binomial transform is A186948.
Second binomial transform is A186947.
Inverse binomial transform is (-1)^n * A168277(n).
Essentially the same as A000079, A151821, A155559, A171449, and A171559.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2).

Concatenation([1,0], List([2..30], n-> 2^n )); # G. C. Greubel, Dec 01 2019

[n lt 2 select 1-n else 2^n: n in [0..30]]; // G. C. Greubel, Dec 01 2019

seq( `if`(n<2, 1-n, 2^n), n=0..30); # G. C. Greubel, Dec 01 2019

Table[If[n<2, 1-n, 2^n], {n, 0, 30}] (* G. C. Greubel, Dec 01 2019 *)

vector(31, n, if(n<3, 2-n, 2^(n-1))) \\ G. C. Greubel, Dec 01 2019

[1,0]+[2^n for n in (2..30)] # G. C. Greubel, Dec 01 2019

G.f.: (1 - 2*x + 4*x^2)/(1-2*x).
a(n) = Sum_{k=0..n} binomial(n,k)*(3^k - 2*k).
E.g.f.: exp(2*x) - 2*x. - G. C. Greubel, Dec 01 2019

A276914 Subsequence of triangular numbers obtained by adding a square and two smaller triangles, a(n) = n^2 + 2*A000217(A052928(n)).

Original entry on oeis.org

0, 1, 10, 15, 36, 45, 78, 91, 136, 153, 210, 231, 300, 325, 406, 435, 528, 561, 666, 703, 820, 861, 990, 1035, 1176, 1225, 1378, 1431, 1596, 1653, 1830, 1891, 2080, 2145, 2346, 2415, 2628, 2701, 2926, 3003, 3240, 3321, 3570, 3655, 3916, 4005, 4278, 4371, 4656

Offset: 0

Daniel Poveda Parrilla, Sep 22 2016

All terms of this sequence are triangular numbers. Graphically, for each term of the sequence, one corner of the square will be part of the corresponding triangle's hypotenuse if the term is an odd number. Otherwise, it will not be part of it.
a(A276915(n)) is a triangular pentagonal number.
a(A079291(n)) is a triangular square number, as A275496 is a subsequence of this.

Daniel Poveda Parrilla, Table of n, a(n) for n = 0..10000
Daniel Poveda Parrilla, Illustration of initial terms.
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A000217, A004526, A016813, A042948, A052928, A079291, A168277, A275496, A276915.

[n*(2*n+(-1)^n): n in [0..40]]; // G. C. Greubel, Aug 19 2022

Table[n (2 n + (-1)^n), {n, 0, 48}] (* Michael De Vlieger, Sep 23 2016 *)

concat(0, Vec(x*(1+9*x+3*x^2+3*x^3)/((1-x)^3*(1+x)^2) + O(x^50))) \\ Colin Barker, Sep 23 2016

[n*(2*n+(-1)^n) for n in (0..40)] # G. C. Greubel, Aug 19 2022

a(n) = n^2 + 2*A000217(A052928(n)).
a(n) = A000217(A042948(n)).
a(n) = n*(2*n + (-1)^n).
a(n) = n*A168277(n + 1).
a(n) = n*A016813(A004526(n)).
From Colin Barker, Sep 23 2016: (Start)
G.f.: x*(1 + 9*x + 3*x^2 + 3*x^3) / ((1 - x)^3*(1 + x)^2).
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n>4.
a(n) = n*(2*n+1) for n even.
a(n) = n*(2*n-1) for n odd. (End)
E.g.f.: x*( 2*(1+x)*exp(x) - exp(-x) ). - G. C. Greubel, Aug 19 2022
Sum_{n>=1} 1/a(n) = 2 - log(2). - Amiram Eldar, Aug 21 2022

A292576 Permutation of the natural numbers partitioned into quadruples [4k-1, 4k-3, 4k-2, 4k], k > 0.

Original entry on oeis.org

3, 1, 2, 4, 7, 5, 6, 8, 11, 9, 10, 12, 15, 13, 14, 16, 19, 17, 18, 20, 23, 21, 22, 24, 27, 25, 26, 28, 31, 29, 30, 32, 35, 33, 34, 36, 39, 37, 38, 40, 43, 41, 42, 44, 47, 45, 46, 48, 51, 49, 50, 52, 55, 53, 54, 56, 59, 57, 58, 60, 63, 61, 62

Offset: 1

Guenther Schrack, Sep 19 2017

nonn

Partition the natural number sequence into quadruples starting with (1,2,3,4); swap the second and third elements, then swap the first and the second element; repeat for all quadruples.

Guenther Schrack, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).
Index entries for sequences that are permutations of the natural numbers

Inverse: A056699(n+1) - 1 for n > 0.
Sequence of fixed points: A008586(n) for n > 0.
Subsequences:
   elements with odd index: A042964(A103889(n)) for n > 0.
   elements with even index: A042948(n) for n > 0.
   odd elements: A166519(n) for n>0.
   indices of odd elements: A042963(n) for n > 0.
   even elements: A005843(n) for n>0.
   indices of even elements: A014601(n) for n > 0.
Sum of pairs of elements:
   a(n+2) + a(n) = A163980(n+1) = A168277(n+2) for n > 0.
Difference between pairs of elements:
   a(n+2) - a(n) = (-1)^A011765(n+3)*A091084(n+1) for n > 0.
Compound relations:
   a(n) = A284307(n+1) - 1 for n > 0.
   a(n+2) - 2*a(n+1) + a(n) = (-1)^A011765(n)*A132400(n+1) for n > 0.
Compositions:
   a(n) = A116966(A080412(n)) for n > 0.
   a(A284307(n)) = A256008(n) for n > 0.
   a(A042963(n)) = A166519(n-1) for n > 0.
   A256008(a(n)) = A056699(n) for n > 0.

a = [3 1 2 4]; % Generate b-file
max = 10000;
for n := 5:max
   a(n) = a(n-4) + 4;
end;

for(n=1, 10000, print1(n + ((-1)^(n*(n-1)/2)*(2 - (-1)^n) - (-1)^n)/2, ", "))

a(1)=3, a(2)=1, a(3)=2, a(4)=4, a(n) = a(n-4) + 4 for n > 4.
O.g.f.: (2*x^3 + x^2 - 2*x + 3)/(x^5 - x^4 - x + 1).
a(n) = n + ((-1)^(n*(n-1)/2)*(2-(-1)^n) - (-1)^n)/2.
a(n) = n + (cos(n*Pi/2) - cos(n*Pi) + 3*sin(n*Pi/2))/2.
a(n) = n + n mod 2 + (ceiling(n/2)) mod 2 - 2*(floor(n/2) mod 2).
Linear recurrence: a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
First Differences, periodic: (-2, 1, 2, 3), repeat; also (-1)^A130569(n)*A068073(n+2) for n > 0.

A213041 Number of triples (w,x,y) with all terms in {0..n} and 2*|w-x| = max(w,x,y) - min(w,x,y).

Original entry on oeis.org

1, 2, 7, 12, 21, 30, 43, 56, 73, 90, 111, 132, 157, 182, 211, 240, 273, 306, 343, 380, 421, 462, 507, 552, 601, 650, 703, 756, 813, 870, 931, 992, 1057, 1122, 1191, 1260, 1333, 1406, 1483, 1560, 1641, 1722, 1807, 1892, 1981, 2070, 2163, 2256

Offset: 0

Clark Kimberling, Jun 10 2012

See A212959 for a guide to related sequences.

For n > 3, a(n-2) is the number of distinct values of the magic constant in a perimeter-magic (n-1)-gon of order n (see A342819). - Stefano Spezia, Mar 23 2021

Terrel Trotter, Perimeter-Magic Polygons, Journal of Recreational Mathematics Vol. 7, No. 1, 1974, pp. 14-20 (see equations 10-13).
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A002620, A004526, A058331, A212959, A168277 (first differences), A342819.

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[Max[w, x, y] - Min[w, x, y] == 2 Abs[w - x],
s = s + 1],
{w, 0, n}, {x, 0, n}, {y, 0, n}]; s)]];
m = Map[t[#] &, Range[0, 45]]   (* A213041 *)

Vec((1+3*x^2)/((1-x)^3*(1+x)) + O(x^99)) \\ Altug Alkan, May 06 2016

a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4) for n > 3.
G.f.: (1 + 3*x^2)/((1 - x)^3 * (1 + x)).
a(n) = (n+1)^2 - 2*A004526(n-1) - 2. - Wesley Ivan Hurt, Jul 15 2013
a(n) = A002620(n+2) + 3*A002620(n). - R. J. Mathar, Jul 15 2013
a(n)+a(n+1) = A058331(n+1). - R. J. Mathar, Jul 15 2013
a(n) = n*(n+1) + (1+(-1)^n)/2. - Wesley Ivan Hurt, May 06 2016
E.g.f.: x*(x + 2)*exp(x) + cosh(x). - Ilya Gutkovskiy, May 06 2016
a(n) = A000384(n+1) - A137932(n+2). - Federico Provvedi, Aug 17 2023

A248825 a(n) = n^2 + 1 - (-1)^n.

Original entry on oeis.org

0, 3, 4, 11, 16, 27, 36, 51, 64, 83, 100, 123, 144, 171, 196, 227, 256, 291, 324, 363, 400, 443, 484, 531, 576, 627, 676, 731, 784, 843, 900, 963, 1024, 1091, 1156, 1227, 1296, 1371, 1444, 1523, 1600, 1683, 1764, 1851, 1936, 2027, 2116

Offset: 0

Paul Curtz, Oct 15 2014

Also, A016742 and A164897 interleaved.
See the spiral in Example field of A054552: after 0, the sequence is given by the terms of the semidiagonals 4, 16, 36, 64, 100, ... and 3, 11, 27, 51, 83, ... sorted into ascending order.
Primes of the sequence are in A056899.

Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A000290, A008586, A008602, A010673, A016742, A042963, A056899, A164897, A166519, A168277, A248800.

[n^2+1-(-1)^n: n in [0..60]]; // Vincenzo Librandi, Oct 16 2014

Table[n^2 + 1 - (-1)^n, {n, 0, 60}] (* Vincenzo Librandi, Oct 16 2014 *)
LinearRecurrence[{2,0,-2,1},{0,3,4,11},60] (* Harvey P. Dale, Jun 30 2019 *)

vector(100,n,(n-1)^2+1+(-1)^n) \\ Derek Orr, Oct 15 2014

[n^2+1-(-1)^n for n in (0..60)] # Bruno Berselli, Oct 16 2014

a(n) = a(-n) = 2*a(n-1) - 2*(n-3) + a(n-4).
a(n) = n^2 + A010673(n) = (n+1)^2 - A168277(n+1).
a(n+1) = A248800(n) + A042963(n+1) = a(n) + A166519(n).
a(n+2) = a(n) + 4*n.
a(n+5) = a(n-5) + A008602(n).
G.f.: x*(3 - 2*x + 3*x^2)/((1 + x)*(1 - x)^3). - Bruno Berselli, Oct 15 2014
Sum_{n>=1} 1/a(n) = Pi^2/24 + tanh(Pi/sqrt(2))*Pi/(4*sqrt(2)). - Amiram Eldar, Aug 21 2022

Edited by Bruno Berselli, Oct 16 2014

cp's OEIS Frontend

A168276 a(n) = 2*n - (-1)^n - 1.

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A186949 a(n) = 2^n - 2*(binomial(1,n) - binomial(0,n)).

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A276914 Subsequence of triangular numbers obtained by adding a square and two smaller triangles, a(n) = n^2 + 2*A000217(A052928(n)).

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A292576 Permutation of the natural numbers partitioned into quadruples [4k-1, 4k-3, 4k-2, 4k], k > 0.

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A213041 Number of triples (w,x,y) with all terms in {0..n} and 2*|w-x| = max(w,x,y) - min(w,x,y).

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A248825 a(n) = n^2 + 1 - (-1)^n.

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