A134519 -id:A134519 - cp's OEIS frontend

A055802 a(n) = T(n,n-2), array T as in A055801.

1, 1, 1, 2, 3, 4, 6, 7, 10, 11, 15, 16, 21, 22, 28, 29, 36, 37, 45, 46, 55, 56, 66, 67, 78, 79, 91, 92, 105, 106, 120, 121, 136, 137, 153, 154, 171, 172, 190, 191, 210, 211, 231, 232, 253, 254, 276, 277, 300, 301, 325, 326, 351, 352, 378, 379, 406, 407, 435

Offset: 2

Clark Kimberling, May 28 2000

For n>2, a(n)+a(n+1) seems to be A002620(n+1)+1.

Colin Barker, Table of n, a(n) for n = 2..1000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A002620, A134519.

Cf. A055801, A055803, A055804, A055805, A055806.

Concatenation([1], List([3..65], n-> (2*n^2 -6*n +11 +(-1)^n*(2*n -11))/16 )); # G. C. Greubel, Jan 23 2020

[1] cat [(2*n^2 -6*n +11 +(-1)^n*(2*n -11))/16: n in [3..65]]; // G. C. Greubel, Jan 23 2020

seq( `if`(n==2, 1, (2*n^2 -6*n +11 +(-1)^n*(2*n -11))/16), n=2..65); # G. C. Greubel, Jan 23 2020

CoefficientList[Series[(1 -2*x^2 +x^3 +2*x^4 -x^5)/((1-x)^3*(1+x)^2), {x,0,65}], x] (* Wesley Ivan Hurt, Jan 20 2017 *)
Table[If[n==2,1, (2*n^2 -6*n +11 +(-1)^n*(2*n -11))/16], {n,2,65}] (* G. C. Greubel, Jan 23 2020 *)

Vec(x^2*(1-2*x^2+x^3+2*x^4-x^5)/((1-x)^3*(1+x)^2) + O(x^65)) \\ Charles R Greathouse IV, Feb 03 2013

vector(65, n, my(m=n+1); if(m==2, 1, (2*m^2 -6*m +11 +(-1)^m*(2*m -11))/16)) \\ G. C. Greubel, Jan 23 2020

[1]+[(2*n^2 -6*n +11 +(-1)^n*(2*n -11))/16 for n in (3..65)] # G. C. Greubel, Jan 23 2020

G.f.: x^2*(1 -2*x^2 +x^3 +2*x^4 -x^5)/((1-x)^3*(1+x)^2).
a(n) = A114220(n-1), n>=3. - R. J. Mathar, Feb 03 2013
From Colin Barker, Jan 27 2016: (Start)
a(n) = (2*n^2 +2*(-1)^n*n -6*n -11*(-1)^n +11)/16 for n>2.
a(n) = (n^2 - 2*n)/8 for n>2 and even.
a(n) = (n^2 - 4*n + 11)/8 for n odd. (End)
E.g.f.: (4*x*(x-2) + x*(x-3)*cosh(x) + (x^2 -x +11)*sinh(x))/8. - G. C. Greubel, Jan 23 2020

A165157 Zero followed by partial sums of A133622.

Original entry on oeis.org

0, 1, 3, 4, 7, 8, 12, 13, 18, 19, 25, 26, 33, 34, 42, 43, 52, 53, 63, 64, 75, 76, 88, 89, 102, 103, 117, 118, 133, 134, 150, 151, 168, 169, 187, 188, 207, 208, 228, 229, 250, 251, 273, 274, 297, 298, 322, 323, 348, 349, 375, 376, 403, 404, 432, 433, 462, 463, 493, 494, 525

Offset: 0

Jaroslav Krizek, Sep 05 2009

A133622 is a toothed sequence.

Interleaving of A055998 and A034856.

			From _Stefano Spezia_, Jul 10 2020: (Start)
Illustration of the initial terms for n > 0:
o    o      o      o         o        o
     o o    o o    o o       o o      o o
            o      o         o        o
                   o o o     o o o    o o o
                             o        o
                                      o o o o
(1)  (3)   (4)    (7)       (8)      (12)
(End)

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Equals -1+A101881.
a(n) = A117142(n+2)-2 = A055802(n+6)-3 = A114220(n+5)-3 = A134519(n+3)-3.
Cf. A133622, A055998, A034856.

a165157 n = a165157_list !! n
a165157_list = scanl (+) 0 a133622_list
-- Reinhard Zumkeller, Feb 20 2015

m:=60; T:=[ 1+(1+(-1)^n)*n/4: n in [1..m] ]; [0] cat [ n eq 1 select T[1] else Self(n-1)+T[n]: n in [1..m] ]; // Klaus Brockhaus, Sep 06 2009

[ n le 2 select n-1 else n le 4 select n else 2*Self(n-2)-Self(n-4)+1: n in [1..61] ]; // Klaus Brockhaus, Sep 06 2009

a(0) = 0, a(2*n) = a(2*n-1) + n + 1, a(2*n+1) = a(2*n) + 1.
a(n) = (n^2+10*n)/8 if n is even, a(n) = (n^2+8*n-1)/8 if n is odd.
a(2*k) = A055998(k) = k*(k+5)/2; a(2*k+1) = A034856(k+1) = k*(k+5)/2+1.
a(n) = 2*a(n-2)-a(n-4)+1 for n > 3; a(0)=0, a(1)=1, a(2)=3, a(3)=4. - Klaus Brockhaus, Sep 06 2009
a(n) = (2*n*(n+9)-1+(2*n+1)*(-1)^n)/16. - Klaus Brockhaus, Sep 06 2009
a(n) = n+binomial(1+floor(n/2),2). - Mircea Merca, Feb 18 2012
G.f.: x*(1+2*x-x^2-x^3)/((1-x)^3*(1+x)^2). - Klaus Brockhaus, Sep 06 2009
From Stefano Spezia, Jul 10 2020: (Start)
E.g.f.: (x*(9 + x)*cosh(x) + (-1 + 11*x + x^2)*sinh(x))/8.
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n > 4. (End)

Edited and extended by Klaus Brockhaus, Sep 06 2009

A082742 Indices of occurrences of 2 in A004738.

Original entry on oeis.org

2, 4, 6, 8, 12, 14, 20, 22, 30, 32, 42, 44, 56, 58, 72, 74, 90, 92, 110, 112, 132, 134, 156, 158, 182, 184, 210, 212, 240, 242, 272, 274, 306, 308, 342, 344, 380, 382, 420, 422, 462, 464, 506, 508, 552, 554, 600, 602, 650, 652, 702, 704, 756, 758, 812, 814, 870, 872, 930, 932, 992, 994, 1056, 1058, 1122, 1124, 1190, 1192, 1260, 1262, 1332, 1334, 1406, 1408, 1482, 1484, 1560, 1562

Offset: 1

Amarnath Murthy, Apr 15 2003

Indices of occurrences of 1 in A004738 are given by A002061, b(n) = n^2 - n + 1 (the central polygonal numbers). All entries are even.

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

Cf. A004738.

A004738 := proc(n)
    local f ;
    f := floor(sqrt(n)+1/2) ;
    f+1-abs(n-1-f^2) ;
end proc:
for n from 1 to 1600 do
    if A004738(n) = 2 then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Feb 03 2013

LinearRecurrence[{1,2,-2,-1,1},{2,4,6,8,12},80] (* Harvey P. Dale, Jun 16 2017 *)

a(n)=(n^2+2*n+8+if(n%2,2*n-5))/4 \\ Charles R Greathouse IV, Feb 03 2013

G.f.: 2*x*(1+x-x^2-x^3+x^4)/((1+x)^2*(1-x)^3). - Charles R Greathouse IV, Feb 03 2013

a(n) = 2*A134519(n). - R. J. Mathar, Feb 03 2013

More terms from R. J. Mathar, Feb 03 2013

cp's OEIS Frontend

A055802 a(n) = T(n,n-2), array T as in A055801.

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A165157 Zero followed by partial sums of A133622.

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A082742 Indices of occurrences of 2 in A004738.

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