A225144 -id:A225144 - cp's OEIS frontend

A064455 a(2n) = 3n, a(2n-1) = n.

1, 3, 2, 6, 3, 9, 4, 12, 5, 15, 6, 18, 7, 21, 8, 24, 9, 27, 10, 30, 11, 33, 12, 36, 13, 39, 14, 42, 15, 45, 16, 48, 17, 51, 18, 54, 19, 57, 20, 60, 21, 63, 22, 66, 23, 69, 24, 72, 25, 75, 26, 78, 27, 81, 28, 84, 29, 87, 30, 90, 31, 93, 32, 96, 33, 99, 34, 102, 35, 105, 36, 108

Offset: 1

N. J. A. Sloane, Oct 02 2001

Also number of 1's in n-th row of triangle in A071030. - Hans Havermann, May 26 2002
Number of ON cells at generation n of 1-D CA defined by Rule 54. - N. J. A. Sloane, Aug 09 2014
a(n)*A098557(n) equals the second right hand column of A167556. - Johannes W. Meijer, Nov 12 2009
Given a(1) = 1, for all n > 1, a(n) is the least positive integer not equal to a(n-1) such that the arithmetic mean of the first n terms is an integer. The sequence of arithmetic means of the first 1, 2, 3, ..., terms is 1, 2, 2, 3, 3, 4, 4, ... (A004526 disregarding its first three terms). - Rick L. Shepherd, Aug 20 2013

			a(13) = a(2*7 - 1) = 7, a(14) = a(2*7) = 21.
a(8) = 8-9+10-11+12-13+14-15+16 = 12. - _Bruno Berselli_, Jun 05 2013

Harry J. Smith, Table of n, a(n) for n = 1..1000
A. J. Macfarlane, Generating functions for integer sequences defined by the evolution of cellular automata with even rule numbers, 2016.
S. Wolfram, Statistical mechanics of cellular automata, Rev. Mod. Phys., 55 (1983), 601--644.
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1)

Interleaving of A000027 and A008585 (without first term).

Cf. A004526, A052928, A064433, A071030, A080512, A098557, A123684, A137501, A167556, A225144, A265888.

maxarg := 75; for n := 1 to maxarg do if n mod 2 = 1 then write((n+1) div 2, " ") else write((n div 2)*3," "); end; end;

a:=[];;  for n in [1..75] do if n mod 2 = 0 then Add(a,3*n/2); else Add(a,(n+1)/2); fi; od; a; # Muniru A Asiru, Oct 28 2018

import Data.List (transpose)
a064455 n = n + if m == 0 then n' else - n'  where (n',m) = divMod n 2
a064455_list = concat $ transpose [[1 ..], [3, 6 ..]]
-- Reinhard Zumkeller, Oct 12 2013

[(1/2)*n*(-1)^n+n+(1/4)*(1-(-1)^n): n in [1..80]]; // Vincenzo Librandi, Aug 10 2014

A064455 := proc(n)
    if type(n,'even') then
        3*n/2 ;
    else
        (n+1)/2 ;
    end if;
end proc: # R. J. Mathar, Aug 03 2015

Table[ If[ EvenQ[n], 3n/2, (n + 1)/2], {n, 1, 70} ]

a(n) = { if (n%2, (n + 1)/2, 3*n/2) } \\ Harry J. Smith, Sep 14 2009

a(n)=if(n<3,2*n-1,((n-1)*(n-2))%(2*n-1)) \\ Jim Singh, Oct 14 2018

def A064455(n): return (3*n - (2*n-1)*(n%2))//2
print([A064455(n) for n in range(1,81)]) # G. C. Greubel, Jan 30 2025

a(n) = (1/2)*n*(-1)^n + n + (1/4)*(1 - (-1)^n). - Stephen Crowley, Aug 10 2009
G.f.: x*(1+3*x) / ( (1-x)^2*(1+x)^2 ). - R. J. Mathar, Mar 30 2011
From Jaroslav Krizek, Mar 22 2011: (Start)
a(n) = n - A123684(n-1) for odd n.
a(n) = n + a(n-1) for even n.
a(n) = A123684(n) + A137501(n).
Abs( a(n) - A123684(n) ) =  A052928(n). (End)
a(n) = Sum_{i=n..2*n} i*(-1)^i. - Bruno Berselli, Jun 05 2013
a(n) = n + floor(n/2)*(-1)^(n mod 2). - Bruno Berselli, Dec 14 2015
a(n) = (n^2-3*n+2) mod (2*n-1) for n>2. - Jim Singh, Oct 31 2018
E.g.f.: (1/2)*(x*cosh(x) + (1+3*x)*sinh(x)). - G. C. Greubel, Jan 30 2025

A050409 Truncated square pyramid numbers: a(n) = Sum_{k = n..2*n} k^2.

Original entry on oeis.org

0, 5, 29, 86, 190, 355, 595, 924, 1356, 1905, 2585, 3410, 4394, 5551, 6895, 8440, 10200, 12189, 14421, 16910, 19670, 22715, 26059, 29716, 33700, 38025, 42705, 47754, 53186, 59015, 65255, 71920, 79024, 86581, 94605, 103110, 112110, 121619

Offset: 0

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Dec 22 1999

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
M. Janjic and B. Petkovic, A Counting Function, arXiv preprint arXiv:1301.4550 [math.CO], 2013. - From _N. J. A. Sloane_, Feb 13 2013
M. Janjic, B. Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014) # 14.3.5.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000330, A033994, A129371, A132112, A132121, A132124.
Cf. A225144. - Bruno Berselli, Jun 06 2013
Cf. A045943: Sum_{k = n..2*n} k.
Cf. A304993: Sum_{k = n..2*n} k*(k+1)/2.

List([0..40], n-> n*(n+1)*(14*n+1)/6); # G. C. Greubel, Oct 30 2019

[&+[k^2: k in [n..2*n]]: n in [0..40]]; // Bruno Berselli, Feb 11 2011

I:=[0, 5, 29, 86]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..40]]; // Vincenzo Librandi, Jun 22 2012

seq(add((n+k)^2,k=0..n),n=0..40); # Zerinvary Lajos, Dec 01 2006

LinearRecurrence[{4,-6,4,-1},{0,5,29,86},40] (* Vincenzo Librandi, Jun 22 2012 *)
Table[(n(n+1)(14n+1))/6,{n,0,40}] (* Harvey P. Dale, Mar 08 2020 *)

PARI
```
a(n)=sum(k=n,n+n,k^2)
```

vector(40, n, n*(n-1)*(14*n-13)/6) \\ G. C. Greubel, Oct 30 2019

[n*(n+1)*(14*n+1)/6 for n in (0..40)] # G. C. Greubel, Oct 30 2019

a(n) = n*(n+1)*(14*n+1)/6.
a(n) = A132121(n,4) for n>3. - Reinhard Zumkeller, Aug 12 2007
From Bruno Berselli, Feb 11 2011: (Start)
G.f.: x*(5+9*x)/(1-x)^4.
a(n) = A129371(2*n). (End)
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Vincenzo Librandi, Jun 22 2012
E.g.f.: x*(30 + 57*x + 14*x^2)*exp(x)/6. - G. C. Greubel, Oct 30 2019

A089594 Alternating sum of squares to n.

Original entry on oeis.org

-1, 3, -6, 10, -15, 21, -28, 36, -45, 55, -66, 78, -91, 105, -120, 136, -153, 171, -190, 210, -231, 253, -276, 300, -325, 351, -378, 406, -435, 465, -496, 528, -561, 595, -630, 666, -703, 741, -780, 820, -861, 903, -946, 990, -1035, 1081, -1128, 1176, -1225, 1275

Offset: 1

Jon Perry, Dec 30 2003

Let A be the Hessenberg n X n matrix defined by: A[1,j]=j mod 2, A[i,i]:=1, A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=3, a(n-1)=(-1)^(n-1)*coeff(charpoly(A,x),x^(n-2)). - Milan Janjic, Jan 24 2010

Also triangular numbers with alternating signs. - Stanislav Sykora, Nov 26 2013

			a(6) = 1 + 4 - 9 + 16 - 25 + 36 = 3 + 7 + 11 = 21.

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (-3,-3,-1).

Cf. A059841 (p=0,q=-1), A130472 (p=1,q=-1), this sequence (p=2,q=-1), A232599 (p=3,q=-1), A126646 (p=0,q=2), A036799 (p=1,q=2), A036800 (p=2,q=2), A036827 (p=3,q=2), A077925 (p=0,q=-2), A232600 (p=1,q=-2), A232601 (p=2,q=-2), A232602 (p=3,q=-2), A232603 (p=2,q=-1/2), A232604 (p=3,q=-1/2).
Cf. A000217.
Cf. A225144. [Bruno Berselli, Jun 06 2013]

[(-1)^n*n*(n+1)/2: n in [1..50]]; // Vincenzo Librandi, Nov 16 2011

seq(sum(binomial(n,m), m=1..2)-n^2,n=2..51); # Zerinvary Lajos, Jun 19 2008
A089594 := n -> (-1)^n*n*(n+1)/2; # Peter Luschny, Jul 08 2011

nn = Range[50]; Accumulate[(-1)^nn*nn^2] (* Jayanta Basu, Jun 06 2013 *)

for(i=1,50, print1(","sum(j=1,i,(-1)^j*j^2)))

a(n)=(-1)^n*n*(n+1)/2 \\ Charles R Greathouse IV, Jul 08 2011

[(-1)^n*binomial(n+1,2) for n in (1..50)] # G. C. Greubel, Mar 31 2021

From R. J. Mathar, Nov 05 2011: (Start)
a(n) = Sum_{i=1..n} (-1)^i*i^2 = (-1)^n*n*(n+1)/2.
G.f.: -x / (1+x)^3. (End)
a(n) = (-1)^n*det(binomial(i+2,j+1), 1 <= i,j <= n-1). - Mircea Merca, Apr 06 2013
G.f.: -W(0)/(2+2*x), where W(k) = 1 + 1/( 1 - x*(k+2)/( x*(k+2) - (k+1)/W(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 19 2013
E.g.f.: (1/2)*x*(x-2)*exp(-x). - G. C. Greubel, Mar 31 2021
Sum_{n>=1} 1/a(n) = 2 - 4*log(2). - Amiram Eldar, Jan 31 2023

A065679 If n is even, a(n) = n^2 else a(n) = n.

Original entry on oeis.org

0, 1, 4, 3, 16, 5, 36, 7, 64, 9, 100, 11, 144, 13, 196, 15, 256, 17, 324, 19, 400, 21, 484, 23, 576, 25, 676, 27, 784, 29, 900, 31, 1024, 33, 1156, 35, 1296, 37, 1444, 39, 1600, 41, 1764, 43, 1936, 45, 2116, 47, 2304, 49, 2500, 51, 2704, 53, 2916, 55, 3136, 57

Offset: 0

Robert G. Wilson v, Dec 03 2001

Harry J. Smith, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,3,0,-3,0,1).

Cf. A000217, A065599, A016742 (bisection), A005408 (bisection).

Cf. A225144.

/* By the third formula: */ A000217:=func; [A000217(n)+(-1)^n*A000217(n-1): n in [0..60]]; // Bruno Berselli, Jun 06 2013

Mathematica

Table[ n^ (Mod[n + 1, 2] + 1), {n, 0, 60} ]

PARI

a(n) = { if (n%2, n, n^2) } \\ Harry J. Smith, Oct 26 2009

Formula

a(n) = n^( (n+1) (mod 2) + 1 ).

O.g.f.: x*(1+x^2)*(1+4*x-x^2)/(1-x^2)^3. - Len Smiley, Dec 05 2001

a(n) = A000217(n)+(-1)^n*A000217(n-1) with A000217(-1)=0. - Bruno Berselli, Jun 06 2013

cp's OEIS Frontend

A064455 a(2n) = 3n, a(2n-1) = n.

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A050409 Truncated square pyramid numbers: a(n) = Sum_{k = n..2*n} k^2.

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A089594 Alternating sum of squares to n.

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A065679 If n is even, a(n) = n^2 else a(n) = n.

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