A028552 -id:A028552 - cp's OEIS frontend

A172176 Triangle T(n, k) = 1 + (n + k - nk)(2n - k - n(n-k)), read by rows.

1, 2, 2, 1, 2, 1, -8, 0, 0, -8, -31, -4, 5, -4, -31, -74, -10, 22, 22, -10, -74, -143, -18, 57, 82, 57, -18, -143, -244, -28, 116, 188, 188, 116, -28, -244, -383, -40, 205, 352, 401, 352, 205, -40, -383, -566, -54, 330, 586, 714, 714, 586, 330, -54, -566

Offset: 0

Roger L. Bagula, Jan 28 2010

			Triangle begins as:
     1;
     2,   2;
     1,   2,   1;
    -8,   0,   0,  -8;
   -31,  -4,   5,  -4,  -31;
   -74, -10,  22,  22,  -10,  -74;
  -143, -18,  57,  82,   57,  -18, -143;
  -244, -28, 116, 188,  188,  116,  -28, -244;
  -383, -40, 205, 352,  401,  352,  205,  -40, -383;
  -566, -54, 330, 586,  714,  714,  586,  330,  -54, -566;
  -799, -70, 497, 902, 1145, 1226, 1145,  902,  497,  -70, -799;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A027620, A028552, A033445.

[1 + (n-(n-1)*k)*(n-(n-1)*(n-k)): k in [0..n], n in [0..12]]; // G. C. Greubel, Apr 26 2022

A172176:= proc(n,m) 1+(n+m-n*m)*(2*n-m-n*(n-m)); end proc:
seq(seq(A172176(n,m), m=0..n), n=0..12);

T[n_, k_]= 1 + (n-(n-1)*k)*(n-(n-1)*(n-k));
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten

def A172176(n,k): return 1 + (n-(n-1)*k)*(n-(n-1)*(n-k))
flatten([[A172176(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Apr 26 2022

T(n, k) = 1 + (n-(n-1)*k)*(n-(n-1)*(n-k)).
T(n, n-k) = T(n, k).
T(n, 0) = 1 - A027620(n-3).
T(n, 1) = -A028552(n-3).
T(n, 2) = A033445(n-2).
Sum_{k=0..n} T(n, k) = (n+1)*(n^4 - 9*n^3 + 15*n^2 - n + 6)/6.

A183570 a(n) = n + floor(sqrt(n + 1)).

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 109

Offset: 0

Clark Kimberling, Jan 05 2011

Cf. A028552 (complement, except for initial term).

A183570 := proc(n) n+floor(sqrt(n+1)) ; end proc:
seq(A183570(n),n=0..100) ; # R. J. Mathar, Jan 29 2011

All terms corrected by R. J. Mathar, Jan 29 2011

A203553 Lodumo_2 of A118175, which is n 1's followed by n 0's.

Original entry on oeis.org

1, 0, 3, 5, 2, 4, 7, 9, 11, 6, 8, 10, 13, 15, 17, 19, 12, 14, 16, 18, 21, 23, 25, 27, 29, 20, 22, 24, 26, 28, 31, 33, 35, 37, 39, 41, 30, 32, 34, 36, 38, 40, 43, 45, 47, 49, 51, 53, 55, 42, 44, 46, 48, 50, 52, 54, 57, 59, 61, 63, 65, 67, 69, 71, 56, 58, 60, 62, 64, 66, 68, 70

Offset: 0

Philippe Deléham, Jan 02 2012

nonn

Permutation of nonnegative numbers.
Lodumo_k of sequences is defined in A159970.
The sequence has a natural decomposition into irregular triangle (see example). The length of row n is A008619 (n). Row sums are the cubes (A000578) interspersed with A007531.

			Triangle begins :
1
0
3, 5
2, 4
7, 9, 11
6, 8, 10
13, 15, 17, 19
12, 14, 16, 18
21, 23, 25, 27, 29
20, 22, 24, 26, 28
31, 33, 35, 37, 39, 41
30, 32, 34, 36, 38, 40
43, 45, 47, 49, 51, 53, 55
42, 44, 46, 48, 50, 52, 54
57, 59, 61, 63, 65, 67, 69, 71
56, 58, 60, 62, 64, 66, 68, 70
73, 75, 77, 79, 81, 83, 85, 87, 89
72, 74, 76, 78, 80, 82, 84, 86, 88 ...
Row sums : 1 = 1^3 ; 0 = 1^3 - 1 ; 3 + 5 = 2^3 ; 2 + 4 = 2^3 - 2 = 6 ; 7 + 9 + 11 = 3^3 = 27 ; 6 + 8 + 10 = 3^3 - 3 = 24 ; 13 + 15 + 17 + 19 = 4^3 = 64 ; 12 + 14 + 16 + 18 = 4^4 - 4 = 60 ; ...

Cf. A118175, A159970, A203554.

Cf. A000578, A007531, A002061, A002378, A028387, A028552.

a(n)= Lodumo_2(A118175(n)).

A203554 Lodumo_2 of A079813, which is n 0's followed by n 1's.

Original entry on oeis.org

0, 1, 2, 4, 3, 5, 6, 8, 10, 7, 9, 11, 12, 14, 16, 18, 13, 15, 17, 19, 20, 22, 24, 26, 28, 21, 23, 25, 27, 29, 30, 32, 34, 36, 38, 40, 31, 33, 35, 37, 39, 41, 42, 44, 46, 48, 50, 52, 54, 43, 45, 47, 49, 51, 53, 55

Offset: 0

Philippe Deléham, Jan 02 2012

nonn

Permutation of nonnegative numbers.
Lodumo_k of sequences is defined in A159970.
The sequence has a natural decomposition into irregular triangle (see example). The length of row n is A008619(n). Rows sums are the cubes (A000578) interspersed with A007531. First column: A002378 interspersed with A002061.

			Triangle begins :
0
1
2, 4
3, 5
6, 8, 10
7, 9, 11
12, 14, 16, 18
13, 15, 17, 19
20, 22, 24, 26, 28
21, 23, 25, 27, 29
30, 32, 34, 36, 38, 40
31, 33, 35, 37, 39, 41
42, 44, 46, 48, 50, 52, 54
43, 45, 47, 49, 51, 53, 55
56, 58, 60, 62, 64, 66, 68, 70
57, 59, 61, 63, 65, 67, 69, 71
72, 74, 76, 78, 80, 82, 84, 86, 88
73, 75, 77, 79, 81, 83, 85, 87, 89...
Row sums : 0 = 1^3 -1 = 0 ; 1 = 1^3 = 1 ; 2 + 4 = 2^3 - 2 = 6 ; 3 + 5 = 2^3 = 8 ; 6 + 8 + 10 = 3^3 - 3 = 24 ; 7 + 9 + 11 = 3^3 = 27 ; 12 + 14 + 16 + 18 = 4^3 - 4 = 60 ; 13 + 15 + 17 + 19 = 4^3 = 64 ; 20 + 22 + 24 + 26 + 28 = 5^3 - 5 = 120 ; 21 + 23 + 25 + 27 + 29 = 5^3 = 125 ; etc...

Cf. A079813, A159970, A203553.

Cf. A028387, A028552, A002061, A002378, A000578, A007531.

A214870 Natural numbers placed in table T(n,k) layer by layer. The order of placement: at the beginning filled odd places of layer clockwise, next - even places counterclockwise. T(n,k) read by antidiagonals.

Original entry on oeis.org

1, 2, 3, 5, 4, 7, 10, 9, 8, 13, 17, 16, 6, 14, 21, 26, 25, 11, 12, 22, 31, 37, 36, 18, 15, 20, 32, 43, 50, 49, 27, 24, 23, 30, 44, 57, 65, 64, 38, 35, 19, 33, 42, 58, 73, 82, 81, 51, 48, 28, 29, 45, 56, 74, 91, 101, 100, 66, 63, 39, 34, 41, 59, 72, 92, 111

Offset: 1

Boris Putievskiy, Mar 11 2013

Permutation of the natural numbers.
a(n) is a pairing function: a function that reversibly maps Z^{+} x Z^{+} onto Z^{+}, where Z^{+} is the set of integer positive numbers.
Layer is pair of sides of square from T(1,n) to T(n,n) and from T(n,n) to T(n,1).
Enumeration table T(n,k) layer by layer. The order of the list:
  T(1,1)=1;
  T(1,2), T(2,1), T(2,2);
  . . .
  T(1,n), T(3,n), ... T(n,3), T(n,1); T(n,2), T(n,4), ... T(4,n), T(2,n);
  . . .

			The start of the sequence as table:
   1   2   5  10  17  26 ...
   3   4   9  16  25  36 ...
   7   8   6  11  18  27 ...
  13  14  12  15  24  35 ...
  21  22  20  23  19  28 ...
  31  32  30  33  29  34 ...
  ...
The start of the sequence as triangle array read by rows:
   1;
   2,  3;
   5,  4,  7;
  10,  9,  8, 13;
  17, 16,  6, 14, 21;
  26, 25, 11, 12, 22, 31;
  ...

Boris Putievskiy, Rows n = 1..140 of triangle, flattened
Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.
Eric Weisstein's World of Mathematics, Pairing functions
Index entries for sequences that are permutations of the natural numbers

Cf. A060734, A060736, A185725, A213921, A213922; table T(n,k) contains: in rows A002522, A000290, A059100, A005563, A117950, A008865, A087475, A028872, A117951, A028347, A114949, A028875, A117619, A028878, A189833, A028881, A189834, A028884, A114948, A028560, A189836; in columns A002061, A014206, A002378, A027688, A028387, A027689, A028552, A027690, A014209, A027691, A027692, A082111, A027693, A028557, A027694, A108195, A187710, A048058, A048840.

t=int((math.sqrt(8*n-7) - 1)/ 2)
i=n-t*(t+1)/2
j=(t*t+3*t+4)/2-n
if i > j:
   result=i*i-i+(j%2)*(2-(j+1)/2)+((j+1)%2)*(j/2+1)
else:
   result=j*j-2*(i%2)*j + (i%2)*((i+1)/2+1) + ((i+1)%2)*(-i/2+1)

As table
T(n,k) = k*k-2*(n mod 2)*k+(n mod 2)*((n+1)/2+1)+((n+1) mod 2)*(-n/2+1), if n<=k;
T(n,k) = n*n-n+(k mod 2)*(2-(k+1)/2)+((k+1) mod 2)*(k/2+1), if n>k.
As linear sequence
a(n) = j*j-2*(i mod 2)*j+(i mod 2)*((i+1)/2+1)+((i+1) mod 2)*(-i/2+1), if i<=j;
a(n) = i*i-i+(j mod 2)*(2-(j+1)/2)+((j+1) mod 2)*(j/2+1), if i>j; where i=n-t*(t+1)/2, j=(t*t+3*t+4)/2-n, t=floor((-1+sqrt(8*n-7))/2).

A220588 a(n) = 2^n - n^2 - n.

Original entry on oeis.org

1, 0, -2, -4, -4, 2, 22, 72, 184, 422, 914, 1916, 3940, 8010, 16174, 32528, 65264, 130766, 261802, 523908, 1048156, 2096690, 4193798, 8388056, 16776616, 33553782, 67108162, 134216972, 268434644, 536870042, 1073740894, 2147482656, 4294966240, 8589933470, 17179867994

Offset: 0

Dario Piazzalunga, Dec 16 2012

			a(3) = -4 because 2^3 - 3^2 - 3 = 8 - 9 - 3 = -4.
a(4) = -4 because 2^4 - 4^2 - 4 = 16 - 16 - 4 = -4.
a(5) = 2 because 2^5 - 5^2 - 5 = 32 - 25 - 5 = 2.
a(6) = 22 because 2^6 - 6^2 - 6 = 64 - 36 - 6 = 22.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-9,7,-2).

Cf. A015519, A024012, A001580.

Table[2^n - n^2 - n, {n, 0, 32}] (* Alonso del Arte, Dec 16 2012 *)

A220588(n):=2^n-n^2-n$ makelist(A220588(n),n,0,20); /* Martin Ettl, Dec 18 2012 */

Vec((1 - 5*x + 7*x^2 - x^3) / ((1 - x)^3*(1 - 2*x)) + O(x^40)) \\ Colin Barker, Aug 16 2017

a(n) = 2*a(n - 1) + ((n - 3)^2 + 3(n - 3)) = 2*a(n - 1) + A028552(n - 3) for n > 4.
a(n) = (2*a(n-1) + 7*a(n-2))*2 = A015519/2 for n > 4.
From Colin Barker, Aug 16 2017: (Start)
G.f.: (1 - 5*x + 7*x^2 - x^3) / ((1 - x)^3*(1 - 2*x)).
a(n) = 5*a(n-1) - 9*a(n-2) + 7*a(n-3) - 2*a(n-4) for n>3.
(End)

a(3) corrected by Charles A. Dagino, Aug 16 2017

A222963 a(n) = (p-3)*(p+3)/4 where p is the n-th prime.

Original entry on oeis.org

0, 4, 10, 28, 40, 70, 88, 130, 208, 238, 340, 418, 460, 550, 700, 868, 928, 1120, 1258, 1330, 1558, 1720, 1978, 2350, 2548, 2650, 2860, 2968, 3190, 4030, 4288, 4690, 4828, 5548, 5698, 6160, 6640, 6970, 7480, 8008, 8188

Offset: 2

Vincenzo Librandi, Apr 05 2013

Old name was: Numbers n such that sqrt(4*n+9) is prime.

Vincenzo Librandi, Table of n, a(n) for n = 2..1000

Subsequence of A028552.

Cf. A000040, A001248, A005097, A168669.

[(p^2-9)/4: p in PrimesInInterval(3, 200)];

Select[Range[0, 10000], PrimeQ[Sqrt[4 # + 9]]&]

4*a(n)+9 = A001248(n).

a(n) = A166011(n)/2. - Michel Marcus, Apr 22 2016

A222964 Numbers k such that 25*k+36 is a square.

Original entry on oeis.org

0, 13, 37, 76, 124, 189, 261, 352, 448, 565, 685, 828, 972, 1141, 1309, 1504, 1696, 1917, 2133, 2380, 2620, 2893, 3157, 3456, 3744, 4069, 4381, 4732, 5068, 5445, 5805, 6208, 6592, 7021, 7429, 7884, 8316, 8797, 9253, 9760, 10240, 10773, 11277, 11836, 12364, 12949, 13501, 14112, 14688

Offset: 1

Vincenzo Librandi, Apr 07 2013

Also, numbers of the form 25m^2+12*m, where m = 0,-1,1,-2,2,-3,3,... - Bruno Berselli, Apr 07 2013

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

Cf. numbers n such that k^2*n+(k+1)^2 is a square: A028552 (k=2), A218864 (k=3), A165717 (k=4).

Cf. numbers of the form k^2*m^2+floor(k^2/2)*m, where m=0,-1,1,-2,2,-3,3,...: A002378 (k=2), A185039 (k=3), A033996 (k=4), this sequence (k=5), A163758 (k=6).

[n: n in [0..15000] | IsSquare(25*n+36)];

I:=[0, 13, 37, 76, 124]; [n le 5 select I[n] else Self(n-1)+2*Self(n-2)-2*Self(n-3)-Self(n-4)+Self(n-5): n in [1..50]];

[0] cat [25*m^2+12*m where m is n*t: t in [-1, 1], n in [1..20]]; // Bruno Berselli, Apr 07 2013

Select[Range[0, 10000], IntegerQ[Sqrt[25 # + 36]]&] (* or *) CoefficientList[Series[x (13 + 24 x + 13 x^2)/((1+x)^2(1-x)^3), {x, 0, 40}], x]
LinearRecurrence[{1,2,-2,-1,1},{0,13,37,76,124},50] (* Harvey P. Dale, Jan 23 2025 *)

G.f.: x^2*(13+24*x+13*x^2)/((1+x)^2*(1-x)^3).
a(n) = (50*n*(n-1)+(2*n-1)*(-1)^n+1)/8.
a(n) = a(n-1)+2*a(n-2)-2*a(n-3)-a(n-4)+a(n-5).
Sum_{n>=2} 1/a(n) = 25/144 - tan(Pi/50)*Pi/12. - Amiram Eldar, Feb 16 2023

A288994 a(n) = n(n+3) when n is congruent to 0 or 3 (mod 4), and n(n+3)/2 otherwise.

Original entry on oeis.org

0, 2, 5, 18, 28, 20, 27, 70, 88, 54, 65, 154, 180, 104, 119, 270, 304, 170, 189, 418, 460, 252, 275, 598, 648, 350, 377, 810, 868, 464, 495, 1054, 1120, 594, 629, 1330, 1404, 740, 779, 1638, 1720, 902, 945, 1978, 2068, 1080, 1127, 2350, 2448, 1274, 1325

Offset: 0

Jean-François Alcover and Paul Curtz, Jun 21 2017

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-6,10,-12,12,-10,6,-3,1).

Cf. A000096, A014695, A028552, A060819, A130658, A145979, A227316.

a[n_] := n (n+3) Switch[Mod[n, 4], 0|3, 1, _, 1/2]; Table[a[n], {n, 0, 50}]
Table[If[MemberQ[{0,3},Mod[n,4]],n(n+3),(n(n+3))/2],{n,0,50}] (* or *) LinearRecurrence[{3,-6,10,-12,12,-10,6,-3,1},{0,2,5,18,28,20,27,70,88},60] (* Harvey P. Dale, Jun 05 2021 *)

concat(0, Vec(x*(2 - x + 15*x^2 - 16*x^3 + 18*x^4 - 9*x^5 + 5*x^6 - 2*x^7) / ((1 - x)^3*(1 + x^2)^3) + O(x^60))) \\ Colin Barker, Jun 21 2017

i=I; a(n) = (1/8 + i/8)*(((3 - 3*i) - i*(-i)^n + i^n)*n*(3 + n)) \\ Colin Barker, Jun 21 2017

a(n) = n*(n+3)/2 * (2 - floor((n+1)/2) mod 2), where n*(n+3)/2 is A000096(n).
a(n) = A060819(n+3)*A145979(n-2).
a(n) = (2*n*(n+3))/(GCD(4, n+2)*GCD(4, n+3)).
a(n) = A227316(n+1) - (period 4 repeat 2,1,1,2).
From Colin Barker, Jun 21 2017: (Start)
G.f.: x*(2 - x + 15*x^2 - 16*x^3 + 18*x^4 - 9*x^5 + 5*x^6 - 2*x^7) / ((1 - x)^3*(1 + x^2)^3).
a(n) = (1/8 + i/8)*(((3 - 3*i) - i*(-i)^n + i^n)*n*(3 + n)), where i=sqrt(-1). (End)
Sum_{n>=1} 1/a(n) = 17/18 + log(2)/6. - Amiram Eldar, Aug 12 2022

A383641 a(n) is the difference between the sum of even composites and the sum of the odd composites in the first n positive integers.

Original entry on oeis.org

0, 0, 0, 4, 4, 10, 10, 18, 9, 19, 19, 31, 31, 45, 30, 46, 46, 64, 64, 84, 63, 85, 85, 109, 84, 110, 83, 111, 111, 141, 141, 173, 140, 174, 139, 175, 175, 213, 174, 214, 214, 256, 256, 300, 255, 301, 301, 349, 300, 350, 299, 351, 351, 405, 350, 406, 349, 407, 407

Offset: 1

Felix Huber, May 08 2025

nonn

			Of the first 9 positive integers, 4, 6, and 8 are even composites and 9 is an odd composite, so a(9) = 4 + 6 + 8 - 9 = 9.

Felix Huber, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Composite Number

Cf. A000720, A002808, A004526, A010701, A028552, A034387, A066247, A071904, A101256, A193356, A262044, A383259.

A383641:=n->`if`(n=1,0,floor((n-2)/2)-n*(n mod 2)+add(ithprime(i),i=2..NumberTheory:-pi(n)));seq(A383641(n),n=1..59);

lim=59;cn=Select[Range[lim],CompositeQ];a[n_]:=Total[Select[cn,EvenQ[#]&&#<=n&]]-Total[Select[cn,OddQ[#]&&#<=n&]];Array[a,lim] (* James C. McMahon, May 14 2025 *)

a(n) = floor((n-2)/2) - n*(n mod 2) + Sum_{i=2..pi(n)} prime(i) for n > 1.
a(n) = A004526(n) - A193356(n) - A010701(n) + A034387(A000720(n)) for n > 1.
a(n) = Sum_{i=1..n} ((-1)^i*i*A066247(i)).

cp's OEIS Frontend

A172176 Triangle T(n, k) = 1 + (n + k - n*k)*(2*n - k - n*(n-k)), read by rows.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

SageMath

Formula

A183570 a(n) = n + floor(sqrt(n + 1)).

Views

Author

Keywords

Crossrefs

Programs

Maple

Extensions

A203553 Lodumo_2 of A118175, which is n 1's followed by n 0's.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A203554 Lodumo_2 of A079813, which is n 0's followed by n 1's.

Views

Author

Keywords

Comments

Examples

Crossrefs

A214870 Natural numbers placed in table T(n,k) layer by layer. The order of placement: at the beginning filled odd places of layer clockwise, next - even places counterclockwise. T(n,k) read by antidiagonals.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Python

Formula

A220588 a(n) = 2^n - n^2 - n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Maxima

PARI

Formula

Extensions

A222963 a(n) = (p-3)*(p+3)/4 where p is the n-th prime.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A222964 Numbers k such that 25*k+36 is a square.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

A172176 Triangle T(n, k) = 1 + (n + k - nk)(2n - k - n(n-k)), read by rows.

A288994 a(n) = n(n+3) when n is congruent to 0 or 3 (mod 4), and n(n+3)/2 otherwise.