A083884 -id:A083884 - cp's OEIS frontend

A377879 Deficiency of squares: a(n) = 2n^2 - sigma(n^2).

1, 1, 5, 1, 19, -19, 41, 1, 41, -17, 109, -115, 155, -7, 47, 1, 271, -199, 341, -161, 141, 37, 505, -499, 469, 71, 365, -199, 811, -1021, 929, 1, 449, 163, 683, -1159, 1331, 221, 663, -737, 1639, -1659, 1805, -251, 299, 361, 2161, -2035, 2001, -467, 1211, -265, 2755, -1819, 1927, -967, 1545, 631, 3421, -5293, 3659, 737

Offset: 1

Antti Karttunen, Nov 23 2024

sign

It is conjectured that 1's occur only when n is two's power (A000079), and that there are no -1's in this sequence. See comments in A033879 and in A337339.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences related to sigma(n).

Cf. A000290, A000079 (conjectured to give positions of all 1's), A033879, A378231 [= a(A003961(n))].
Cf. A000012, A083884.
Cf. also square array A083064.

Table[2n^2-DivisorSigma[1, n^2], {n, 62}] (* James C. McMahon, Nov 24 2024 *)

A033879(n) = (n+n-sigma(n));
A377879(n) = A033879(n*n);

a(n) = A033879(A000290(n)).

A083885 (4^n+2^n+0^n+(-2)^n)/4.

Original entry on oeis.org

1, 1, 6, 16, 72, 256, 1056, 4096, 16512, 65536, 262656, 1048576, 4196352, 16777216, 67117056, 268435456, 1073774592, 4294967296, 17180000256, 68719476736, 274878431232, 1099511627776, 4398048608256, 17592186044416, 70368752566272

Offset: 0

Paul Barry, May 09 2003

Binomial transform of A083884.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (4,4,-16).

[(4^n+2^n+0^n+(-2)^n)/4: n in [0..20]]; // Vincenzo Librandi, Jun 16 2011

Join[{1},Table[(4^n+2^n+(-2)^n)/4,{n,30}]] (* or *) Join[{1}, LinearRecurrence[ {4,4,-16},{1,6,16},30]] (* Harvey P. Dale, Dec 12 2011 *)

a(n) = (4^n+2^n+0^n+(-2)^n)/4.
G.f.: (4*x^3-2*x^2-3*x+1)/((2*x+1)*(2*x-1)*(4*x-1)).
E.g.f.: exp(4*x)+exp(2*x)+exp(0)+exp(-2*x).
A007814(a(n)) = A022998(n-1). - Ralf Stephan, Feb 14 2004
a(0)=1, a(1)=1, a(2)=6, a(3)=16, a(n)=4*a(n-1)+4*a(n-2)-16*a(n-3) [From Harvey P. Dale, Dec 12 2011]

A191687 Table T(n,k) = ceiling((1/2)*((k+1)^n+(1+(-1)^k)/2)) read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 1, 4, 5, 2, 1, 1, 8, 14, 8, 3, 1, 1, 16, 41, 32, 13, 3, 1, 1, 32, 122, 128, 63, 18, 4, 1, 1, 64, 365, 512, 313, 108, 25, 4, 1, 1, 128, 1094, 2048, 1563, 648, 172, 32, 5, 1

Offset: 1

Adi Dani, Jun 11 2011

T(n,k) is the number of compositions of even natural numbers into n parts <= k.

			Top left corner:
  1, 1, 1,  1,  1,...
  1, 1, 2,  2,  3,...
  1, 2, 5,  8, 13,...
  1, 4,14, 32, 63,...
  1, 8,41,128,313,...
T(2,4)=13: there are 13 compositions of even natural numbers into 2 parts <=4
0: (0,0);
2: (0,2), (2,0), (1,1);
4: (0,4), (4,0), (1,3), (3,1), (2,2);
6: (2,4), (4,2), (3,3);
8: (4,4).

Adi Dani, Restricted compositions of natural numbers

Rows sums gives: A000012, A004526, A000982, A036486, A171714, A191484, A191489, A191494, A191495, A191496

Columns sums gives: A000012, A000079, A007051, A004171, A034478, A081341, A034494, A092811, A083884, A093143

Table[Table[Ceiling[1/2*((k+1)^n+(1+(-1)^k)/2)],{n,0,9},{k,0,9}]]

cp's OEIS Frontend

A377879 Deficiency of squares: a(n) = 2n^2 - sigma(n^2).

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A083885 (4^n+2^n+0^n+(-2)^n)/4.

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A191687 Table T(n,k) = ceiling((1/2)*((k+1)^n+(1+(-1)^k)/2)) read by antidiagonals.

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