A017101 -id:A017101 - cp's OEIS frontend

A343099 Sums of 3 distinct odd squares.

35, 59, 75, 83, 91, 107, 115, 131, 139, 147, 155, 171, 179, 195, 203, 211, 219, 227, 235, 243, 251, 259, 275, 283, 291, 299, 307, 315, 323, 331, 339, 347, 355, 363, 371, 379, 387, 395, 403, 411, 419, 427, 435, 443, 451, 459, 467, 475, 483, 491, 499, 507, 515, 523, 531

Offset: 1

Wesley Ivan Hurt, Apr 05 2021

nonn

From Robert Israel, Apr 06 2021: (Start)
All terms == 3 (mod 8).
Conjecture: contains all numbers == 3 (mod 8) except 3, 11, 19, 27, 43, 51, 67, 99, 123, 163, 187, 267, 627. (End)

			107 is in the sequence since 107 = 1^2 + 5^2 + 9^2.

Robert Israel, Table of n, a(n) for n = 1..10000
Index entries for sequences related to sums of squares

Subsequence of A017101.

Cf. A004432, A016754 (odd squares).

N:= 10^4: # for terms <= N
S:= {seq(seq(seq(x^2+y^2+z^2, z = 1 .. min(y-2, floor(sqrt(N-x^2-y^2))), 2),y = 1 .. min(x-2, floor(sqrt(N-x^2))), 2), x = 1 .. floor(sqrt(N)),2)}:
sort(convert(S,list)); # Robert Israel, Apr 05 2021

A350052 Third part of the trisection of A017077: a(n) = 17 + 24*n.

Original entry on oeis.org

17, 41, 65, 89, 113, 137, 161, 185, 209, 233, 257, 281, 305, 329, 353, 377, 401, 425, 449, 473, 497, 521, 545, 569, 593, 617, 641, 665, 689, 713, 737, 761, 785, 809, 833, 857, 881, 905, 929, 953, 977, 1001, 1025, 1049, 1073

Offset: 0

Wolfdieter Lang, Dec 11 2021

The trisection of A017077 = {1 + 8*k}A103214%20=%20%7B1%20+%2024*n%7D">{k>=0} gives A103214 = {1 + 24*n}{n>=0}, 3*A017101 = {3*(3 + 8*n)}{n >= 0} and {a(n)}{n>=0}. These three sequences are congruent to 1 modulo 8 and to 1, 3, and 5 modulo 6, respectively.

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

Cf. A008606, A017077, 3*A017101, A103214.

24 * Range[0, 44] + 17 (* Amiram Eldar, Dec 18 2021 *)

a(n) = 17 + 24*n \\ Winston de Greef, Jan 28 2024

a(n) = 17 + 24*n = 17 + A008606(n), for n >= 0
a(n) = 2*a(n-1) - a(n-2), for n >= 1, with a(-1) = -7, a(0) = 17.
G.f.: (17 + 7*x)/(1-x)^2.
E.g.f.: (17 + 24*x)*exp(x).

A361226 Square array T(n,k) = k((1+2n)*k - 1)/2; n>=0, k>=0, read by antidiagonals upwards.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 0, 2, 5, 3, 0, 3, 9, 12, 6, 0, 4, 13, 21, 22, 10, 0, 5, 17, 30, 38, 35, 15, 0, 6, 21, 39, 54, 60, 51, 21, 0, 7, 25, 48, 70, 85, 87, 70, 28, 0, 8, 29, 57, 86, 110, 123, 119, 92, 36, 0, 9, 33, 66, 102, 135, 159, 168, 156, 117, 45

Offset: 0

Paul Curtz, Mar 05 2023

The main diagonal is A002414.
The first upper diagonal is A160378(n+1).
The antidiagonals sums are A034827(n+2).
b(n) = (A034827(n+3) = 0, 2, 10, 30, 70, ...) - (A002414(n) = 0, 1, 9, 30, 70, ...) = 0, 1, 1, 0, 0, 5, 21, 56, ... .
b(n+2) = A299120(n). b(n+4) = A033275(n). b(n+4) - b(n) = A002492(n).

			The rows are
  0, 0,  1,  3,  6,  10,  15,  21, ...   = A161680
  0, 1,  5, 12, 22,  35,  51,  70, ...   = A000326
  0, 2,  9, 21, 38,  60,  87, 119, ...   = A005476
  0, 3, 13, 30, 54,  85, 123, 168, ...   = A022264
  0, 4, 17, 39, 70, 110, 159, 217, ...   = A022266
  ... .
Columns: A000004, A001477, A016813, A017197=3*A016777, 2*A017101, 5*A016873, 3*A017581, 7*A017017, ... (coefficients from A026741).
Difference between two consecutive rows: A000290. Hence A143844.
This square array read by antidiagonals leads to the triangle
  0
  0   0
  0   1   1
  0   2   5   3
  0   3   9  12   6
  0   4  13  21  22  10
  0   5  17  30  38  35  15
  ... .

Cf. A000004, A000290, A000326, A001477, A002414, A005476, A016777, A016813, A016873, A017017, A017101, A017197, A017581, A022264, A022266, A026741, A034827, A160378, A161680, A360962.

Cf. A002492, A033275, A143844, A299120.

T[n_, k_] := k*((2*n + 1)*k - 1)/2; Table[T[n - k, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, Mar 05 2023 *)

a(n) = { my(row = (sqrtint(8*n+1)-1)\2, column = n - binomial(row + 1, 2)); binomial(column, 2) + column^2 * (row - column) } \\ David A. Corneth, Mar 05 2023

# Seen as a triangle:
from functools import cache
@cache
def Trow(n: int) -> list[int]:
    if n == 0: return [0]
    r = Trow(n - 1)
    return [r[k] + k * k if k < n else r[n - 1] + n - 1 for k in range(n + 1)]
for n in range(7): print(Trow(n)) # Peter Luschny, Mar 05 2023

Take successively sequences n*(n-1)/2, n*(3*n-1)/2, n*(5*n-1)/2, ... listed in the EXAMPLE section.
G.f.: y*(x + y)/((1 - y)^3*(1 - x)^2). - Stefano Spezia, Mar 06 2023
E.g.f.: exp(x+y)*y*(2*x + y + 2*x*y)/2. - Stefano Spezia, Feb 21 2024

A367882 Table T(n, k) read by downward antidiagonals: T(n, k) = floor((4T(n, k-1)+3)/3) starting with T(n, 0) = 4n.

Original entry on oeis.org

0, 1, 4, 2, 6, 8, 3, 9, 11, 12, 5, 13, 15, 17, 16, 7, 18, 21, 23, 22, 20, 10, 25, 29, 31, 30, 27, 24, 14, 34, 39, 42, 41, 37, 33, 28, 19, 46, 53, 57, 55, 50, 45, 38, 32, 26, 62, 71, 77, 74, 67, 61, 51, 43, 36, 35, 83, 95, 103, 99, 90, 82, 69, 58, 49, 40

Offset: 0

Philippe Deléham, Dec 04 2023

Permutation of nonnegative numbers.

Let b(m) be the row n in which m appears, this sequence would start: 1, 1, 1, 1, 2, 1, 2, 1, 3, 2, 1, 3,... . If we would remove in this sequence the first appearance of each number then we would obtain again the same sequence, hence b(m) is a fractal sequence. - Thomas Scheuerle, Dec 04 2023

			Square array starts:
  0,   1,   2,   3,   5,   7, ...
  4,   6,   9,  13,  18,  25, ...
  8,  11,  15,  21,  29,  39, ...
 12,  17,  23,  31,  42,  57, ...
 16,  22,  30,  41,  55,  74, ...
 ...

Paolo Xausa, Table of n, a(n) for n = 0..11324 (first 150 antidiagonals, flattened).
Index entries for sequences that are permutations of the nonnegative integers

Cf. A008586, A017101, A106839, A158057.

A367882[n_, k_] := A367882[n, k] = If[k == 0, 4*n, Floor[4*A367882[n, k-1]/3 + 1]];
Table[A367882[k, n-k], {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, Apr 03 2024 *)

T(n, k) = if(k==0, 4*n, (4*T(n, k-1)+3)\3) \\ Thomas Scheuerle, Dec 04 2023

T(n, 0) = 4*n = A008586(n).
T(3*n, 1) = 16*n + 1 = A158057(n).
T(3*n+1, 1) = 16*n + 6 = 2*A017101(n).
T(3*n+2, 1) = 16*n + 11 = A106839(n).
T(3^k+n, k) = 4^(k+1) + T(n, k). - Thomas Scheuerle, Dec 04 2023

More terms from Paolo Xausa, Apr 03 2024

A047473 Numbers that are congruent to {2, 3} mod 8.

Original entry on oeis.org

2, 3, 10, 11, 18, 19, 26, 27, 34, 35, 42, 43, 50, 51, 58, 59, 66, 67, 74, 75, 82, 83, 90, 91, 98, 99, 106, 107, 114, 115, 122, 123, 130, 131, 138, 139, 146, 147, 154, 155, 162, 163, 170, 171, 178, 179, 186, 187, 194, 195, 202, 203, 210, 211, 218, 219, 226, 227, 234

Offset: 1

N. J. A. Sloane

Numbers k such that k and k+2 have the same digital binary sum. - Benoit Cloitre, Dec 01 2002

Also, numbers k such that k*(3*k + 1)/8 + 1/4 is a nonnegative integer. - Bruno Berselli, Feb 14 2017

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

Union of A017089 and A017101.

Cf. A047467, A047468, A047470, A047471.

Flatten[# + {2,3} &/@ (8 Range[0, 30])] (* or *) LinearRecurrence[{1, 1, -1}, {2, 3, 10}, 60] (* Harvey P. Dale, Sep 28 2012 *)

a(n) = 8*n - a(n-1) - 11 for n>1, a(1)=2. - Vincenzo Librandi, Aug 06 2010
From R. J. Mathar, Oct 08 2011: (Start)
a(n) = 4*n - 7/2 - 3*(-1)^n/2.
G.f.: x*(2 + x + 5*x^2)/((1 + x)*(1 - x)^2). (End)
a(1)=2, a(2)=3, a(3)=10; for n>3, a(n) = a(n-1) + a(n-2) - a(n-3). - Harvey P. Dale, Sep 28 2012
Sum_{n>=1} (-1)^(n+1)/a(n) = (2-sqrt(2))*Pi/16 + sqrt(2)*log(sqrt(2)+1)/8 - log(2)/8. - Amiram Eldar, Dec 18 2021

More terms from Vincenzo Librandi, Aug 06 2010

A305553 Numbers that are not the sum of 2 squares and a 4th power.

Original entry on oeis.org

7, 12, 15, 22, 23, 28, 31, 39, 43, 44, 47, 55, 60, 63, 67, 70, 71, 76, 78, 79, 87, 92, 93, 95, 103, 108, 111, 112, 119, 124, 127, 135, 140, 143, 151, 156, 159, 167, 168, 172, 175, 177, 183, 184, 188, 191, 192, 199, 204, 207, 214, 215, 220, 223, 231, 236

Offset: 1

XU Pingya, Jun 20 2018

nonn

Numbers of the form 4*A017101(k) are terms of this sequence.

m is a term iff 16m is also.

W. Jagy and I. Kaplansky, Sums of Squares, Cubes and Higher Powers, Experimental Mathematics, vol. 4 (1995) pp. 169-173.

Subsequence of A000037, A140823 and A022544.
A004215 and A214891 are subsequences.
Cf. A017101, A022552.

n=239;
t=Union@Flatten@Table[x^2+y^2+z^4, {x,0,n^(1/2)}, {y,x,(n-x^2)^(1/2)}, {z,0,(n-x^2-y^2)^(1/4)}];
Complement[Range[0,n], t]

A306970 Numbers of the form 8k+3 not represented by 2x^2+4y^2+4yz+9z^2.

Original entry on oeis.org

3, 43, 163, 907

Offset: 1

N. J. A. Sloane, Mar 26 2019

There may be no further terms.

Irving Kaplansky, The first nontrivial genus of positive definite ternary forms, Mathematics of Computation, Vol. 64 (1995): 341-345.

Cf. A017101 (numbers of the form 8*k+3).

cp's OEIS Frontend

A343099 Sums of 3 distinct odd squares.

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A350052 Third part of the trisection of A017077: a(n) = 17 + 24*n.

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A361226 Square array T(n,k) = k*((1+2*n)*k - 1)/2; n>=0, k>=0, read by antidiagonals upwards.

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A367882 Table T(n, k) read by downward antidiagonals: T(n, k) = floor((4*T(n, k-1)+3)/3) starting with T(n, 0) = 4*n.

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A047473 Numbers that are congruent to {2, 3} mod 8.

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A305553 Numbers that are not the sum of 2 squares and a 4th power.

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A306970 Numbers of the form 8*k+3 not represented by 2*x^2+4*y^2+4*y*z+9*z^2.

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A361226 Square array T(n,k) = k((1+2n)*k - 1)/2; n>=0, k>=0, read by antidiagonals upwards.

A367882 Table T(n, k) read by downward antidiagonals: T(n, k) = floor((4T(n, k-1)+3)/3) starting with T(n, 0) = 4n.

A306970 Numbers of the form 8k+3 not represented by 2x^2+4y^2+4yz+9z^2.