A131093 -id:A131093 - cp's OEIS frontend

A096777 a(n) = a(n-1) + Sum_{k=1..n-1}(a(k) mod 2), a(1) = 1.

1, 2, 3, 5, 8, 11, 15, 20, 25, 31, 38, 45, 53, 62, 71, 81, 92, 103, 115, 128, 141, 155, 170, 185, 201, 218, 235, 253, 272, 291, 311, 332, 353, 375, 398, 421, 445, 470, 495, 521, 548, 575, 603, 632, 661, 691, 722, 753, 785, 818, 851, 885, 920, 955, 991, 1028

Offset: 1

Reinhard Zumkeller, Jul 09 2004

a(n) = a(n-1) + (number of odd terms so far in the sequence). Example: 15 is 11 + 4 odd terms so far in the sequence (they are 1,3,5,11). See A007980 for the same construction with even integers. - Eric Angelini, Aug 05 2007

A016789 and A032766 give positions where even and odd terms occur; a(3*n)=A056106(n); a(3*n-1)=A077588(n); a(3*n-2)=A056108(n). - Reinhard Zumkeller, Dec 29 2007

			G.f. = x + 2*x^2 + 3*x^3 + 5*x^4 + 8*x^5 + 11*x^6 + 15*x^7 + 20*x^8 + ... - _Michael Somos_, Apr 18 2020

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
J.-L. Baril, T. Mansour, A. Petrossian, Equivalence classes of permutations modulo excedances, 2014.
Eric Weisstein's World of Mathematics, Odd Number
Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).

Cf. A004396, A007980, A016789, A032766, A056106, A056108, A061347, A077588, A097602, A131093.

a096777 n = a096777_list !! (n-1)
a096777_list = 1 : zipWith (+) a096777_list
                               (scanl1 (+) (map (`mod` 2) a096777_list))
-- Reinhard Zumkeller, Mar 11 2014

[Floor((n-2)^2/3)+n: n in [1..60]]; // Vincenzo Librandi, Dec 27 2015

A096777:=n->n + floor((n-2)^2/3); seq(A096777(n), n=1..100); # Wesley Ivan Hurt, Mar 06 2014

Table[n + Floor[(n-2)^2/3], {n, 100}] (* Wesley Ivan Hurt, Mar 06 2014 *)

a(n)=(n-2)^2\3+n \\ Charles R Greathouse IV, Mar 06 2014

a(n+1) - a(n) = A004396(n).
a(n) = floor(n/3) * (3*floor(n/3) + 2*(n mod 3) - 1) + n mod 3 + 0^(n mod 3). - Reinhard Zumkeller, Dec 29 2007
a(n) = floor((n-2)^2/3) + n. - Christopher Hunt Gribble, Mar 06 2014
G.f.: -x*(x^4+1) / ((x-1)^3*(x^2+x+1)). - Colin Barker, Mar 07 2014
Euler transform of finite sequence [2, 0, 1, 1, 0, 0, 0, -1]. - Michael Somos, Apr 18 2020
a(n) = (10 + 3*n*(n - 1) - A061347(n+1))/9. - Stefano Spezia, Sep 22 2022

A101135 a(1)=1. a(n) = a(n-1) + sum of the squares which are among the first (n-1) terms of the sequence.

Original entry on oeis.org

1, 2, 3, 4, 9, 23, 37, 51, 65, 79, 93, 107, 121, 256, 647, 1038, 1429, 1820, 2211, 2602, 2993, 3384, 3775, 4166, 4557, 4948, 5339, 5730, 6121, 6512, 6903, 7294, 7685, 8076, 8467, 8858, 9249, 9640, 10031, 10422, 10813, 11204, 11595, 11986, 12377, 12768

Offset: 1

Leroy Quet, Jun 07 2007

nonn

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for sequences related to sums of squares.

Cf. A097602.

Cf. A131093.

Block[{a = {1}}, Do[AppendTo[a, a[[n - 1]] + Total@ Select[a, IntegerQ@ Sqrt@ # &]], {n, 2, 46}]; a] (* Michael De Vlieger, Aug 16 2017 *)

vector(100, n, if(n==1, a_n_1=sum_=1, a_n_1+=sum_; if(ispolygonal(a_n_1, 4), sum_+=a_n_1)); a_n_1) \\ Colin Barker, Feb 20 2015

More terms from Reinhard Zumkeller, Jun 14 2007

A374602 Array of successive integer solutions to sqrt((d-c)b^2 + c(b+1)^2) for nonsquare integers d >= 2 (d=A000037(n) for n >= 1), where b and c are positive integers and c < d, read by antidiagonals.

Original entry on oeis.org

5, 29, 3, 169, 11, 5, 985, 41, 13, 3, 5741, 153, 34, 7, 4, 33461, 571, 89, 18, 5, 10, 195025, 2131, 233, 29, 11, 11, 4, 1136689, 7953, 610, 69, 28, 23, 5, 7, 6625109, 29681, 1597, 178, 62, 58, 13, 8, 6, 38613965, 110771, 4181, 287, 79, 338, 14, 13, 22, 4

Offset: 1

Charles L. Hohn, Jul 13 2024

T(n,k) is the diagonal lengths of increasingly nearly regular d-dimensional Pythagorean hyperrectangles.
Each row n divides into equal length, geometrically periodic subsequences, each with its own subsequence period length (A377290) and geometric growth factor (A377291); it is conjectured that this is the case for all n, and that all solutions conform as such and that there are no solutions that do not, but these are not proven.
It is also not known if there is an algorithm for generating values for all rows other than testing all possible values for a row until a subsequence pattern emerges.
Square d produce solutions following a different pattern, shown as A375336.

			n=row index; d=nonsquare integer of index n (A000037(n)):
 n    d   T(n,k)
---+----+-------------------------------------------------------------
 1 |  2 |  5, 29, 169, 985, 5741, 33461, 195025, 1136689, 6625109, ...
 2 |  3 |  3, 11,  41, 153,  571,  2131,   7953,   29681,  110771, ...
 3 |  5 |  5, 13,  34,  89,  233,   610,   1597,    4181,   10946, ...
 4 |  6 |  3,  7,  18,  29,   69,   178,    287,     683,    1762, ...
 5 |  7 |  4,  5,  11,  28,   62,    79,    175,     446,     988, ...
 6 |  8 | 10, 11,  23,  58,  338,   373,    781,    1970,   11482, ...
 7 | 10 |  4,  5,  13,  14,   25,    62,    111,     148,     185, ...
 8 | 11 |  7,  8,  13,  32,   57,   139,    158,     259,     638, ...
 9 | 12 |  6, 22,  39,  69,   82,   125,    306,     543,    1142, ...
10 | 13 |  4,  5,   7,  17,   30,    43,     53,      76,     185, ...
11 | 14 |  9, 11,  14,  19,   46,    81,    267,     329,     418, ...
12 | 15 |  6, 10,  21,  23,   30,    39,     94,     165,     362, ...
13 | 17 | 25, 27,  34,  41,   98,   171,    260,    1649,    1779, ...
14 | 18 |  6, 13,  15,  18,   21,    50,     87,     132,     198, ...
15 | 19 |  5,  7,   8,   9,   11,    31,     34,      37,      56, ...
16 | 20 | 10, 26,  68, 125,  159,   178,    197,     466,     807, ...
17 | 21 |  6,  9,  12,  13,   14,    33,     57,      86,     134, ...
18 | 22 |  5,  7,   8,  17,   18,    19,     31,      64,      77, ...
19 | 23 | 16, 19,  27,  28,   29,    68,    117,     176,     764, ...
20 | 24 |  6,  9,  11,  14,   36,    39,     57,      58,      59, ...
...
sqrt((2-1)*1^2 + 1*(1+1)^2) = sqrt(5) -> not an integer so not included.
sqrt((2-1)*3^2 + 1*(3+1)^2) = 5 -> T(1,1).
sqrt((2-1)*20^2 + 1*(20+1)^2) = 29 -> T(1,2).
sqrt((3-2)*1^2 + 2*(1+1)^2) = 3 -> T(2,1).
sqrt((6-2)*7^2 + 2*(7+1)^2) = 18 -> T(4,3).

Row 1 is A001653 starting at n=2.
Row 2 is A079935 starting at n=2.
Bisection of row 2 starting with the first term is A189356 starting at n=1.
Bisection of row 2 starting with the second term is A122769 starting at n=2.
Row 3 is A001519 starting at n=3.
Bisection of row 3 starting with the first term is A033889 starting at n=1.
Bisection of row 3 starting with the second term is A033891 starting at n=1.
Row 4 is A131093 starting at n=3.
Cf. A000037, A373666.
Cf. A377290, A377291, A375336.

row(n, c)=my(v=List(), d=n+floor(sqrt(n)+1/2) /* d=A000037(n) */, t=ceil(sqrt(d))); while(#v

T(n, 1) = A373666(A000037(n)).

A381149 a(1) = 2, a(2) = 3; thereafter, a(n) = a(n-1) + sum of prior prime terms.

Original entry on oeis.org

2, 3, 8, 13, 31, 80, 129, 178, 227, 503, 1282, 2061, 2840, 3619, 4398, 5177, 5956, 6735, 7514, 8293, 17365, 26437, 61946, 97455, 132964, 168473, 203982, 239491, 275000, 310509, 346018, 381527, 798563, 1215599, 1632635, 2049671, 2466707, 5350450, 8234193, 11117936

Offset: 1

James C. McMahon, Feb 15 2025

nonn

			For n=5, a(5) = a(4) + sum of prior primes = 13 + (2 + 3 + 13) = 31, so that 13 is counted twice.

James C. McMahon, Table of n, a(n) for n = 1..10000

Cf. A101135, A119746, A131093, A381150.

Nest[Append[#,#[[-1]]+Total[Select[#,PrimeQ]]]&,{2,3},38]

from sympy import isprime
from itertools import count, islice
def agen(): # generator of terms
    yield from [2, 3]
    primesum, an = 5, 3
    while True:
        an += primesum
        if isprime(an): primesum += an
        yield an
print(list(islice(agen(), 40))) # Michael S. Branicky, Feb 19 2025

A381150 a(0) = 1, a(1) = 2, a(2) = 3; thereafter, a(n) = a(n-1) + (sum of prior prime terms or whose negatives are prime) - (sum of prior composite terms or whose negatives are composite).

Original entry on oeis.org

1, 2, 3, 8, 5, 7, 16, 9, -7, -30, -23, -39, -16, 23, 85, 62, -23, -131, -370, -239, -347, -802, -455, 347, 1496, 1149, -347, -2190, -1843, 347, 2884, 2537, -347, -3578, -3231, 347, 4272, 3925, -347, -4966, -4619, 347, 5660, 5313, -347, -6354, -6007, -11667, -5660

Offset: 0

James C. McMahon, Feb 15 2025

sign

			For n=5, a(5) = 5 + (2 + 3 + 5) - 8 = 7.
For n=9, a(9) = -7 + (2 + 3 + 5 + 7 -7) - (8 + 16 + 9) = -7 + 10 - 33 = -30

Michael De Vlieger, Table of n, a(n) for n = 0..10003 (a(n) for n = 1..1000 from James C. McMahon)
Michael De Vlieger, Scatterplot of m*log_10(m*a(n)), n = 1..2^10, where m = 1 of a(n) > 0 (shown in green) and m = -1 if a(n) < 0 (shown in red).
Michael De Vlieger, Scatterplot of m*log_10(m*a(n)), n = 1..2^16, where m = 1 of a(n) > 0 (shown in green) and m = -1 if a(n) < 0 (shown in red).

Cf. A000040, A002808, A101135, A119746, A131093, A381149.

b:= proc(n) option remember; `if`(n<1, 0, b(n-1)+(t->
     `if`(isprime(abs(t)), t, `if`(abs(t)>1, -t, 0)))(a(n)))
    end:
a:= proc(n) option remember; `if`(n<3, n+1, a(n-1)+b(n-1)) end:
seq(a(n), n=0..48);  # Alois P. Heinz, Feb 15 2025

Nest[Append[#,#[[-1]]+Total[Select[#,PrimeQ]]-Total[Select[#,CompositeQ]]]&,{1,2,3},46]

cp's OEIS Frontend

A096777 a(n) = a(n-1) + Sum_{k=1..n-1}(a(k) mod 2), a(1) = 1.

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A101135 a(1)=1. a(n) = a(n-1) + sum of the squares which are among the first (n-1) terms of the sequence.

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A374602 Array of successive integer solutions to sqrt((d-c)*b^2 + c*(b+1)^2) for nonsquare integers d >= 2 (d=A000037(n) for n >= 1), where b and c are positive integers and c < d, read by antidiagonals.

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Formula

A381149 a(1) = 2, a(2) = 3; thereafter, a(n) = a(n-1) + sum of prior prime terms.

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A381150 a(0) = 1, a(1) = 2, a(2) = 3; thereafter, a(n) = a(n-1) + (sum of prior prime terms or whose negatives are prime) - (sum of prior composite terms or whose negatives are composite).

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Author

Keywords

Examples

Links

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Programs

Maple

Mathematica

A374602 Array of successive integer solutions to sqrt((d-c)b^2 + c(b+1)^2) for nonsquare integers d >= 2 (d=A000037(n) for n >= 1), where b and c are positive integers and c < d, read by antidiagonals.