A101135 -id:A101135 - cp's OEIS frontend

A097602 a(n+1) = a(n) + number of squares so far; a(1) = 1.

1, 2, 3, 4, 6, 8, 10, 12, 14, 16, 19, 22, 25, 29, 33, 37, 41, 45, 49, 54, 59, 64, 70, 76, 82, 88, 94, 100, 107, 114, 121, 129, 137, 145, 153, 161, 169, 178, 187, 196, 206, 216, 226, 236, 246, 256, 267, 278, 289, 301, 313, 325, 337, 349, 361, 374, 387, 400, 414, 428

Offset: 1

Reinhard Zumkeller, Aug 30 2004

nonn

Conjecture: a(n) = m^2 iff m mod 3 > 0.

a(n) is a square iff n is congruent to {1, 4} mod 9. - Vladeta Jovovic, Aug 30 2004

			a(2) = a(1) + #{1} = 1 + 1 = 2;
a(3) = a(2) + #{1} = 2 + 1 = 3;
a(4) = a(3) + #{1} = 3 + 1 = 4;
a(5) = a(4) + #{1,4} = 4 + 2 = 6;
a(6) = a(5) + #{1,4} = 6 + 2 = 8;
a(7) = a(6) + #{1,4} = 8 + 2 = 10;
a(8) = a(7) + #{1,4} = 10 + 2 = 12;
a(9) = a(8) + #{1,4} = 12 + 2 = 14;
a(10) = a(9) + #{1,4} = 14 + 2 = 16;
a(11) = a(10) + #{1,4,16} = 16 + 3 = 19;
a(12) = a(11) + #{1,4,16} = 19 + 3 = 22.

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

Cf. A101135, A096777, A131073.

a:=[1,2,3,4,6,8,10,12,14,16,19];; for n in [12..70] do a[n]:= 2*a[n-1]-a[n-2]+a[n-9]-2*a[n-10]+a[n-11]; od; a; # G. C. Greubel, Jan 14 2019

a097602 n = a097602_list !! (n-1)
a097602_list = 1 : f 1 1 where
   f x c = y : f y (c + a010052 y) where y = x + c
-- Reinhard Zumkeller, Nov 15 2011

m:=70; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( x*(1+x^4-x^9+x^10)/((1+x+x^2)*(1+x^3+x^6)*(1-x)^3) )); // G. C. Greubel, Jan 14 2019

LinearRecurrence[{2,-1,0,0,0,0,0,0,1,-2,1}, {1,2,3,4,6,8,10,12,14,16,19}, 70] (* G. C. Greubel, Jan 14 2019 *)

my(x='x+O('x^70)); Vec(x*(1+x^4-x^9+x^10)/((1+x+x^2)*(1+x^3+x^6)*(1-x)^3)) \\ G. C. Greubel, Jan 14 2019

a=(x*(1+x^4-x^9+x^10)/((1+x+x^2)*(1+x^3+x^6)*(1-x)^3)).series(x, 70).coefficients(x, sparse=False); a[1:] # G. C. Greubel, Jan 14 2019

a(9*n+1) = (3*n+1)^2; a(9*n+4) = (3*n+2)^2. - Vladeta Jovovic, Aug 30 2004
G.f.: x*(1+x^4-x^9+x^10)/((1+x+x^2)*(1+x^3+x^6)*(1-x)^3). - Vladeta Jovovic, Aug 30 2004
a(n+1) = a(n) + Sum_{k=1..n} A010052(a(k)). - Reinhard Zumkeller, Nov 15 2011

A131093 a(1)=1, a(n) = a(n-1) + sum of odd numbers which are among the first (n-1) terms of the sequence.

Original entry on oeis.org

1, 2, 3, 7, 18, 29, 69, 178, 287, 683, 1762, 2841, 6761, 17442, 28123, 66927, 172658, 278389, 662509, 1709138, 2755767, 6558163, 16918722, 27279281, 64919121, 167478082, 270037043, 642633047, 1657862098, 2673091149

Offset: 1

Reinhard Zumkeller, Jun 14 2007

nonn

a(n) mod 2 = A011655(n+1);

a(n) = a(n-1) + Sum(a(k)*(a(k) mod 2): 1<=k

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Odd Number

Cf. A096777, A101135.

nxt[{ot_,a_}]:=Module[{x=a+ot},{If[OddQ[x],ot+x,ot],x}]; NestList[nxt,{1,1},30][[All,2]] (* Harvey P. Dale, Dec 18 2020 *)

Empirical g.f.: -x*(x-1)*(x^4+3*x^3+6*x^2+3*x+1) / (x^6-10*x^3+1). - Colin Barker, Mar 29 2013

A100963 a(1)=1. a(n) = a(n-1) + sum of the triangular numbers which are among the first (n-1) terms of the sequence.

Original entry on oeis.org

1, 2, 3, 7, 11, 15, 34, 53, 72, 91, 201, 311, 421, 531, 641, 751, 861, 1832, 2803, 3774, 4745, 5716, 6687, 7658, 8629, 9600, 10571, 11542, 12513, 13484, 14455, 15426, 16397, 17368, 18339, 19310, 20281, 21252, 22223, 23194, 24165, 25136, 26107, 27078, 28049

Offset: 1

Leroy Quet, Jun 07 2007

nonn

Colin Barker, Table of n, a(n) for n = 1..1000

Cf. A129895.

Cf. A101135, A255260, A255261.

Nest[Append[#, Last@ # + Total@ Select[#, IntegerQ@ First@ Values@ Last@ Solve[x (x + 1)/2 == #, x] &]] &, {1}, 44] (* Michael De Vlieger, Sep 24 2017 *)

vector(100, n, if(n==1, a_n_1=sum_=1, a_n_1+=sum_; if(ispolygonal(a_n_1, 3), sum_+=a_n_1)); a_n_1) \\ Colin Barker, Feb 19 2015

More terms from Robert Gerbicz, Jun 10 2007

A255260 a(n) = a(n-1) + sum of the pentagonal numbers which are among the first (n-1) terms of the sequence, with a(1)=1.

Original entry on oeis.org

1, 2, 3, 4, 5, 11, 17, 23, 29, 35, 76, 117, 275, 433, 591, 749, 907, 1065, 1223, 1381, 1539, 1697, 1855, 2013, 2171, 2329, 2487, 2645, 2803, 2961, 3119, 3277, 3435, 3593, 3751, 3909, 4067, 4225, 4383, 4541, 4699, 4857, 5015, 5173, 5331, 5489, 5647, 5805

Offset: 1

Colin Barker, Feb 19 2015

nonn

			a(7) = 17 because a(6) = 11, pentagonal numbers already in the sequence are 1 and 5, and 11+1+5 = 17.

Colin Barker, Table of n, a(n) for n = 1..1000

Cf. A100963, A101135, A255261.

vector(100, n, if(n==1, a_n_1=sum_=1, a_n_1+=sum_; if(ispolygonal(a_n_1, 5), sum_+=a_n_1)); a_n_1)

A255261 a(n) = a(n-1) + sum of the hexagonal numbers which are among the first (n-1) terms of the sequence, with a(1)=1.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 13, 20, 27, 34, 41, 48, 55, 62, 69, 76, 83, 90, 97, 104, 111, 118, 125, 132, 139, 146, 153, 313, 473, 633, 793, 953, 1113, 1273, 1433, 1593, 1753, 1913, 2073, 2233, 2393, 2553, 2713, 2873, 3033, 3193, 3353, 3513, 3673, 3833, 3993, 4153

Offset: 1

Colin Barker, Feb 20 2015

nonn

			a(8) = 20 because a(7) = 13, hexagonal numbers already in the sequence are 1 and 6, and 13+1+6 = 20.

Colin Barker, Table of n, a(n) for n = 1..1000

Cf. A100963, A101135, A255260.

vector(100, n, if(n==1, a_n_1=sum_=1, a_n_1+=sum_; if(ispolygonal(a_n_1, 6), sum_+=a_n_1)); a_n_1)

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]

Showing 1-7 of 7 results.

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A097602 a(n+1) = a(n) + number of squares so far; a(1) = 1.

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A131093 a(1)=1, a(n) = a(n-1) + sum of odd numbers which are among the first (n-1) terms of the sequence.

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A100963 a(1)=1. a(n) = a(n-1) + sum of the triangular numbers which are among the first (n-1) terms of the sequence.

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A255260 a(n) = a(n-1) + sum of the pentagonal numbers which are among the first (n-1) terms of the sequence, with a(1)=1.

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A255261 a(n) = a(n-1) + sum of the hexagonal numbers which are among the first (n-1) terms of the sequence, with a(1)=1.

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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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