cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A010003 a(0) = 1, a(n) = 11*n^2 + 2 for n>0.

Original entry on oeis.org

1, 13, 46, 101, 178, 277, 398, 541, 706, 893, 1102, 1333, 1586, 1861, 2158, 2477, 2818, 3181, 3566, 3973, 4402, 4853, 5326, 5821, 6338, 6877, 7438, 8021, 8626, 9253, 9902, 10573, 11266, 11981, 12718, 13477, 14258, 15061, 15886, 16733, 17602, 18493, 19406
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Apart from the first term, numbers of the form (r^2+2*s^2)*n^2+2 = (r*n)^2+(s*n-1)^2+(s*n+1)^2: in this case is r=3, s=1. After 13, all terms are in A000408. - Bruno Berselli, Feb 06 2012

Links

Crossrefs

Cf. A206399.

Programs

Formula

G.f.: (1+x)*(1+9*x+x^2)/(1-x)^3. - Bruno Berselli, Feb 06 2012
E.g.f.: (x*(x+1)*11+2)*e^x-1. - Gopinath A. R., Feb 14 2012
Sum_{n>=0} 1/a(n) = 3/4+sqrt(22)/44*Pi*coth( Pi*sqrt(22)/11) = 1.134242719070... - R. J. Mathar, May 07 2024

Extensions

More terms from Bruno Berselli, Feb 06 2012

A010007 a(0) = 1, a(n) = 17*n^2 + 2 for n>0.

Original entry on oeis.org

1, 19, 70, 155, 274, 427, 614, 835, 1090, 1379, 1702, 2059, 2450, 2875, 3334, 3827, 4354, 4915, 5510, 6139, 6802, 7499, 8230, 8995, 9794, 10627, 11494, 12395, 13330, 14299, 15302, 16339, 17410, 18515, 19654, 20827, 22034, 23275, 24550, 25859, 27202, 28579
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Apart from the first term, numbers of the form (r^2+2*s^2)*n^2+2 = (r*n)^2+(s*n-1)^2+(s*n+1)^2: in this case is r=3, s=2. After 1, all terms are in A000408. - Bruno Berselli, Feb 06 2012

Links

Crossrefs

Cf. A206399.

Programs

  • Magma
    [1] cat [17*n^2+2: n in [1..50]]; // Vincenzo Librandi, Aug 03 2015
  • Mathematica
    Join[{1}, 17 Range[41]^2 + 2] (* Bruno Berselli, Feb 06 2012 *)
Join[{1}, LinearRecurrence[{3, -3, 1}, {19, 70, 155}, 50]] (* Vincenzo Librandi, Aug 03 2015 *)

Formula

G.f.: (1+x)*(1+15*x+x^2)/(1-x)^3. - Bruno Berselli, Feb 06 2012
E.g.f. : (x*(x+1)*17+2)*e^x-1. - Gopinath A. R., Feb 14 2012
Sum_{n>=0} 1/a(n) = 3/4+sqrt(34)/68*Pi*coth(Pi*sqrt(34)/17) = 1.09001290652... - R. J. Mathar, May 07 2024
a(n) = A069130(n)+A069130(n+1). - R. J. Mathar, May 07 2024

Extensions

More terms from Bruno Berselli, Feb 06 2012

A010009 a(0) = 1, a(n) = 19*n^2 + 2 for n>0.

Original entry on oeis.org

1, 21, 78, 173, 306, 477, 686, 933, 1218, 1541, 1902, 2301, 2738, 3213, 3726, 4277, 4866, 5493, 6158, 6861, 7602, 8381, 9198, 10053, 10946, 11877, 12846, 13853, 14898, 15981, 17102, 18261, 19458, 20693, 21966, 23277, 24626, 26013, 27438, 28901, 30402, 31941
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Apart from the first term, numbers of the form (r^2+2*s^2)*n^2+2 = (r*n)^2+(s*n-1)^2+(s*n+1)^2: in this case is r=1, s=3. After 1, all terms are in A000408. - Bruno Berselli, Feb 06 2012

Links

Crossrefs

Cf. A206399.

Programs

  • Magma
    [1] cat [19*n^2+2: n in [1..50]]; // Vincenzo Librandi, Aug 03 2015
  • Mathematica
    Join[{1}, 19 Range[41]^2 + 2]  (* Harvey P. Dale, Feb 07 2011 *)
Join[{1}, LinearRecurrence[{3, -3, 1}, {21, 78, 173}, 50]] (* Vincenzo Librandi, Aug 03 2015 *)

Formula

G.f.: (1+x)*(1+17*x+x^2)/(1-x)^3. - Bruno Berselli, Feb 06 2012
E.g.f.: (x*(x+1)*19+2)*e^x-1. - Gopinath A. R., Feb 14 2012
Sum_{n>=0} 1/a(n) = 3/4 + sqrt(38)/76*Pi*coth(Pi*sqrt(38)/19) = 1.08111673149128.. - R. J. Mathar, May 07 2024
a(n) = A069132(n)+A069132(n+1). - R. J. Mathar, May 07 2024

A010012 a(0) = 1, a(n) = 22*n^2 + 2 for n>0.

Original entry on oeis.org

1, 24, 90, 200, 354, 552, 794, 1080, 1410, 1784, 2202, 2664, 3170, 3720, 4314, 4952, 5634, 6360, 7130, 7944, 8802, 9704, 10650, 11640, 12674, 13752, 14874, 16040, 17250, 18504, 19802, 21144, 22530, 23960, 25434, 26952, 28514, 30120, 31770, 33464, 35202, 36984
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

From Bruno Berselli, Feb 06 2012: (Start)
First trisection of A008259.
Apart from the first term, numbers of the form (r^2+2*s^2)*n^2+2 = (r*n)^2+(s*n-1)^2+(s*n+1)^2: in this case is r=2, s=3. After 1, all terms are in A000408. (End)

Links

Crossrefs

Cf. A206399.

Programs

  • Magma
    [1] cat [22*n^2+2: n in [1..50]]; // Vincenzo Librandi, Aug 03 2015
  • Mathematica
    Join[{1}, 22 Range[41]^2 + 2] (* Bruno Berselli, Feb 06 2012 *)
Join[{1},LinearRecurrence[{3,-3,1},{24,90,200},50]] (* Harvey P. Dale, Jul 20 2013 *)
CoefficientList[Series[(1 + x) (1 + 20 x + x^2)/(1 - x)^3, {x, 0, 50}], x] (* Vincenzo Librandi, Aug 03 2015 *)

Formula

G.f.: (1+x)*(1+20*x+x^2)/(1-x)^3. - Bruno Berselli, Feb 06 2012
E.g.f.: (x*(x+1)*22+2)*e^x-1. - Gopinath A. R., Feb 14 2012
Sum_{n>=0} 1/a(n) = 3/4 + sqrt(11)/44*Pi*coth( Pi/sqrt(11)) = 1.0706480516966... - R. J. Mathar, May 07 2024
a(n) = A069173(n)+A069173(n+1). - R. J. Mathar, May 07 2024

A010017 a(0) = 1, a(n) = 27*n^2 + 2 for n>0.

Original entry on oeis.org

1, 29, 110, 245, 434, 677, 974, 1325, 1730, 2189, 2702, 3269, 3890, 4565, 5294, 6077, 6914, 7805, 8750, 9749, 10802, 11909, 13070, 14285, 15554, 16877, 18254, 19685, 21170, 22709, 24302, 25949, 27650, 29405, 31214, 33077, 34994, 36965, 38990, 41069, 43202
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

From Bruno Berselli, Feb 06 2012: (Start)
First trisection of A005918.
Apart from the first term, numbers of the form (r^2+2*s^2)*n^2+2 = (r*n)^2+(s*n-1)^2+(s*n+1)^2: in this case is r=5, s=1.
After 1, all terms are in A000408. (End)

Links

Crossrefs

Cf. A206399.

Programs

  • Magma
    [1] cat [27*n^2+2: n in [1..50]]; // Vincenzo Librandi, Aug 03 2015
  • Mathematica
    Join[{1}, 27 Range[40]^2 + 2] (* Bruno Berselli, Feb 06 2012 *)
RecurrenceTable[{a[1]==29, a[2]==110, a[3]==245, a[n]== 3*a[n-1] - 3*a[n-2] + a[n-3]}, a, {n, 1,30}] (* G. C. Greubel, Aug 02 2015 *)
  • PARI
    first(m)=my(v=vector(m));for(i=1,m,v[i]=27*(i)^2+2);concat([1],v); /* Anders Hellström, Aug 02 2015 */

Formula

G.f.: (1+x)*(1+25*x+x^2)/(1-x)^3. - Bruno Berselli, Feb 06 2012
E.g.f.: (x*(x+1)*27+2)*e^x-1. - Gopinath A. R., Feb 14 2012
a(n) = 3*a(n-1)-3*a(n-2)+a(n-3), n>=4, a(1)=29, a(2)=110, a(3)=245. - G. C. Greubel, Aug 02 2015
Sum_{n>=0} 1/a(n) = 3/4+sqrt(6)/36*Pi*coth(Pi*sqrt(6)/9) = 1.0581468172342... - R. J. Mathar, May 07 2024

A104078 Numbers which are the sum of three nonzero squares and are also divisible by 31.

Original entry on oeis.org

62, 93, 155, 186, 217, 248, 310, 341, 372, 403, 434, 465, 558, 589, 620, 651, 682, 713, 744, 806, 837, 868, 899, 930, 961, 992, 1054, 1085, 1147, 1178, 1209, 1240, 1302, 1333, 1364, 1395, 1426, 1457, 1488, 1550, 1581, 1612, 1643, 1674, 1705, 1736, 1798
Offset: 1

Views

Author

Jun Mizuki (suzuki32(AT)sanken.osaka-u.ac.jp), Mar 02 2005

Keywords

Examples

			a(1)=62 because 62=1^2+5^2+6^2=2^2+3^2+7^2.
a(2)=93 because 93=2^2+5^2+8^2.
a(3)=155 because 155=5^2+7^2+9^2=3^2+5^2+11^2....

Links

Crossrefs

Cf. A000408.

A125112 Numbers which are not the sum of 3 nonzero squares, but which can be expressed as the product of two numbers that are the sum of 3 nonzero squares.

Original entry on oeis.org

63, 87, 135, 156, 159, 183, 207, 231, 252, 279, 303, 319, 327, 348, 351, 375, 399, 423, 444, 447, 471, 476, 495, 519, 540, 543, 551, 567, 572, 583, 591, 615, 624, 636, 639, 663, 671, 687, 700, 711, 732, 735, 759, 783, 807, 828, 831, 847, 855, 879, 903, 924
Offset: 1

Views

Author

Artur Jasinski, Nov 21 2006

Keywords

Comments

Intersection of A004214 with products of pairs of terms of A000408.

Examples

			a(2) = 87 = 3 * 29 = (1^2+1^2+1^2) * (4^2+3^2+2^2)
87 does not have a partition as a sum x^2+y^2+z^2 with x,y,z>0
63=3*21; 87=3*29; 135=3*45; 156=6*26; 572=22*26;

Links

Crossrefs

Cf. A000408 (sums of 3 nonzero squares), A004214 (not sums of 3 nonzero squares).

Programs

  • Maple
    isA000408 := proc(n) local a,b,c2 ; a:=1; while a^2A125112 := proc(n) local d,i; if isA000408(n) then RETURN(false) ; else d := numtheory[divisors](n) ; for i from 1 to nops(d) do if isA000408(op(i,d)) and isA000408(n/op(i,d)) then RETURN(true) ; fi ; od ; RETURN(false) ; fi ; end: for an from 1 to 1600 do if isA125112(an) then printf("%d,",an) ; fi ; od ; # R. J. Mathar, Nov 23 2006
  • Mathematica
    isA000408[n_] := Module[{a, b, c2}, a = 1; While[a^2 < n, b = 1; While[b <= a && a^2 + b^2 < n, c2 = n - a^2 - b^2; If[IntegerQ@Sqrt@c2, Return[True]]; b++]; a++]; Return[False]];
isA125112[n_] := Module[{d, i}, If[isA000408[n], Return[False], d = Divisors[n]; For[i = 1, i <= Length[d], i++, If[isA000408[d[[i]]] && isA000408[n/d[[i]]], Return[True]]]; Return[False]]];
Select[Range[1600], isA125112] (* Jean-François Alcover, Jul 22 2024, after R. J. Mathar *)

Extensions

Edited and extended by R. J. Mathar and Ray Chandler, Nov 23 2006

A164098 Numbers of the form m * (k_1^2 + k_2^2 + ... + k_m^2).

Original entry on oeis.org

1, 4, 9, 10, 16, 18, 20, 25, 26, 27, 28, 33, 34, 36, 40, 42, 48, 49, 50, 51, 52, 54, 55, 57, 58, 60, 63, 64, 65, 66, 68, 70, 72, 74, 76, 78, 80, 81, 82, 84, 85, 87, 88, 90, 91, 92, 95, 99, 100, 102, 104, 105, 106, 108, 110, 112, 114, 115, 116, 120, 121, 122, 123, 124, 125
Offset: 1

Views

Author

Jonas Wallgren, Aug 10 2009, Aug 17 2009

Keywords

Comments

From Franklin T. Adams-Watters, Aug 29 2009: (Start)
The k_i must all be positive integers.
Note that every integer > 33 is the sum of 5 positive squares, and for n > 5, every integer > n+13 is the sum of n positive squares. (End)
The complement of this sequence includes: A000040, A037074, A143206, 2 * A002145, and 3 * A094712. - Robert Israel, Jan 27 2025

Examples

			34 = 2*(4^2 + 1^2), 42 = 3*(3^2 + 2^2 + 1^2), thus 34 and 42 are in the sequence.

Links

Crossrefs

Cf. A000290, A000404, A000408, A000414, A047700, A111178. [From Franklin T. Adams-Watters, Aug 29 2009]
Cf. A000040, A002145, A037074, A094712, A143206.

Programs

  • Maple
    g:= proc(y,m)
  # can we write y as sum of m positive squares?
   option remember;
   local x;
   if y < m then return false fi;
   if m = 1 then return issqr(y) fi;
   if issqr(y-m+1) then return true fi;
   for x from 1 while x^2 + m-1 < y do
     if procname(y-x^2,m-1) then return true fi
   od;
   false
end proc:
filter:= proc(n)
  ormap(t -> g(n/t, t), numtheory:-divisors(n))
end proc:
select(filter, [$1..1000]); # Robert Israel, Jan 26 2025
  • PARI
    issumsqs(n,k) = if(n<=0||k<=0,return(k==0&&n==0)); forstep(j=sqrtint(n),max(sqrtint(n\k),1),-1,if(issumsqs(n-j^2,k-1),return(1)));0
isa(n)=local(ds);ds=divisors(n);for(k=1,(#ds+1)\2,if(issumsqs(n\ds[k],ds[k]),return(1)));0
for(n=1,200,if(isa(n),print1(n","))) \\ Franklin T. Adams-Watters, Aug 29 2009

Extensions

More terms from Franklin T. Adams-Watters, Aug 29 2009

A166265 Numbers of the form 1+x^2+y^2, x, y integers >= 1.

Original entry on oeis.org

3, 6, 9, 11, 14, 18, 19, 21, 26, 27, 30, 33, 35, 38, 41, 42, 46, 51, 53, 54, 59, 62, 66, 69, 73, 74, 75, 81, 83, 86, 90, 91, 98, 99, 101, 102, 105, 107, 110, 114, 117, 118, 123, 126, 129, 131, 137, 138, 146, 147, 149, 150, 154, 158, 161, 163, 165, 170, 171, 174, 179, 181, 182
Offset: 1

Views

Author

N. J. A. Sloane, Mar 05 2010

Keywords

Crossrefs

Cf. A000408, A000378, A005767, A169580, A166687.

Programs

  • Mathematica
    With[{nn=40},Take[Union[Total/@Tuples[Range[nn]^2,2]+1],2*nn]] (* Harvey P. Dale, Mar 12 2015 *)

A215537 Lowest k such that k is representable as both the sum of n and of n+1 nonzero squares.

Original entry on oeis.org

25, 17, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79
Offset: 1

Views

Author

Jon Perry, Aug 15 2012

Keywords

Examples

			25 = 5^2 = 3^2 + 4^2
17 = 4^2 + 1^2 = 3^2 + 2^2 + 2^2
12 = 2^2 + 2^2 + 2^2 = 3^2 + 1^2 + 1^2 + 1^2
after this just add 1^2 to both sides.

Crossrefs

Cf. A000290 (representable as sum of 1 square), A000404 (sum of 2 positive squares), A000408 (sum of 3 positive squares), A000414 (sum of 4 positive squares), A047700 (sum of 5 positive squares)

Programs

  • Maple
    # true if a is representable as a sum of n squares, each square >= m^2.
isRepnSqrsMin := proc(a,n,m)
    local mpr ;
    if a < n*m^2 then
        return false;
    end if;
    if n = 1 then
        if a>= m^2 and issqr(a) then
            true;
        else
            false;
        end if;
    else
        for mpr from m to a do
            if a-mpr^2 < 1 then
                return false;
            elif procname(a-mpr^2,n-1,mpr) then
                return true;
            end if;
        end do:
    end if;
end proc:
# true if a is representable as a sum of n positive squares.
isRepnSqrs := proc(a,n)
    isRepnSqrsMin(a,n,1) ;
end proc:
A215537 := proc(n)
    local k;
    for k from 1 do
        if isRepnSqrs(k,n) and isRepnSqrs(k,n+1) then
            return k;
        end if;
    end do:
end proc: # R. J. Mathar, Sep 11 2012
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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