A243578 -id:A243578 - cp's OEIS frontend

A243577 Integers of the form 8k+7 that can be written as a sum of four distinct 'almost consecutive' squares.

39, 71, 87, 119, 191, 255, 287, 351, 471, 567, 615, 711, 879, 1007, 1071, 1199, 1415, 1575, 1655, 1815, 2079, 2271, 2367, 2559, 2871, 3095, 3207, 3431, 3791, 4047, 4175, 4431, 4839, 5127, 5271, 5559, 6015, 6335, 6495, 6815, 7319, 7671, 7847, 8199, 8751, 9135, 9327, 9711

Offset: 1

Walter Kehowski, Jun 08 2014

By Lagrange's Four Square Theorem, any integer n of the form 8k+7 (A004771) can be written as sum of no fewer than four squares. The initial terms 7,15,23,31 are the generating set for A004771 in the sense that if n = a^2 + b^2+ c^2 + d^2, then (a mod 4)^2 + (b mod 4)^2 + (c mod 4)^2 + (d mod 4)^2 is one of 7,15,23,31.
From now on assume that n is of the form 8k+7 and is a sum of distinct squares a,b,c,d, sorted.
We say that [a,b,c,d] is almost consecutive if the differences b-a, c-b, d-c are 1 or 2. The generating set for this sequence is
   39, [1,2,3,5], with gap pattern 112,
   71, [1,3,5,6], with gap pattern 221,
   87, [2,3,5,7], with gap pattern 122,
  119, [3,5,6,7], with gap pattern 211,
in the sense that adding [4*i,4*i,4*i,4*i], i >= 0, preserves the gap pattern. It should be noted that the four generators are all obtainable from [1,1,2,3] or [1,2,3,3] by addition of suitable vectors. Let's write it out:
  [1,2,3,5] = [1,1,2,3] + [4,0,0,0] or
  [1,2,3,5] = [1,1,2,3] + [0,4,0,0]
  [1,3,5,6] = [1,1,2,3] + [0,4,4,0]
  [2,3,5,7] = [1,2,3,3] + [4,0,4,0] or
  [2,3,5,7] = [1,2,3,3] + [4,0,0,4]
  [3,5,6,7] = [1,2,3,3] + [4,4,4,0] or
  [3,5,6,7] = [1,2,3,3] + [4,4,0,4].
There are generators for other gap patterns, but the minimal gap patterns are of the most interest. - Walter Kehowski, Jul 07 2014

			For n=1, a(n) = 4*1^2 + 14*1 + 21 =  39 and  39 = 1^2 + 2^2 + 3^2 + 5^2.
For n=2, a(n) = 4*2^2 + 14*2 + 27 =  39 and  71 = 1^2 + 3^2 + 5^2 + 6^2.
For n=3, a(n) = 4*3^2 + 10*3 + 21 =  87 and  87 = 2^2 + 3^2 + 5^2 + 7^2.
For n=4, a(n) = 4*4^2 + 10*4 + 15 = 119 and 119 = 3^2 + 5^2 + 6^2 + 7^2.

Walter Kehowski, Table of n, a(n) for n = 1..1728
J. O. Sizemore, Lagrange's Four Square Theorem
R. C. Vaughan, Lagrange's Four Square Theorem
Eric Weisstein's World of Mathematics, Lagrange's Four-Square Theorem
Wikipedia, Lagrange's four-square theorem
Index entries for linear recurrences with constant coefficients, signature (3,-5,7,-7,5,-3,1).

Cf. A004771, A008586, A016813, A016825, A243578, A243579, A243580, A243581, A243582.

A243577 := proc(n::posint)
   if n mod 4 = 1 then
      return [4*n^2+14*n+21, [n,n+1,n+2,n+4]]
   elif n mod 4 = 2 then
      return [4*n^2+14*n+27, [n-1,n+1,n+3,n+4]]
   elif n mod 4 = 3 then
      return [4*n^2+10*n+21, [n-1,n,n+2,n+4]]
   else
      return [4*n^2+10*n+15, [n-1,n+1,n+2,n+3]]
   fi;
end:
# Walter A. Kehowski, Jun 08 2014

Rest@ CoefficientList[Series[-x (15 x^6 - 30 x^5 + 45 x^4 - 60 x^3 + 69 x^2 - 46 x + 39)/((x - 1)^3*(x^2 + 1)^2), {x, 0, 48}], x] (* Michael De Vlieger, Feb 19 2019 *)
LinearRecurrence[{3,-5,7,-7,5,-3,1},{39,71,87,119,191,255,287},50] (* Harvey P. Dale, Jul 05 2021 *)

Vec(-x*(15*x^6-30*x^5+45*x^4-60*x^3+69*x^2-46*x+39)/((x-1)^3*(x^2+1)^2) + O(x^100)) \\ Colin Barker, Jun 09 2014

If n mod 4 = 1, then a(n) = 4*n^2 + 14*n + 21.
If n mod 4 = 2, then a(n) = 4*n^2 + 14*n + 27.
If n mod 4 = 3, then a(n) = 4*n^2 + 10*n + 21.
If n mod 4 = 0, then a(n) = 4*n^2 + 10*n + 15.
a(n) = -3*(-7 + (-i)^n+i^n) - (1-i)*((-6-6*i) + (-i)^n + i*i^n)*n + 4*n^2 where i=sqrt(-1). - Colin Barker, Jun 09 2014
G.f.: -x*(15*x^6 - 30*x^5 + 45*x^4 - 60*x^3 + 69*x^2 - 46*x + 39) / ((x-1)^3*(x^2+1)^2). - Colin Barker, Jun 09 2014

A243579 Integers of the form 8k+7 that can be written as a sum of four distinct squares of the form m, m+2, m+4, m+5, where m == 1 (mod 4).

Original entry on oeis.org

71, 255, 567, 1007, 1575, 2271, 3095, 4047, 5127, 6335, 7671, 9135, 10727, 12447, 14295, 16271, 18375, 20607, 22967, 25455, 28071, 30815, 33687, 36687, 39815, 43071, 46455, 49967, 53607, 57375, 61271, 65295, 69447, 73727, 78135, 82671, 87335, 92127, 97047, 102095, 107271, 112575, 118007, 123567, 129255, 135071, 141015, 147087

Offset: 1

Walter Kehowski, Jun 08 2014

If n is of the form 8k+7 and n = a^2+b^2+c^2+d^2 with gap pattern 221, then [a,b,c,d] = [1,3,5,6]+[4*i,4*i,4*i,4*i] for i>=0.

			a(5) = 64*5^2-8*5+15 = 1575 and m = 4*5-3 = 17 so 1575 = 17^2+19^2+21^2+22^2.

Walter Kehowski, Table of n, a(n) for n = 1..20737
J. Owen Sizemore, Lagrange's Four Square Theorem
R. C. Vaughan, Lagrange's Four Square Theorem
Eric Weisstein's World of Mathematics, Lagrange's Four-Square Theorem
Wikipedia, Lagrange's four-square theorem
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A008586, A016813, A016825, A004767, A243577, A243578, A243579, A243580, A243581, A243582.

[64*n^2-8*n+15 : n in [1..50]]; // Wesley Ivan Hurt, Nov 28 2021

A243579 := proc(n::posint) return 64*n^2-8*n+15 end;

Vec(-x*(15*x^2+42*x+71)/(x-1)^3 + O(x^100)) \\ Colin Barker, Sep 13 2015

a(n) = 64*n^2-8*n+15.
From Colin Barker, Sep 13 2015: (Start)
a(n) = 3*a(n-1)-3*a(n-2)+a(n-3) for n>3.
G.f.: x*(15*x^2+42*x+71) / (1-x)^3. (End)

A243580 Integers of the form 8k + 7 that can be written as a sum of four distinct squares of the form m, m + 1, m + 3, m + 5, where m == 2 (mod 4).

Original entry on oeis.org

87, 287, 615, 1071, 1655, 2367, 3207, 4175, 5271, 6495, 7847, 9327, 10935, 12671, 14535, 16527, 18647, 20895, 23271, 25775, 28407, 31167, 34055, 37071, 40215, 43487, 46887, 50415, 54071, 57855, 61767, 65807, 69975, 74271, 78695, 83247, 87927, 92735, 97671, 102735, 107927, 113247, 118695, 124271, 129975, 135807, 141767, 147855

Offset: 1

Walter Kehowski, Jun 08 2014

If n is of the form 8k + 7 and n = a^2 + b^2 + c^2 + d^2 where [a, b, c, d] has gap pattern 122, then [a, b, c, d] = [2, 3, 5, 7] + [4*i, 4*i, 4*i, 4*i], i >= 0.

			a(5) = 64*n^2 + 8*5 + 15 = 1655 and m = 4*5 - 2 = 18 so 1655 = 18^2 + 19^2 + 21^2 + 23^2.

Walter Kehowski, Table of n, a(n) for n = 1..20737
J. Owen Sizemore, Lagrange's Four Square Theorem (web.archive)
R. C. Vaughan, Lagrange's Four Square Theorem
Eric Weisstein's World of Mathematics, Lagrange's Four-Square Theorem
Wikipedia, Lagrange's four-square theorem
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A008586, A016813, A016825, A004767, A243577, A243578, A243579, A243580, A243581, A243582

A243580 := proc(n::posint) return 64*n^2+8*n+15 end;

Table[64n^2 + 8n + 15, {n, 50}] (* Alonso del Arte, Jun 08 2014 *)
LinearRecurrence[{3,-3,1},{87,287,615},50] (* Harvey P. Dale, Mar 27 2019 *)

Vec(-x*(15*x^2+26*x+87)/(x-1)^3 + O(x^100)) \\ Colin Barker, Sep 13 2015

a(n) = 64*n^2 + 8*n + 15.
From Colin Barker, Sep 13 2015: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>3.
G.f.: -x*(15*x^2+26*x+87) / (x-1)^3. (End)

A243581 Integers of the form 8k + 7 that can be written as a sum of four distinct squares of the form m, m + 2, m + 3, m + 4, where m == 2 (mod 4).

Original entry on oeis.org

119, 351, 711, 1199, 1815, 2559, 3431, 4431, 5559, 6815, 8199, 9711, 11351, 13119, 15015, 17039, 19191, 21471, 23879, 26415, 29079, 31871, 34791, 37839, 41015, 44319, 47751, 51311, 54999, 58815, 62759, 66831, 71031, 75359, 79815

Offset: 1

Walter Kehowski, Jun 08 2014

If n is of the form 8k + 7 and n = a^2 + b^2 + c^2 + d^2 where [a, b, c, d] has gap pattern 122, then [a, b, c, d] = [3, 5, 6, 7] + [4*i, 4*i, 4*i, 4*i], i >= 0.

			a(5) = 64*5^2 + 40*5 + 15 = 1815 and 4*5 - 1 = 19 so 1815 = 19^2 + 21^2 + 22^2 + 23^2.

Walter Kehowski, Table of n, a(n) for n = 1..20737
J. Owen Sizemore, Lagrange's Four Square Theorem
R. C. Vaughan, Lagrange's Four Square Theorem
Eric Weisstein's World of Mathematics, Lagrange's Four-Square Theorem
Wikipedia, Lagrange's four-square theorem
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A008586, A016813, A016825, A004767, A243577, A243578, A243579, A243580, A243581, A243582.

[ 64*n^2 + 40*n + 15 : n in [1..50] ]; // Wesley Ivan Hurt, Jun 11 2014

A243581 := proc(n::posint) return 64*n^2+40*n+15 end;

Table[64n^2 + 40n + 15, {n, 50}] (* Alonso del Arte, Jun 08 2014 *)
LinearRecurrence[{3,-3,1},{119,351,711},50] (* Harvey P. Dale, Jul 23 2014 *)

Vec(-x*(15*x^2-6*x+119)/(x-1)^3 + O(x^100)) \\ Colin Barker, Jun 09 2014

a(n) = 64*n^2 + 40*n + 15.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Colin Barker, Jun 09 2014
G.f.: -x*(15*x^2-6*x+119) / (x-1)^3. - Colin Barker, Jun 09 2014

A243582 Integers of the form 8k+7 (A004771) that cannot be written as sum of four distinct squares.

Original entry on oeis.org

7, 15, 23, 31, 47, 55, 103

Offset: 1

Walter Kehowski, Jun 08 2014

			a(6) = 55 since 55 == 7 (mod 8) and all its representations as a sum of squares have duplicates, namely, 55=1^2+1^2+2^2+7^2, 55=1^2+2^2+5^2+5^2, 55=1^2+3^2+3^2+6^2.

J. O. Sizemore, Lagrange's Four Square Theorem
R. C. Vaughan, Lagrange's four-square theorem
Eric Weisstein's World of Math, Lagrange's Four-Square Theorem
Wikipedia, Lagrange's four-square theorem

Cf. A001948, A004771, A008586, A016813, A016825, A004767, A243577, A243578, A243579, A243580, A243581.

cp's OEIS Frontend

A243577 Integers of the form 8k+7 that can be written as a sum of four distinct 'almost consecutive' squares.

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A243579 Integers of the form 8k+7 that can be written as a sum of four distinct squares of the form m, m+2, m+4, m+5, where m == 1 (mod 4).

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A243580 Integers of the form 8k + 7 that can be written as a sum of four distinct squares of the form m, m + 1, m + 3, m + 5, where m == 2 (mod 4).

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A243581 Integers of the form 8k + 7 that can be written as a sum of four distinct squares of the form m, m + 2, m + 3, m + 4, where m == 2 (mod 4).

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Magma

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A243582 Integers of the form 8k+7 (A004771) that cannot be written as sum of four distinct squares.

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