A181318 -id:A181318 - cp's OEIS frontend

A377851 Smallest multiplier which can complete the square for n-polygonal numbers, together with a constant offset.

8, 1, 24, 8, 40, 3, 56, 16, 72, 5, 88, 24, 104, 7, 120, 32, 136, 9, 152, 40, 168, 11, 184, 48, 200, 13, 216, 56, 232, 15, 248, 64, 264, 17, 280, 72, 296, 19, 312, 80, 328, 21, 344, 88, 360, 23, 376, 96, 392, 25, 408, 104, 424, 27, 440, 112, 456, 29, 472

Offset: 3

Jonathan Dushoff, Nov 09 2024

This smallest multiplier is also the only multiplier that is relatively prime to the offset.
The n-polygonal numbers, indexed by x, are P(n,x) = (n-2)*(x-1)*x/2 + x = A139601(n-3,x).
S(x) = P(n,x)*a(n) + A181318(n-4) completes the square in that quadratic, ensuring S(x) is a square for all x.

			For n=7, the heptagonal numbers are h(x) = x*(5*x-3)/2 and with multiplier a(7) = 40 and offset A181318(7-4) = 9 become 40*h(x)+9 = (10*x - 3)^2.

Paolo Xausa, Table of n, a(n) for n = 3..10000
Wikipedia, Completing the square.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,2,0,0,0,-1).

Cf. A181318 (offsets).

Table[8*(n - 2)/GCD[n, 4]^2, {n, 3, 100}] (* Paolo Xausa, Dec 07 2024 *)

a(n) = 8*(n-2)/gcd(n,4)^2 \\ Andrew Howroyd, Nov 10 2024

a(n) = 8*(n-2)/gcd(n,4)^2. - Andrew Howroyd, Nov 10 2024
From Stefano Spezia, Nov 13 2024: (Start)
G.f.: x^3*(8 + x + 24*x^2 + 8*x^3 + 24*x^4 + x^5 + 8*x^6)/(1 - x^4)^2.
E.g.f.: (4 + 32*x + 6*cos(x) + 2*(16*x - 5)*cosh(x) + 3*x*sin(x) + (5*x - 64)*sinh(x))/4. (End)

A215189 Array t(n,k) of the family ((n+k)/gcd(n+k,4))*(n/gcd(n,4)), read by antidiagonals.

Original entry on oeis.org

0, 1, 0, 1, 1, 0, 9, 3, 3, 0, 1, 3, 1, 1, 0, 25, 5, 15, 5, 5, 0, 9, 15, 3, 9, 3, 3, 0, 49, 21, 35, 7, 21, 7, 7, 0, 4, 14, 6, 10, 2, 6, 2, 2, 0, 81, 18, 63, 27, 45, 9, 27, 9, 9, 0, 25, 45, 10, 35, 15, 25, 5, 15, 5, 5, 0, 121, 55, 99, 22, 77, 33, 55, 11, 33, 11, 11, 0, 9, 33, 15, 27, 6, 21, 9, 15, 3, 9, 3, 3, 0

Offset: 0

Jean-François Alcover, Jun 12 2013

Identification of rows and columns:
Row 2, n=1: A060819,
row 3, n=2: A060819 (shifted),
row 4, n=3: A068219,
row 5, n=4: A060819 (shifted),
row 6, n=5: A060819 (shifted and multiplied by 5),
row 7, n=6: A068219 (shifted),
row 8, n=7: A060819 (shifted and multiplied by 7);
column 1, k=0: A181318,
column 2, k=1: A064038,
column 3, k=2: A198148,
column 4, k=3: A160050,
column 5, k=4: A061037,
column 6, k=5: A178242,
column 7, k=6: A217366,
column 8, k=7: A217367.
This array is the transposition of the array given by Paul Curtz in the comments in A181318.

			Array begins:
   0,  0,  0,  0,  0,  0,  0, ...
   1,  1,  3,  1,  5,  3,  7, ...
   1,  3,  1,  5,  3,  7,  2, ...
   9,  3, 15,  9, 21,  6, 27, ...
   1,  5,  3,  7,  2,  9,  5, ...
  25, 15, 35, 10, 45, 25, 55, ...
   9, 21,  6, 27, 15, 33,  9, ...
  49, 14, 63, 35, 77, 21, 91, ...
  ...
Triangle begins:
    0;
    1,  0;
    1,  1,  0;
    9,  3,  3,  0;
    1,  3,  1,  1,  0;
   25,  5, 15,  5,  5,  0;
    9, 15,  3,  9,  3,  3,  0;
   49, 21, 35,  7, 21,  7,  7,  0;
    4, 14,  6, 10,  2,  6,  2,  2,  0;
   81, 18, 63, 27, 45,  9, 27,  9,  9,  0;
   25, 45, 10, 35, 15, 25,  5, 15,  5,  5,  0;
  121, 55, 99, 22, 77, 33, 55, 11, 33, 11, 11,  0;
    9, 33, 15, 27,  6, 21,  9, 15,  3,  9,  3,  3,  0;
  ...

G. C. Greubel, Antidiagonals n=0..100 of triangle, flattened

Cf. A060819, A181318, A064038, A198148, A160050, A061037, A178242, A217366, A217367.

/* As triangle: */ [[(n-k)/GCD(n-k, 4)*n/GCD(n, 4): k in [0..n]]: n in [0..12]]; // Bruno Berselli, Jun 13 2013

t[n_, k_] := (n+k)/GCD[n+k, 4]*n/GCD[n, 4];  Table[t[n-k, k], {n, 0, 12}, {k, 0, n}] // Flatten

t(n,k) = ((n+k)/gcd(n+k,4))*(n/gcd(n,4)).

A227168 a(n) = gcd(2n, n(n+1)/2)^2.

Original entry on oeis.org

1, 1, 36, 4, 25, 9, 196, 16, 81, 25, 484, 36, 169, 49, 900, 64, 289, 81, 1444, 100, 441, 121, 2116, 144, 625, 169, 2916, 196, 841, 225, 3844, 256, 1089, 289, 4900, 324, 1369, 361, 6084, 400

Offset: 1

Paul Curtz, Jul 03 2013

a(n) is defined as A062828(n)^2 for n >= 1. If we extend the sequence to n=0 and negative n by use of the recurrence that relates a(n) to a(n+12), a(n+8) and a(n+4), we obtain a(0)=0, a(-1)=4 and a(-n) = A176743(n-2)^2 for n >= 2.
Define c(n) = a(n+2) - a(n-2) for c >= 0. Because a(n) is a shuffle of three interleaved 2nd-order polynomials, c(n) is a shuffle of three interleaved 1st-order polynomials: c(n) = 4* A062828(n)*(periodically repeated 1, 8, 1, 1).
The sequence a(n) is case p=0 of the family A062828(n)*A062828(n+p):
0, 1, 1, 36,  4, 25,  9, 196, ... = a(n).
0, 1, 6, 12, 10, 15, 42,  56, ... = A130658(n)*A000217(n) = A177002(n-1)*A064038(n+1).
0, 6, 2, 30,  6, 70, 12, 126, ... = 2*A198148(n)
0, 2, 5, 18, 28, 20, 27,  70, ... = A177002(n+2)*A160050(n+1) = A014695(n+2)*A000096(n).

G. C. Greubel, Table of n, a(n) for n = 1..5000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,3,0,0,0,-3,0,0,0,1).

Cf. A062828, A016814, A017138, A005565, A006752, A061037, A061038.

[GCD(2*n, n*(n+1)/2)^2: n in [1..50]]; // G. C. Greubel, Sep 20 2018

A227168 := proc(n)
    A062828(n)^2 ;
end proc: # R. J. Mathar, Jul 25 2013

a[n_] := GCD[2*n, n*(n + 1)/2]^2; Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Jul 03 2013 *)

a(n)=if(n%2, n*if(n%4>2, 2, 1), n/2)^2 \\ Charles R Greathouse IV, Jul 07 2013

a(n) = A062828(n)^2.
a(4n) = (4*n+1)^2; a(2n+1) = (n+1)^2; a(4n+2) = 4*(4*n+3)^2.
a(n) = 3*a(n-4) - 3*a(n-8) + a(n-12).
a(n) * (period 4: repeat 4, 1, 1, 4) = A061038(n).
A005565(n-3) = a(n+1) * A061037(n). - Corrected by R. J. Mathar, Jul 25 2013
a(n) = A130658(n-1)^2 * A181318(n). - Corrected by R. J. Mathar, Aug 01 2013
G.f.: -x*(1 + x + 36*x^2 + 4*x^3 + 22*x^4 + 6*x^5 + 88*x^6 + 4*x^7 + 9*x^8 + x^9 + 4*x^10) / ( (x-1)^3*(1+x)^3*(x^2+1)^3 ). - R. J. Mathar, Jul 20 2013
Sum_{n>=1} 1/a(n) = 47*Pi^2/192 + 3*G/8, where G is Catalan's constant (A006752). - Amiram Eldar, Aug 21 2022

cp's OEIS Frontend

A377851 Smallest multiplier which can complete the square for n-polygonal numbers, together with a constant offset.

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A215189 Array t(n,k) of the family ((n+k)/gcd(n+k,4))*(n/gcd(n,4)), read by antidiagonals.

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A227168 a(n) = gcd(2n, n(n+1)/2)^2.

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cp's OEIS Frontend

A377851 Smallest multiplier which can complete the square for n-polygonal numbers, together with a constant offset.

Views

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Mathematica

PARI

Formula

A215189 Array t(n,k) of the family ((n+k)/gcd(n+k,4))*(n/gcd(n,4)), read by antidiagonals.

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Author

Keywords

Comments

Examples

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Magma

Mathematica

Formula

A227168 a(n) = gcd(2*n, n*(n+1)/2)^2.

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Magma

Maple

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Formula

A227168 a(n) = gcd(2n, n(n+1)/2)^2.