A355759 Sums of the first ceiling((n+1)/2) entries on the diagonals of a square spiral with a starting value of 1 in the center, where the diagonal and the antidiagonal are used alternately.

1, 4, 6, 11, 15, 24, 32, 45, 57, 76, 94, 119, 143, 176, 208, 249, 289, 340, 390, 451, 511, 584, 656, 741, 825, 924, 1022, 1135, 1247, 1376, 1504, 1649, 1793, 1956, 2118, 2299, 2479, 2680, 2880, 3101, 3321, 3564, 3806, 4071, 4335, 4624, 4912, 5225, 5537, 5876, 6214, 6579, 6943

Offset

1,2

Links

Karl-Heinz Hofmann, Table of n, a(n) for n = 1..10000

Karl-Heinz Hofmann, Examples of a(1..18)

Project Euler, Problem 28. Number spiral diagonals

Index entries for linear recurrences with constant coefficients, signature (2,0,-2,2,-2,0,2,-1).

Formula

G.f.: -(x^7 - 2*x^6 + x^4 - x^3 + 2*x^2 - 2*x-1)/((x - 1)^4 * (x + 1)^2 * (x^2 +1)).

a(n) = (24 + 20*n + 6*n^2 + n^3) / 24 for n even.

a(n) = (12 + 17*n + 6*n^2 + n^3) / 24 for n odd and n (mod 4) == 3.

a(n) = (17*n + 6*n^2 + n^3) / 24 for n odd and n (mod 4) == 1.

a(2*n) = A006527(n+1).

a(2*n-1) = A208995(n) - 1.

E.g.f.: ((30 + 45*x + 12*x^2 + x^3)*cosh(x) + (51 + 42*x + 12*x^2 + x^3)*sinh(x) - 6*cos(x))/24. - Stefano Spezia, Aug 19 2022

Example

See the PDF in links.

Mathematica

CoefficientList[Series[-(x^7 - 2*x^6 + x^4 - x^3 + 2*x^2 - 2*x - 1)/((x - 1)^4*(x + 1)^2*(x^2 + 1)), {x, 0, 50}], x] (* Amiram Eldar, Aug 19 2022 *)

Prog

(Python)
def A355759(n):  # polynomial way.
    if   n % 2 == 0: return((24 + 20*n + 6*n**2 + n**3)//24)
    elif n % 4 == 3: return((12 + 17*n + 6*n**2 + n**3)//24)
    elif n % 4 == 1: return((     17*n + 6*n**2 + n**3)//24)
(PARI) a(n) = (n^2 + 6*n + if(n%2, 17, 20))*n \ 24 + (n%4!=1); \\ Kevin Ryde, Aug 19 2022

Crossrefs

Cf. A200975, A106607, A114254.

Cf. A006527 and A208995 (bisections, see formulas).

Sequence in context: A232807 A309160 A266795 * A060180 A190499 A190564

Adjacent sequences: A355756 A355757 A355758 * A355760 A355761 A355762

Keyword

nonn,easy

Author

Karl-Heinz Hofmann, Aug 14 2022

Status

approved