A227906 Coins left after packing heart patterns (fixed orientation) into n X n coins.

0, 1, 2, 5, 4, 9, 6, 13, 8, 17, 10, 21, 12, 25, 14, 29, 16, 33, 18, 37, 20, 41, 22, 45, 24, 49, 26, 53, 28, 57, 30, 61, 32, 65, 34, 69, 36, 73, 38, 77, 40, 81, 42, 85, 44, 89, 46, 93, 48, 97, 50, 101, 52, 105, 54, 109, 56, 113, 58, 117, 60, 121, 62, 125, 64

Offset

0,3

Comments

On the Japanese TV show "Tsuki no Koibito", a girl told her boyfriend that she saw a heart in 4 coins. Actually there are a total of 6 distinct patterns appearing in 2 X 2 coins in which each pattern consists of a part of the perimeter of each coin and forms a continuous area.

a(n) is the number of coins left after packing fixed orientation heart patterns (type 4c2s1: 4-curve cover 2 coins and symmetry) into n X n coins. The total number of hearts is A093005 and the number of voids left is A093353. See illustration in links.

Same as A114752 with a(0) also defined. - R. J. Mathar, Feb 27 2026

Links

Table of n, a(n) for n=0..64.

Kival Ngaokrajang, Illustration of initial terms (U)

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

Formula

From Colin Barker, Oct 30 2013: (Start)

a(n) = (-1 + (-1)^n - (-3 + (-1)^n)*n)/2.

a(n) = n for n even.

a(n) = 2*n-1 for n odd.

a(n) = 2*a(n-2) - a(n-4) for n >= 6.

G.f.: x*(3*x^2+2*x+1)/((x-1)^2*(x+1)^2). (End)

Mathematica

With[{nn=60}, Join[Riffle[Range[0, nn, 2], Range[1, 2nn+1, 4]]]] (* Harvey P. Dale, Feb 11 2015 *)

Prog

(Small Basic)
For n = 0 To 100
  If Math.Remainder(n, 2) = 0 then
    a = n
  Else
    a = a + n
  EndIf
  TextWindow.Write(a+", ")
EndFor
(PARI) Vec(x*(3*x^2+2*x+1)/((x-1)^2*(x+1)^2) + O(x^100)) \\ Colin Barker, Oct 30 2013
(PARI) a(n) = (-1 + (-1)^n - (-3 + (-1)^n)*n)/2 \\ David Cleaver, Feb 14 2026

Crossrefs

Cf. A008795, A230370 (3-curves), A074148, A229093, A229154 (4-curves), A001399, A230267, A230276 (5-curves), A229593, A228949, A229598 (6-curves).

Bisections give: A005843, A016813.

Sequence in context: A065221 A373053 A283419 * A114752 A204923 A123302

Adjacent sequences: A227903 A227904 A227905 * A227907 A227908 A227909

Keyword

nonn,easy

Author

Kival Ngaokrajang, Oct 19 2013

Extensions

a(3) corrected, formulas and programs edited by David Cleaver, Feb 14 2026

Status

approved