The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation. Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A265282 Number of triangles in a certain geometric structure: see "Illustration of initial terms" link for precise definition. 2
 0, 1, 3, 5, 10, 13, 22, 26, 41, 46, 68, 74, 105, 112, 153, 161, 214, 223, 289, 299, 380, 391, 488, 500, 615, 628, 762, 776, 931, 946, 1123, 1139, 1340, 1357, 1583, 1601, 1854, 1873, 2154, 2174, 2485, 2506, 2848, 2870, 3245, 3268, 3677, 3701, 4146, 4171, 4653 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS In words: This sequence gives the number of triangles of all sizes in a ((2*n^2+8*n-1+(-1)^n)/8-polyiamond with a (7*n-2-(n-2)*(-1)^n)/4-gon: we have (2*n^3+9*n^2+31*n+21+3*(n^2-5*n-7)*(-1)^n)/96 triangles in a direction and (2*n^3+27*n^2+109*n-66+3*(n^2+9*n+18)*(-1)^n+12*(-1)^((2*n-1+(-1)^n)/4))/192 triangles in the other direction. (But the Illustration link is far more informative. - N. J. A. Sloane, Jan 23 2016) At stage n, we count ((2*n^2 + 6*n + 3 - 2*n + 3*(-1)^n)/16 triangles of size 1 in one direction and (2*n^2 + 10*n - 5 + (2*n+5)*(-1)^n)/16 triangles of size 1 in the opposite direction. The total number of triangles of size 1 in both directions is A024206(n). a(n) = A045947(n) duplicated + A024206(n). Note that A045947(n) duplicated = (2*n^3-n^2-7*n+(3*n^2-n-4)*(-1)^n+4*(-1)^((2*n-1+(-1)^n)/4))/64. We observe that a(4)=10 strengthens the Pythagorean relation between 4 and 10 (Tetraktys): cf. triangular numbers, A000217; and that it is from n = 4 we can see and count hexagonal and dodecagonal forms, for example, in a reticular system (incomplete with hexagonal holes) by opposition to the compact shape obtained from A002717. We can obtain this reticular system from A248851. LINKS G. C. Greubel, Table of n, a(n) for n = 0..10000 Luce ETIENNE, Illustration of initial terms Luce ETIENNE, A265282 from A248851 Index entries for linear recurrences with constant coefficients, signature (1,2,-2,0,0,-2,2,1,-1). FORMULA a(n) = (2*n^3 + 15*n^2 + 57*n - 8 + (3*n^2 - n + 4)*(-1)^n + 4*(-1)^((2*n - 1 + (-1)^n)/4))/64. G.f.: x*(1+2*x+x^3-x^4-x^5+x^7) / ((1-x)^4*(1+x)^3*(1+x^2)). - Colin Barker, Dec 07 2015 MATHEMATICA Table[(2*n^3 + 15*n^2 + 57*n - 8 + (3*n^2 - n + 4)*(-1)^n +     4*(-1)^((2*n - 1 + (-1)^n)/4))/64, {n, 0, 100}] (* G. C. Greubel, Dec 20 2015 *) LinearRecurrence[{1, 2, -2, 0, 0, -2, 2, 1, -1}, {0, 1, 3, 5, 10, 13, 22, 26, 41}, 60] (* Harvey P. Dale, Aug 07 2019 *) PROG (PARI) vector(100, n, n--; (2*n^3+15*n^2+57*n-8+(3*n^2-n+4)*(-1)^n+4*(-1)^((2*n-1+(-1)^n)/4))/64) \\ Altug Alkan, Dec 06 2015 (MAGMA) [(2*n^3 + 15*n^2 + 57*n - 8 + (3*n^2 - n + 4)*(-1)^n + 4*(-1)^((2*n - 1 + (-1)^n) div 4)) / 64: n in [0..50]]; // Vincenzo Librandi, Dec 07 2015 (PARI) concat(0, Vec(x*(1+2*x+x^3-x^4-x^5+x^7)/((1-x)^4*(1+x)^3*(1+x^2)) + O(x^100))) \\ Colin Barker, Dec 07 2015 CROSSREFS Cf. A000217, A000292, A001477, A001859, A002620, A002717, A004006, A004526, A045947, A132337. Cf. A248851. Sequence in context: A309270 A165718 A031878 * A160792 A308759 A137395 Adjacent sequences:  A265279 A265280 A265281 * A265283 A265284 A265285 KEYWORD nonn,easy AUTHOR Luce ETIENNE, Dec 06 2015 EXTENSIONS a(26) corrected by Altug Alkan, Dec 06 2015 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified August 8 09:16 EDT 2020. Contains 336293 sequences. (Running on oeis4.)