|
|
A279946
|
|
Numbers that are both dodecagonal and centered heptagonal.
|
|
1
|
|
|
1, 10396, 326656, 2619897841, 82318050361, 660219495802336, 20744313326831116, 166376633378560463881, 5227608446905776928921, 41927244364003774523222476, 1317367783816405284315203776, 10565749434051302554022550018121, 331979316252074156011094205115681
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,2
|
|
COMMENTS
|
Intersection of dodecagonal numbers A051624 and centered heptagonal numbers A069099. A051624(j) = j(5j - 4), A069099(k) = (7*k^2 - 7^k + 2)/2, and the table below gives indices j and k at which A051624(j) = A069099(k):
.
n a(n) j k
= ================= ======== ========
1 1 1 0, 1
2 10396 46 55
3 326656 256 306
4 2619897841 22891 27360
5 82318050361 128311 153361
6 660219495802336 11491036 13734415
7 20744313326831116 64411666 76986666
... (End)
|
|
REFERENCES
|
F. Tapson (1999). The Oxford Mathematics Study Dictionary (2nd ed.). Oxford University Press. pp. 88-89.
|
|
LINKS
|
|
|
FORMULA
|
Empirical: a(1)=1, a(2)=10396, a(3)=326656, a(4)=2619897841, a(n) = 252002*a(n-2) - a(n-4) + 85050 for n > 4. - Jon E. Schoenfield, Dec 24 2016
G.f.: x*(1 + 10395*x + 64258*x^2 + 10395*x^3 + x^4) / ((1 - x)*(1 - 502*x + x^2)*(1 + 502*x + x^2)). - Colin Barker, Dec 24 2016
|
|
EXAMPLE
|
10396 is both the 46th dodecagonal number and the 55th centered heptagonal number: A051624(46) = 46(5*46 - 4) = 10396 and A069099(55) = (7*55^2 - 7*55 + 2)/2 = 10396.
A051624(256) = 256(5*256 - 4) = 326656 = (7*306^2 - 7*306 + 2)/2 = A069099(306). (End)
|
|
MATHEMATICA
|
LinearRecurrence[{1, 252002, -252002, -1, 1}, {1, 10396, 326656, 2619897841, 82318050361}, 20] (* Harvey P. Dale, Jul 06 2021 *)
|
|
PROG
|
(PARI) Vec(x*(1 + 10395*x + 64258*x^2 + 10395*x^3 + x^4) / ((1 - x)*(1 - 502*x + x^2)*(1 + 502*x + x^2)) + O(x^20)) \\ Colin Barker, Dec 24 2016
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn,easy
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|