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A194454
a(n) = 12*n^2 + 2*n + 1.
4
1, 15, 53, 115, 201, 311, 445, 603, 785, 991, 1221, 1475, 1753, 2055, 2381, 2731, 3105, 3503, 3925, 4371, 4841, 5335, 5853, 6395, 6961, 7551, 8165, 8803, 9465, 10151, 10861, 11595, 12353, 13135, 13941, 14771, 15625, 16503, 17405, 18331, 19281
OFFSET
0,2
COMMENTS
A142241 gives the first differences.
Inverse binomial transform of this sequence: 1, 14, 24, 0, 0 (0 continued).
a(n)*a(n-1)-11 is a square, precisely 4*A051866(n)^2.
Sequence found by reading the line from 1, in the direction 1, 15, ..., in the square spiral whose vertices are the generalized octagonal numbers A001082. - Omar E. Pol, Jul 18 2012
FORMULA
G.f.: (1+x)*(1+11*x)/(1-x)^3.
a(n) = A154106(-n-1).
a(n) = 2*A049453(n) + 1.
a(n) = A051866(n) + A051866(n+1). - Charlie Marion, Nov 15 2019
E.g.f.: exp(x)*(1 + 14*x + 12*x^2). - Stefano Spezia, Nov 15 2019
EXAMPLE
Using these numbers we can write:
1, 15, 53, 115, 201, 311, 445, 603, 785, 991, 1221, ...
0, 0, 1, 15, 53, 115, 201, 311, 445, 603, 785, ...
0, 0, 0, 0, 1, 15, 53, 115, 201, 311, 445, ...
0, 0, 0, 0, 0, 0, 1, 15, 53, 115, 201, ...
0, 0, 0, 0, 0, 0, 0, 0, 1, 15, 53, ...
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, ...
======================================================
The sums of the columns give the sequence A172073 (after 0):
1, 15, 54, 130, 255, 441, 700, 1044, 1485, 2035, 2706, ...
MATHEMATICA
Table[12 n^2 + 2 n + 1, {n, 0, 50}] (* Vincenzo Librandi, Mar 26 2013 *)
PROG
(Magma) [12*n^2+2*n+1: n in [0..40]];
(PARI) for(n=0, 40, print1(12*n^2+2*n+1", "));
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Bruno Berselli, Aug 24 2011
STATUS
approved