|
|
A085787
|
|
Generalized heptagonal numbers: m*(5*m - 3)/2, m = 0, +-1, +-2 +-3, ...
|
|
86
|
|
|
0, 1, 4, 7, 13, 18, 27, 34, 46, 55, 70, 81, 99, 112, 133, 148, 172, 189, 216, 235, 265, 286, 319, 342, 378, 403, 442, 469, 511, 540, 585, 616, 664, 697, 748, 783, 837, 874, 931, 970, 1030, 1071, 1134, 1177, 1243, 1288, 1357, 1404, 1476, 1525, 1600, 1651, 1729
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,3
|
|
COMMENTS
|
These numbers appear in a theta function identity. See the Hardy-Wright reference, Theorem 355 on p. 284. See the g.f. of A113429. - Wolfdieter Lang, Oct 28 2016
Numbers k such that the concatenation k225 is a square. - Bruno Berselli, Nov 07 2018
|
|
REFERENCES
|
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fifth ed., Clarendon Press, Oxford, 2003, p. 284.
|
|
LINKS
|
|
|
FORMULA
|
G.f.: x * (1 + 3*x + x^2) / ((1 - x) * (1 - x^2)^2). a(n) = a(-1-n) for all n in Z. - Michael Somos, Oct 17 2006
a(n) = 5*n*(n + 1)/8 - 1/16 + (-1)^n*(2*n + 1)/16. - R. J. Mathar, Jun 29 2009
Sum_{n>=1} 1/a(n) = 10/9 + (2*sqrt(1 - 2/sqrt(5))*Pi)/3. - Vaclav Kotesovec, Oct 05 2016
Sum_{n>=1} (-1)^(n+1)/a(n) = 5*log(5)/3 - 10/9 - 2*sqrt(5)*log(phi)/3, where phi is the golden ratio (A001622). - Amiram Eldar, Feb 28 2022
|
|
EXAMPLE
|
|
|
MATHEMATICA
|
|
|
PROG
|
(PARI) t(n)=n*(n+1)/2
for(i=0, 40, print1(t(i)+t(floor(i/2)), ", "))
(PARI) {a(n) = (5*(-n\2)^2 - (-n\2)*3*(-1)^n) / 2}; /* Michael Somos, Oct 17 2006 */
(Magma) [5*n*(n+1)/8-1/16+(-1)^n*(2*n+1)/16: n in [0..60]]; // Vincenzo Librandi, Sep 11 2011
(Haskell)
a085787 n = a085787_list !! n
a085787_list = scanl (+) 0 a080512_list
|
|
CROSSREFS
|
Sequences of generalized k-gonal numbers: A001318 (k=5), A000217 (k=6), this sequence (k=7), A001082 (k=8), A118277 (k=9), A074377 (k=10), A195160 (k=11), A195162 (k=12), A195313 (k=13), A195818 (k=14), A277082 (k=15), A274978 (k=16), A303305 (k=17), A274979 (k=18), A303813 (k=19), A218864 (k=20), A303298 (k=21), A303299 (k=22), A303303 (k=23), A303814 (k=24), A303304 (k=25), A316724 (k=26), A316725 (k=27), A303812 (k=28), A303815 (k=29), A316729 (k=30).
|
|
KEYWORD
|
nonn,easy
|
|
AUTHOR
|
|
|
EXTENSIONS
|
Formula in sequence name added by Omar E. Pol, May 28 2012
|
|
STATUS
|
approved
|
|
|
|