|
| |
|
|
A062938
|
|
n*(n+1)*(n+2)*(n+3)+1 = (n^2 +3*n + 1)^2.
|
|
10
| |
|
|
1, 25, 121, 361, 841, 1681, 3025, 5041, 7921, 11881, 17161, 24025, 32761, 43681, 57121, 73441, 93025, 116281, 143641, 175561, 212521, 255025, 303601, 358801, 421201, 491401, 570025, 657721, 755161, 863041, 982081, 1113025, 1256641
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,2
|
|
|
COMMENTS
| a(n) = product of first four terms of an arithmetic progression + n^4, where the first term is 1 and the common difference is n. E.g. a(1) = 1*2*3*4 +1^4 =25, a(4) = 1*5*9*13 + 4^4= 841 etc. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Sep 19 2003
Is it possible for one of the squares to be the sum of two or more lesser squares each used only once? - J. M. Bergot, Feb 17 2011
Yes, in fact a(1)-a(11) are examples. [Charles R Greathouse IV, Jun 28 2011]
This sequence demonstrates that the product of any 4 consecutive integers plus 1 is a square. The square roots are in A028387. [From Harvey P. Dale, Oct 19 2011]
|
|
|
LINKS
| Harry J. Smith, Table of n, a(n) for n=0..1000
Index entries for sequences related to linear recurrences with constant coefficients, signature (5,-10,10,-5,1).
|
|
|
FORMULA
| a(n+1)=Numerator of ((n + 2)! + (n - 2)!)/(n!) n=3,4,5,... - Artur Jasinski (grafix(AT)csl.pl), Jan 09 2007
a(n) = A028387(n)^2. [From Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Oct 31 2010]
a(n) = n*(n+1)*(n+2)*(n+3)+1^4 = 1*(1+n)*(1+2*n)*(1+3*n)+n^4=(n^2+3*n+1)^2;
in general, n*(n+k)*(n+2*k)*(n+3*k)+k^4 = k*(k+n)*(k+2*n)*(k+3*n)+n^4 =
(n^2+3*k*n+k^2)^2. - Charles F. Marion, Jan 13 2011
|
|
|
EXAMPLE
| 2*3*4*5 + 1 = 121 = 11^2.
|
|
|
MATHEMATICA
| Table[(n^2+3*n+1)^2, {n, 0, 50}]
Times@@#+1&/@Partition[Range[0, 50], 4, 1] (* From Harvey P. Dale, Apr 02 2011 *)
|
|
|
PROG
| (PARI) j=[]; for(n=0, 70, j=concat(j, (n^2+3*n+1)^2)); j
(PARI) { for (n=0, 1000, write("b062938.txt", n, " ", (n^2 + 3*n + 1)^2) ) } [From Harry J. Smith (hjsmithh(AT)sbcglobal.net), Aug 14 2009]
|
|
|
CROSSREFS
| Cf. A028387 (for the square roots of the terms of this sequence). [From Harvey P. Dale, Oct 19 2011]
Sequence in context: A031151 A016970 A174371 * A190875 A205800 A198040
Adjacent sequences: A062935 A062936 A062937 * A062939 A062940 A062941
|
|
|
KEYWORD
| nonn,easy
|
|
|
AUTHOR
| Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Jul 05 2001
|
|
|
EXTENSIONS
| More terms from Jason Earls (zevi_35711(AT)yahoo.com), Harvey P. Dale (hpd1(AT)nyu.edu) and Dean Hickerson (dean.hickerson(AT)yahoo.com), Jul 06 2001
|
| |
|
|
